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O<K Z 9i-IbV]UPd6lCp

x 1"sin x ' x

K q B<DedI-W-DMG-I-DHV x \<iJDHx q DM\<X%[ q W-DM\ B\<c<iJc<Hc"\<iJDM\<fDHHc1I=\oK EKWZd6DHG-c<Biy[;EU "WJ\<cx<l X UoEUG-GJc<r8\c<W-G-cXM\<f UjJj-i-c<Y2XHIJVI-i-cByUa\f

sin xK 9cN\<c<V q BXed q W6UD x 1

VlsjJcad q W6UDMV XUaV q j-iJc<XM\ q VcHEDNd6fV]Ua\DHV]Ua\I-W-DHXMY2c<HcV]Ua[Z\<G-IJYZU 2

x+ x = 0. OK Q

dDFE q IJV¤j-i-I-xad6I ¡ DHGJI-DHV x q EDM\ VcEDNdf G-DNd6I-G-DHrJG-c<Hc V]Ua[J\G-I-Y;U 2

x+ x− x3

6= 0,

O<K

I§\PUaY=EUPd6DHDK©nU_i-I-XK]O<K O¢j-i-I-BDeEDMG-lªi-DM q dfR\PUa\<l Uj-j-iJc<Y2XHI-V]UkJI-I m q G-Y-k-IJI

sin xY2c<G-DHWJG-lzV W-I-XNd6c<V WZdDMG-c<BTiy[2EUnE6d6[

n = 0, 1, . . . , 43KSd6[ YZU ¡ EcHcTY2c<G

Y-i-DN\<G-cHc G-UoWJDHG-IJ[nVl x q EDMVªj-cad q W6UP\<f G-cB q gEI-G-UVI-W-DMXHY q XMI-XM\DHV q "

BUoEUGJG-c<Vj-iJI-xadI ¡ DHG-I-I c<j-I-XMlzByU q j-i-ck-DHXMX^Y2cadDMxyUG-I-r mIZI-W-DMXHY2c<McV]Ua[Z\<G-I-Y;UJK

% rrf/Y.X-(q3d( i '.(( sinx ./8'*+' q(X-`*/8q`+-'*/gTof :R5d " R` n = 1, . . . , 43

$+ #! $ $/ # #$%"! #$# %!.10*&!('*)+*,

UaXHXHVca\<i-I-V EI-G-UVI-W-DMXHY-I-D XMI-XM\DHVl "zVcHEDedI-i q DHVlzD=Y2c<G-DHWJG-lzV WJI-XNdcVc<x<lzYJG-c<B<DMG-G-lCp EIJmmnDMi-DHG-kJI6UPdf<GJlCp q i-UB<G-DMG-I-rK 9i-IJVDHG-IJ\DNdfG-c Y\PUYJI-VXHIJXM\<DMV]UV XMcpJi-UG-IZdIJXHfj-i-DeEXM\PUBad6DHG-IJ[ I \<DMi-VI-G-cPdc<HIZ[ "yj-DMi-B<c<G6URW6Uodf<G-c9B<ca G-I-Y I-DB¢VDNp2UaG-I-Y2DKA¤i6UXHXMV]Ua\<iJI-ByUDMVc<VXed q W6UD'E6d6[_c<j-iJDeEDedDHG-IZ[_EIJG6UVI W-DMXHY2c<r^XMI-XM\DHVlG-DMc<xpycHEI-Vc q Y;UaUa\<f`c<x*DMYJ\ "aEc<j q XHY;U I-r cj-I-XUaG-I-D XHc<XN\<c[-GJIJ[_RUoEUaG-I-DHVB<DNdIJW-I-G

x1, x2, . . . , xN

B G-DHY2c\c<i-lzrbVc<VDMGJ\ B<i-DHVDMG-It = t0

KACDedI-W-I-GJl

xi

Vc< q \ j-i-IJG-I-V]Ua\f j-iJc<IJBcadf<G-lzD9HG6URW-DHG-IZ[ "j-i-I-WJDHV EB q Vi6UP dI-WJG-lzVG6Uxc<i6UVLB<DNdIJW-I-G

xi

Ix′ica\<B<DMW6U \EByU_i6UaHG-lCp8XHcXM\<cR[JG-IJ[K UY2c<G

HBcadukJI-IEI-G-UVI-W-DMXHY2c<rXHIJXM\<DMVl(Bc B<iJDHVDHGJITRUjJI-XHlzByUaDM\<X%[¢XMI-XM\DHVc<rcx<lzY G-c<BDHG-G-lCp EI-mnmDMi-DHGJk-I6UPdfG-lCp q i-UB<G-DMG-I-r

dxi

dt= xi = fi(x1, x2, . . . , xN ), i = 1, 2, . . . , N.

O<K t!

XNdI(i6UXMXHV]Ua\i-I-ByUa\f B<DedI-W-IJG-lx1, x2, . . . , xN

Y;UY Y2c<c<iEI-G-Ua\<l|\c<W-Y-IxB

N VDHiJG-c<V j-i-c<XN\<i6UGJXM\<BD!"6\<c¢j-cad q W-UDM\X%[1G6Ua d6[;EGJc<D9HDMc<VDM\i-I-W-DMXHY2c<DTjJi-DeE

XM\PUaBadDHGJI-DbXHcXM\<cR[JG-IJ[ EI-G6UVI-WJDHXHY2c<rXHI-XN\<DHVlgB§B<I;ED M\c<r\<cW-Y-IK 9c<XNd6DeE G- G-UalzByUa \ IJc<xi6U ¡ U DMr "U W-U D 2=>:WS@S VE S LS "Uj-i-c<XN\<i6UaG-XM\B<cXHcXM\<cR[JG-I-r -=>:WS O@FDSP EGFL=@? E.@S EI-G6UaVI-W-DHXMY2c<r1XMI-XM\DHVl9K'VDHGJDHG-I-XHcXM\<cR[JG-IJ[(XHI-XN\<DHVl B<c B<i-DMVDHG-I c\<BDHW6UDN\=EB<I ¡ DMG-I-DmUacB<c<r \<c<WJY-I(BoEcadfG-DMY;ca\<c<i-crTdI-G-I-I "G6UaHlzByUDMVc<r -=<:WS@S EGFL=\C1 E SYFBADC6KPA mUacB<c<Vbj-iJc<XM\i6UG XM\B<D XMI-XM\DHVl q i6UB<G-DMG-IJ[JVI O<K t'c<j-i-DFEDNd-[-DM\X%[ B<DHYZ\<c<i-GJc<D j-cad6D XMY;ci-c<XM\DHr "XHcj-c<XM\PUaBad6[- DMDTY;U ¡ Ec<r_\<c<W-Y2D

xB<lCp<cHE[ TI-rbIJG-DMDBDHYJ\c<i1XMY;ci-c<XM\I

F (x)"6Y2c<Vj-c<GJDHGJ\l"Y;ca\<c<i-cHc EU \X%[1jJi6UB<lzVIW6UXN\R[-VI q i6UaB<G-DHGJI-r O<K t 2

[f1(x1, x2, . . . , xN), f2(x1, x2, . . . , xN), . . . , fN(x1, x2, . . . , xN)]. O<K w

STI-G6UVIJW-DHXMYZUP[XMI-XM\DHV]U O<K tCVc ¡ DM\ x<l \f Uj-I-XHUG6U¢B B<DMYJ\<ci-G-c<rmnc<i-VD2

x = F(x), OK

EDF(x)

^B<DMYJ\<ci m q G-Y-k-IZ[1i6UPVDHiJG-c<XM\I N K©9DHc<xp<cEI-Vc q \<c<WJG-IJ\<f=BUI-Vc<XMB[JHf1j-cGJ[J\I-r W-I-XedU1XN\<DHjJDHG-DMr(XMB<c<x<cHEl Ii6UaHVDHi-GJc<XM\I mÙPc<B<cHc1jJi-c<XM\i6UG-XN\<ByUEI-G6UVIJW-DHXMY;cr XHI-XN\<DHVl9K ncHE A PXHS IE C ODCI?DC -WS>R<ST; j-cG-I-V]UDN\<X%[G6UI-VDMG-f! DMDnW-I-XNd6c G-DNRUBI-XHI-VlCpY2c<ci<EI-G6Ua\ "G-DMc<xpycHEI-VlCp E6d6[8cHEG-caG6URW-G-cHcc<j-i-DFEDNdDMG-IJ[§XHc<XN\<cR[-GJIJ[ XMI-XM\DHVl9K 9cE Y2cc<iEI-G-Ua\PUVI jJDHi-B<cG6URW6UPdf<GJc(j-cG-I-V]UPdIJXHf I-VDHG-GJcj-i-cXM\<i-UG-XM\B<DHGJG-lzD j-DMi-D VDHGJG-lzD!"p2Ui-UYJ\<DMi-IJ q I-D^BRUIJVc<i6UXMj-cadc ¡ DMG-I-D¢\<DNd I§cx*DHYJ\c<B-K*A\<c ¡ DB<i-DMV [1E6d6[=cHEG-caG6URW-G-cHci-DTDHG-IJ[§XHcc\<B<DN\<XN\<B q TIZp q i-UB<G-DMG-I-rbEB<I ¡ DMG-IJ[G-DMc<xpycHEI-Vc"j-c<VI-Vc1Y;cc<iEI-G6Ua\ "RUoEUa\<fbXMc<c\B<DM\XM\<B q I-DG6URW6UPd6f<G-lzD^HG6U W-DMG-IJ[ IJVj q df<XHcB IZdI(XHY2c<i-c<XN\<DMrKzA XMB[JHI X M\<IJVªXHI-XN\<DHV]U X n XN\<DMj-DHGJ[JVIXHBc<x<cHEl(pyUi6UYJ\DHi-IZ q DN\<X%[mUacB<lzV j-iJc<XM\i6UG-XN\<B<c<V i6UPVDHiJG-c<XM\I B'EByUni6UaRUx<cad6f! DMr

N = 2nNK

./## #'"$

XNdI=EI-G6UVIJW-DHXMYZUP[ XHI-XN\<DMV]U1RUoEUaG6U q i6UB<GJDHG-I-DMV O<K "\cbj-c<XN\ q dI-i q DM\ X%[ "WJ\c Y;U ¡ Ec<V q x(t0)

B mUacB<c<V jJi-c<XM\i6UG-XN\<B<D XN\PUB<IJ\X%[B XMc<c\<BDM\<XN\<B<IJDXHcXM\<cR[JG-I-D

x(t) (t > t0)"Y q EUbRU B<i-DMV [ t − t0

j-DHiJDHVDHXN\<IJ\X%[¤mUacByUa[ \c<W Y;U ";EBI ¡ q Ua[-X%[BXHcc\<B<DN\<XN\<B<I-IX q i6UB<G-DMG-I-DMV O<K eK-A¤c<j-DMi6Ua\<ci-G-c<r_mnc<i-VD OK CVc ¡ G-cRUjJI-XUa\f^B¢B<I;ED

x(t) = Ttx(t0), O<K ]

EDTt

RUY2c<G c<jJDHi6Ua\c t0

K UaY Y;UYx(t0)

Ix(t)

j-iJI-G6UoE6d6D ¡ Ua\ cEG-c<V q I \<c<V q ¡ DmUacB<c<V q j-i-cXM\<i-UG-XM\B q EI-G6UVIJW-DHXMY;cr XHI-XN\<DHVl "\<c V]Ua\<DMV]Ua\<I-YJI Hc<B<ci2[Z\ BEUGJG-c<rXHIZ\ q Uak-I-I2-cj-DHi6UP\<c<i Tt

c\<c<xi6U ¡ UDN\mUacB<c<DujJi-c<XM\i6UG-XN\<B<c¢XMI-XM\DHVlG6U XMDHxR[KALXHc<c\B<DM\XM\B<I-I X M\<IJVLVc ¡ G-c G6UaHlzByUa\fcj-DHi6UP\<c 0Is > 0

" \<c8XHcc\<B<DN\<XN\ 0, s > 0.O<K

STI-G6UVIJW-DHXMY-I-D'XMI-XM\DHVlVc ¡ G-cYZdUXMXHI-mnI-kJI-i-c<ByUP\<fBÙB<I-XMI-Vc<XN\<Ic\TB<I EU9c<j-DMi6Ua\c<i6Unc\<c<xi6U ¡ DMG-IJ[¢I¢XM\i q YJ\ q i-lmUacB<c<Hc9jJi-c<XM\i6UG-XN\<ByUZK XNd6I c<jJD i6Ua\c<i^jJi-DeE q XHV]Ua\i-I-ByUaDM\I-XMYZd W-IJ\<Dedf<G-cdI-G-DMr-G-lzDujJi-DHc<xi6UacByUG-IJ[ G6URW6UodfG-c<Mc^XMc<XM\co[JG-IJ[ "\c^c<GG6UaHlzByUDN\<X%[d6I-G-DHrJG-lzVzKI-G-DMr-G-lzrc<j-DHi-Ua\<c<i1c<xadUoEU DM\bXHB<c<rJXM\<Bc<V"X q j-DMi-j-cHI-k-I-I2 T[x(t) + y(t)] = Tx(t) + Ty(t)

KcXNdI c<jJDHi6U \<ci^G-DedI-G-DMr-G-lzr "2\<cI XMc<c\B<DM\XM\A FDCIEK? NFHC UC ?DC nK I-XN\<DHVl "6E6d6[Y2c\<ci-lCpc\c<x<i6U ¡ DHG-IJD

x(t)Xuj-c<Vc f<Lc<j-DHi-Ua\<c<i-U

TVc ¡ DN\x<l \<fTc<j-i-DFEDNdD

G-c9E6d6[duxlCpt > t0

G-DMj-i-DHiJlzB<G-c B<cB<i-DMVDHG-I ";G6UalzB<U \\PUY ¡ DUODSTE SD = "Aj-c¢UG-UPdc<MI-IXHc XN\PUk-I-cG6Ui-G-lzV \<DMW-DHG-IJDHV ¡ I;EY2c<XN\<IKe]XedIc<j-DMi6Ua\c<i_c\<c<xi6U ¡ DMG-IJ[ c<j-i-DFEDNdDMG G6UuEI-XMY-i-DM\G-c<V VG-c ¡ DMXM\B<D HG6URW-DHG-IJr^B<i-DMVDHG-I ";\c`XMc<c\<BDM\XM\B q TI-DÊI-G6UVIJW-DHXMY-I-DXHIJXM\<DMVl G-UalzByUa \ =t =@;>=*A "IZdI XMI-XM\DHV]UVI XEI-XMY-i-DN\<G-lzVB<i-DMVDHG-DMVzK j-c<XMc<x<lRUoEUaG-IJ[c<j-DMi6Ua\<ci6U c\c<x<i6U ¡ DHG-IZ[

T\PUY ¡ DVc< q \ i6UaNdI-W6Ua\f<X%[K

j-DHi-Ua\<c<iTVc ¡ G-cTUoEUa\fBB<I;ED EI-mnmnDHi-DMG-k-I6UPd6f<G-c<Mc I;dI_I-GJ\<DMHi6Uodf<G-c<Mc

j-i-DMc<x<i6UPc<ByUaG-IJ[ "ZBB<I;ED9V]UP\<i-I-kJlLIZd6I_\PUaxadI-k-l "JBB<I;ED'Mi6UmnI-Y;UIZdIm q G-Y k-I-II_\oK EKAUB<I-XMI-Vc<XN\<I=c\^\c<Hc "Y;UY2c<rbiy[2E=G6URW-DMG-I-r Vc< q \j-i-I-GJI-V]Ua\<fmUaHc<B<lzD

Y2c<c<iEI-G-Ua\<l "-cj-i-DeEDNd6[J IJD'XHc<XN\<cR[-GJI-D9XHI-XN\<DMVl "-i6UaNdI-W6U \ ?DC O@FHCPF<?DS>C*A;>A FHC EK?DS>C -=<:WS )C O@FHS5IEGFL=W? IE.W=aK

-!. *$%/.10*&! # #$%"! (',/- ,#$%

A U ¡ G q Hi q jJj q EI-G6UVIJW-DHXMY-IZpXHIJXM\<DMV j-i-DFEXM\PUBPd6[- \XHIJXM\<DMVl "`BY2c\<ci-lCp B<cHVc ¡ G-l Y2cadDMxyUG-IZ[Kh9cadDMxyUa\DNdf<GJlzDXMI-XM\DHVl|X \<c<W-YJI Hi-DHGJIJ[ IZpV]Ua\DHV]Ua\I-W-DHXMY-IZpbVcEDNdDMrbi6UaeEDNd6[J \^G6U cj-i-DeEDNdDMG-G-lzD YZdUXHXMl9KUaMd6I-W6U \XDAB?DCB? )CnI ?DCPXHAB?DC B? )C9Y;cPdDHxyUP\<DNd6f<G-lzD XMI-XM\DHVl " ISPQFHCI;<STE S C ?B? )C AFL= $O@FHC ;WCPXHCI?B? )CNMm LS>? ICPF<=@EKA<? )C A ;>A IABOB=@EKA<? )C M*=&WE ST?DS "? )C A ?DC =<E S?DS "? )CMK XHcx<lzr Y;dUXHX j-i-DFEXM\PUaBad6[- \ \PUY G6UalzB<UDHVlzD =WE SD LSYXHCIR =WE C XDJ ? )CIA E C K XHGJc<B<G-lzD`XHB<c<rJXM\<B<U q Y;UaRUGJG-lCpbXMI-XM\DHVj-cHEi-cx<G-c^cx<X q ¡ EU \X%[bBi6Uxc\PUPpj-c\<DMc<i-I-IY2cadDHx<UG-I-rKh9cadDMxyUa\<Dedf<G6UP[^XMI-XM\DHV]U G6UaHlzByUDN\<X%[ XDAB?DCB?DS TI;dI?DCPXHAB?DC B?DS TBnRUB<IJXHI

Vc<XN\<I c\\<c<Mc"*d6I-G-DHrJG6UbIZdI GJDNdI-GJDHr-G6Ubcj-I-XHlzB<U Ua[ DMD^XHIJXM\<DMV]UEI-mnmnD i-DMG-k-I6UPd6f<G-lCp q i6UBG-DHG-IJrKI-G-DMr-G-lzDXMI-XM\DHVl [JBad6[J \<X%[ W6UXM\G-lzVXNd q W6UDMVG-DedI-G-DMr-G-lCpK EG6UaY;c "6B XMIZd q j-i-IJG-k-I-j-I-UPdf<G-cr1ByU ¡ G-c<XN\<I dI-GJDHr-G-lCpXMI-XM\DHVj-i-IÎ-XHXedDeEcByUG-I-I^B<c<j-i-cXHc<B q XN\<c<r-WJI-B<c<XN\<I_Y2cadDHx<UG-I-r ";Un\PUY ¡ D BTB<caVc ¡ GJcXM\I¢I-XHj-cPdfcByUG-IJ[Tj-i-I-GJk-I-j6U'X q jJDHi-j-caI-k-IJI iJD DMG-I-r\PUY;Ua[Y;dUXHXMI-mnI-Y;Uk-IJ[c<j-i-UBPEUG-UJK

STI-G6UVIJW-DHXMY-I-D8XHI-XN\<DHVl " VcHEDedI-i q DHVlzD Y2c<GJDHW-G-lzV W-I-XNd6c<V c<xlzY-G-c<B<DMG G-lCpnEI-mnmnDHi-DMG-k-I6Uodf<G-lCp q i6UaB<G-DHGJI-r "G-UalzByUa \ ISPQFHCI;<STE S C ?B? *AnIZd6I E S C ? *A XHI-XN\<DHV]UaVIK G-Ic<j-IJXHlzByU \<X%[1XjJc<Vc! fY2c<GJDHW-G-cVDHi-G-cHc^mUaHcB<c<Mcuj-iJc<XM\i6UG-XN\<ByU'Inp2Ui6UaYJ\<DMi-IJ q \<X%[ Y;cG-DHW-GJlzV8WJI-XNdcVbXM\DHj-DMG-DHr¢XHBc<x<cHEl9K EG6U I\PU ¡ DXMI-XM\DHV]U B i6UaNdI-W-G-lCp q XNdc<BIJ[ZpVc ¡ DM\ i6UXMXHV]Ua\i-I-ByUP\<f<X%[ Y;UYXHcXHi-DFEc\<cW-DHG-G-Ua[ d6I-x<c Y;UY i6UXMj-i-DeEDNdDMG-G6Ua[ K Ua\<DMV]Ua\<I-WJDHXHYJI-D1VcHEDedI FL= $O@FHC ;WCPXHCI?B? bXHIJXM\<DMV N\<c_EI-mnmDMi-DHGJk-I6UPdfG-lzD q i6UB<GJDHG-IJ[ BW-UXM\G-lCp jJi-cIJHB<cHEG-lCp "JI-GZ\<DHMi6UPdfG-lzD q i6UB<GJDHG-IJ[ IZdI c<x<lzY-GJc<B<DHGJG-lzD q i6UBG-DHG-IZ[ X RUj6UP ElzByUa IJV Ui- q VDMGJ\<c<VzK 9I-XNd6cTXM\DHj-DMG-DHrXMB<c<x<cHEl(i6UXMj-i-DeEDNdDMG-G-c<rXMI-XM\DHVlx<DMXHY2c<G-DMW-G-c";I \i-DHx q DM\X%[x<DHXMY2c<G-DHWJG-c<D'W-IJXNdc9EUG-G-lCp E-d6[^cj-i-DeEDNdDMG-IJ[ DMD XMcXM\cR[-G-IJ[ KA"\DHc<iJI-I edDHYJ\i-I-W-DMXHY-I;p Y2cadDMxyUG-IJr§XMI-XM\DHV q i6UXMXHV]Ua\i-I-ByU \1Y;UYXHcXHi-DFEc\<cW-DHG-G q B\<DNp^XNd q W-Ua[Zp "JY2c< EUE6dI-G6U`B<cadG-lY;cPdDHxyUaG-I-rX q TDHXM\B<DHG G-c j-i-DHBl UDN\nHDHcVDM\<iJI-W-DHXMY-I-D i6UaHVDHi-l XHUVc<r`XHI-XN\<DHVl9K ]XedI i6UPVDHiJl j-i-I x<c<i-U XHc<IJHVDHi-IJVl XE6d6I-G-c<rB<cad6G-lgHDHGJDHi-I-i q DHVlCpLY2cadDMxyUG-IJr " \c XMI-XM\DHV qG-DMc<xpycHEI-Vc i6UXHXMV]Ua\<iJI-ByUa\f^Y;UYi6UXHjJi-DeEDedDHGJG q TK9c MG-DHiJHDM\I-W-DMXHY2c<V q j-i-IZG6UY q EI-G6UVI-WJDHXHYJI-D XHIJXM\<DMVlªEDNd6[Z\<X%[ G6UY2c<G

XHDMi-ByUa\I-B<G-lzD I G-DHY2c<G-XMDHi-B<Ua\<I-BG-lzDK ST? CQFW=WEKAW? )CXHI-XN\<DHVl p2Uai6UYJ\DHi-I q \<X%[GJDHIJHVDHG-G-lzV Bc(B<iJDHVDHGJIUj6UXMc<V MG-DHi-MI-IK A VDep2UGJI-Y2D IZpG-Ual ByU \ +N=*ALXDJ E ST?DS *A6KSd6[ Y2c<G-XMDHi-B<Ua\<I-BG-lCp(XHI-XN\<DHVªX§XM\<DMj-DHGZ[-VI XMB<cx<cHEl c<j-iJDeEDed6[-DN\<X%[ \PUY G6UaHlzByUDMVlzr +N=*ALXDJ E ST?BAB=@? XHIJXM\<DMVl

H(p,q)" ED

qic<x<c<x DHGJG-lzD§Y2c<ci<EI-G6Ua\l "

pic<x<cx! DMG-G-lzD8I-Vj q df<XMl XHI-XN\<DMVl " i =

1, 2, . . . , nK #ZUVIZdfR\<cG-I6UGj-cadG-cXM\<fsp2Ui-UYJ\<DMi-IJ q DM\ EIJG6UVI-W-DMXHY q jJi-I-i-c

E q XHI-XN\<DMVlI XmnIJI-WJDHXHY2c<r¤\c<W-Y-IHi-DHGJIJ[¤B x<cPdf! IJG-XM\B<DbXNd q W-UDHB j-iJDeE XM\PUaBad6[-DN\ XMc<x<c<r DHD j-cadG q HGJDHi-MI-TK B<cad k-IJ[ B<cB<i-DMVDHG-I Y;cG-XHDMi-ByUa\I-B<G-lCpXHIJXM\<DMV¤c<j-I-XMlzByUDN\<X%[ q i-UB<G-DMG-IJ[-VIVDNpyUG-I-Y-I#JUVI;dfR\<c<G6U

qi = ∂H(p,q)/∂pi, pi = −∂H(p,q)/∂qi.O<K O [

' q i-UB<G-DMG-I-r O<K O [ CXedDeE q DM\n∑

i=1

(∂qi/∂qi + ∂pi/∂pi) = 0. O<K O<O

A \DHi-VI-G-UPp¤c<xc<x! DMG-G-lCp(mUacB<lCp(Y;cc<iEI-G6Ua\ XMc<c\G-c! DMG-I-D O<K O<O Vc ¡ G-cj-i-DFEXM\PUaB<IJ\<f^YZUaY

N∑

i=1

∂xi/∂xi = 0, O<K O Z

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j-cHEW-IJGJ[- DHDMX%[ q i6UBG-DHG-IJ G-DMj-i-DMi-lzB<G-c<XN\<IK \<XM EUXedDeE q DM\ "WJ\<c Nd6DHVDHGZ\mUacB<c<Hc¢c<x*DHV]UB¢Y;cG-XHDMi-ByUa\I-B<G-lCpXHIJXM\<DMV]UPpG-DnIJVDMGJ[-DN\<X%[1B<c¢B<i-DHVDMG-IKSTI-G6UVIJW-DHXMY-I-D XMI-XM\DHVl XIJVDMGJ[- I-VX%[|B<csBi-DHVDMG-I Uj6UXMc<V HGJDHi

HIJI G6UaHlzByU \X%[ XMc<c\<BDM\<XN\<B<DMG-G-c ?DC LST? ICPF<=@EKAW? "A6K I-XN\<DHVl "ZBTY2c\c<i-lCpHGJDHi-MIJ[ q VDHGJf!UDN\<X%[_B<cB<i-DMVDHG-IÎJ RU\i-DHGJIJ[ IZd6Iî6UXMXHDN[-G-IJ[ "ZG-UalzByUa \<X%[;>A ABOB=@EKAW? "A6KoA=XHc<c\B<DM\XM\B<I-I¢X N\<I-V1XHI-XN\<DHVl " HG-DMi-HIZ[ Y2c\<ci-lCp Bc B<iJD VDHGJIG-Ui6UXN\PUDM\ "6G6UPlzByU \<X%[XHIJXM\<DMV]UVI_X9c\<iJI-k6Ua\DNdf<GJlzV(\<iJDHG-I-DMVIZd6Ic\i-I-k-Ua\<Dedf<G-c<rEI-XHXMI-j6Uk-IJDHrK# UY-IJD`XMI-XM\DHVlVc ¡ G-c i6UXMXHV]Ua\i-I-ByUP\<fY;UY_EI-X XHIJj6Ua\<IJB<G-lzDujJi-I XMVDHG-D G6UjJi6UBadDMG-IJ[ c\XHW-DN\PUTB<i-DMVDHG-I G6Unj-i-c\I-B<c<j-cPdc ¡ G-cDK9i-I-G-kJI-j-I6UPd6f<G-c<rcXHc<x<DMG-G-c<XN\<f< EI-XMXHI-j-Ua\<I-BG-lCpXMI-XM\DHV [-BPd6[-DN\<X%[RUB<IJXHI Vc<XN\<f edDHVDMGJ\PUbmUac<Bc<Hcc<x*DMV]Uc\1B<i-DMVDHG-IKAsXMI-XM\DHV]UPp X j-c< dc! TDHG-IJDHVHGJDHi-MI-ImUac<BlzrTc<x*DMV1B<c B<i-DMVDHG-I q VDMG-f!UaDM\<X%[ "B XHI-XN\<DHV]Uop Xc\<iJI-k6Ua\DNdfG-lzV \<i-DMG-I-DHV q B<DedI-W-I-B<UDM\X%[KSTI-G6UVIJW-DHXMY-I-D XHI-XN\<DHVlG6UalzB<U \X%[ =&WE ST?DS "? *A "<DMXNdI c<G-I G-Dzj-cHEBDHi

¡ DMG-l EDMr-XM\B<I- BG-DTG-IZp(XHIZd "jJDHi-DMVDHG-G-lCp(B<c8B<i-DMVDHG-I K]üi6UB<GJDHG-IJ[UB\cG-c<VGJlCpXHI-XN\<DHV([-B<G-crUB<I-XMI-Vc<XM\I1c\ B<iJDHVDHGJIG-DnXMcEDHi ¡ Ua\oKCcad6f! I-GJXM\<Bc i-DHUPdf<GJlCpY2cadDMxyUa\<Dedf<G-lCpXMI-XM\DHV(BmIZI-Y2D!"6i-UoEI-c<mnIJHI-Y2D!"

x<I-cPdc<HIJI "pZI-VI-I I=Ei q HIZp c<xadUaXM\R[Zp§G-UG-I-r G-DMY2c<G-XHDMi-ByUa\I-B<G-cJK i-DeEI GJIZpB<l EDed6[-DN\<X%[^c<XHc<xlzr YZdUXHX \PUY G6UalzB<UDHVlCp=<E SD LSYX C R =@E CPXHJ ? XHI-XN\<DHV ";Y2c\<ci-lzDnj-i-I-G-kJI-j-I6UPd6f<G-c G-DHY2c<G-XMDHi-B<Ua\<I-BG-l I_G-DedI-G-DMr-G-l9K B\c<Y2cadDMxyUa\<Dedf<G-crG6UaHlzByU \=EIJG6UVI-W-DMXHY q XHI-XN\<DHV q " jJi-DHc<xi6Ua q q HGJDHi-MI-IJXM\<cW-G-I-Y;U BHGJDHi-MI-|G-DNRUa\ q pyU TIZp Y2cad6DHxyUGJI-r "j-i-I-W-DMV"c<XHGJc<B<G-lzDp2Ui-UYJ\<DMi-I-XN\<I-Y-I Y2cdDMxyUG-I-r UaVjZdIJ\ q EU "W6UXN\<c\PU "*mnc<i-V]UY2cadDMxyUG-IJr§I \oK EK 'c<jJi-DeEDed6[- \<X%[ j6U i6UVDN\<i6UaVI XHI-XN\<DHVlIbB c<j-i-DFEDNd6DHG-G-lCpbj-i-DFEDNdUopbG-DnRUaB<I-X%[J\ c\ B<lzxc<i6U¢I-X pycHEGJc<Hc G6URW6UPdf<GJc<Hc XHc<XN\<cR[-GJIJ[ w K

- ! - $ $! $%- !(',-!.1*$%/.10*&!(' # #$%

2$92>-:;< ]<>2+-6;, 2+-,5 DM\<cHEUG6UPdIZRU=Y;cPdD xyUa\DNdfG-lCp j-i-c<k-DMXHXMc<B X j-c<Vc! Tf<I-XHXedDeEcByUG-IJ[ mUacB<lCp¢\<i6UaDHYJ\c<i-I-r EIJG6U VI-W-DMXHY2c<r=XHIJXM\<DMVlsxlCd=B<B<DFEDHGbB \<DHci-I- Y2cadDMxyUG-IJrb K K UG;EDedf!`\PUVc<VI K K G2Ei-c<G-cB<lzVI X \DNp jJc<i XM\PUodjJi-I-B<lzW-GJlzVIJG-XM\i q VDHGJ\c<V j-i-I I-X XNd6DeEc<B<UG-I-IXUVlCp_i-UaMdIJW-G-lCpY2cadDHx<Ua\<Dedf<G-lCp[-BPdDHG-IJrK x<X q EIJV G-DMXHY2cadf<Y2c`j-i-c<XN\<lCp "ZGJcn\I-j-I-WJG-lCp j-i-I-VDMi-c<Bj-i-DeEXM\PUBad6DHG-IJ[ EI

G6UVIJW-DHXMY-IZp j-i-c<kJDHXHXMc<BTBB<I;ED \i6UDMYJ\<c<iJI-rIJcx<i6U ¡ Ua DMr \c<W-Y-I B`mUaHc<B<c<Vj-i-cXM\<i-UG-XM\B<DK,<2+->-,?¢ <34>- 5¢UXHXMVc\<iJI-V dI-G-DMr-G-lzr c<XMk-IZd-d-[J\<ci

x<DNbj-c\DHi-f " q i6UBG-DHG-IZ[ Y2c\<ci-c<Hc Vc ¡ G-c8XHmnc<i-V q dIJi-c<ByUa\f G6U8j-i-I-VDMi-DbY2cdDMxyUa\<Dedf<G-cHc

LC Y2c<GJ\ q i-U i-I-XaK O<K Z " U "2j-iJDeEj-cPdc ¡ I-B UVjZdIJ\ q E q Y2cadDMxyUG-IJr

8kEc<XN\PUa\<cW-G-c'V]UPd6c<rKPAClzx<i6UaB B Y;UoWJDHXM\B<D j-DMi-DHVDMG-G-c<r`RUiy[;E

qG6U Y;cG;EDHGJXUa\c<i-D!"

Xnj-c<Vc! Tf< q i6UB<G-DMG-I-r1hnIJiypZMc<mU j-cPd q W-I-V

q + (LC)−1q = 0. O<K ORQ

STc<VG-c ¡ I-B O<K ORQ CG6ULq"6j-cad q W6UaDHV2

d

dt(Lq2

2+

q2

2C) = 0,

O<K O

\oK DKJE6d6[d x<c<Hc¢Vc<VDMGJ\PU Bi-DHVDMG-IBlzj-cadGJ[J \X%[bi6UBDHG-XN\<ByU

E = EL + EC = const, EL = Lq2/2, EC = q2/2C, O<K Oot

c\i6U ¡ U I-DCj-c<XM\cR[-G-XN\<B<cnB<c'Bi-DHVDMG-Ij-cadGJc<r HGJDHi-MI-I¢c<XHk-I;d-d6[J\c<i6U X q VVlV]UMG-IJ\<GJc<r

EL

I edDHYJ\i-I-W-DMXHY2c<rEC

MG-DHi-MI-reKAx<cad6DHD q Ec<x<G-lCp8Y2c<ci<EI-G6U \PUPp q i6UBG-DHG-IZ[Y2c<GJXHDHiJByUa\<IJB<G-c<Mc^c<XMk-IZd-d6[Z\<c<i-U Vc ¡ GJc¢Uj-I-XHUa\<f XNd6DeE q TI-Vc<x<i-Uac<V "6BB<DeE[UVDHG q B<i-DHVDMG-I τ = t/

√LC

2

x+ x = 0, x2 + x2 = a2, a = const. OK ORw

Sd6[_mUacB<lCp_Y2c<c<iEI-G-Ua\x1 = x

Ix2 = x

"-Uj-I TDHV q i6UBG-DHG-IZ[B B<I;ED

x1 = x2, x2 = −x1; x21 + x2

2 = a2.O<K Oo

6 yc./8`+NdghN+-`8'W./8'h i f&08q3/R*+-`(f i +q i f*dJTY'+-'( qs( " ydgQ/8T xr*/8fhNf+h./8`+NdJ'( r*f1( QMdMR5dg''*/8q i f/gTY'+ -'*+-f((

~UacB<lzrLjJc<iJ\<iJDM\(XHI-XN\<DHVl j-i-DeEXM\PUBad-[-DM\XHc<xc<r¤c<Y-i q ¡ G-c<XN\<f i6UoEI q XU a Xk-DMGJ\<i-cV B8G6URW6UPd6D1Y2c<c<iEI-G-Ua\oK c<W-Y;U§B8mUacB<c<V jJi-c<XM\i6UG-XN\<B<D " B Y2c\c<i-c<rB<DMYJ\<c<i mUaHc<B<c<rXHY2c<iJc<XM\I8cx<i6U UDM\X%[bB^G q d6f"G-UalzByUaDM\<X%[bc<XHc<xc<r "IbB¢EUG G-c<V(XNd q W6UaD`G q df¢Y2c<c<iEIJG6Ua\ DHXM\fS5 STR = E S =9\<IJj6U DC ?BEGFK©nUPdI-WJI-DnI-GJ\<DMHi6UodU EB<I ¡ DMG-IJ[ q Y2c<G-XMDHi-B<Ua\<I-BG-c<r1XMI-XM\DHVl Z Hc¢j-ci2[2EY;U "

c\i6U ¡ U DMD9BÈUG-G-cV jJi-I-VDHiJD9mÙaYJ\XHcpZi6UGJDHG-IJ[ MG-DHi-MI-I O<K Oot "yEUDM\¢B<ca Vc ¡ GJc<XM\f c<jJI-XUa\f DMDnXjJc<Vc! f q i6UaB<G-DHGJIJ[8O Mc jJc<iy[;EY;UJKZSTDHrJXM\<BIJ\<Dedf<G-c"c<j-iJDeEDedI-B^GJc<B q j-DHi-DMVDHG-G q ϕ

XHc<c\G-c!TDHG-IJ[JVI

x1 = a cosϕ, x2 = a sinϕ, O<K O ]

j-cad q WJI-V q i6UB<GJDHG-IJ[

ϕ = 1, a = 0, OK OR

Y2c\<ci-lzD9I j-i-DFEXM\PUaBad6[- \¢RUY2c<G EB<I ¡ DMG-IJ[ mUaHc<B<c<r \<c<WJY-IK-A c B<i-DMVDHG-I MB<cd k-I-c<G-IJi q DN\ cHEG6U j-DMi-DHVDMG-G6Ua[ ϕ " I¤mUaHc<B<c<D jJi-c<XM\i6UG-XN\<B<c Y2c<G-XMDHi-ByUP\<I-BG-c<Mc c<XHkJIZd-d6[Z\<c<i6U " \PUY-I-Vc<x<i6UPc<V "cHEG-c<VDMi-G-c-K #ZUi-Vc<GJI-W-DHXMY-I-VY2cadDHx<UG-I [-V cXHk-IZdJd6[J\c<i6Uc\<BDHW6UDN\i6UBG-c<VDHiJG-c<D EBI ¡ DHGJI-D'IJHc<x<i6U ¡ U TDHr^\c<W-Y-Ij-cc<Y-i q ¡ G-c<XN\<Ii6UoEI q XU a "-Y;UY N\<c j-c<Y;UaRUaG-c G-U¢iJI-XKO<K Z " RoKXNdIY2c<GJXHDHiJByUa\<IJB<G6Ua[XMI-XM\DHV]U'G-DedI-G-DMr-G6U "P\<c'DMD mUacB<lzrTj-ciJ\<i-DN\ q Xedc ¡ GJ[JDM\<X%[ K 9i-cIZd-d XM\<iJI-i q DMV M\c^G6Uj-i-IJVDHi-D q i6UB<G-DMG-I-r

x+ sin x = 0. O<K Z@[

A mUac<BlCpj-DHiJDHVDHGJG-lCpx1 = x, x2 = x q i6UBG-DHG-IJD O<K Z@[ Uj-I-XMlzByU \X%[XNd6DeE q TI-V¤c<x<i-Uac<V2x1 = x2, x2 = − sin x1.

O<K Z O c<XN\<cR[-G-IZ[ i6UBG-c<B<DMXHIJ[8G-DedI-G-DMr-G-c<Mc1V]Ua[Z\<G-I-Y;UG6UmUaHc<B<c<r=jZd6c<XHY2c<XN\<I i6UaX j-cad6c ¡ DHG-l BPEcadf=c<XMI

x1 (x2 = 0)B=\c<W-Y;UPp

x1 = 0,±π,±2π, . . .K c<ca\<B<DN\

XM\B q TI-r=mÙPc<B<lzrj-c<iJ\i-DM\ XHI-XN\<DHVlj-i-DFEXM\PUaBadDHG G-U i-I-XaKOK QJKACI2EG-c"WJ\<cc<XMc<x<lzDb\<cW-Y-I

x1 = 0,±2π,±4π, . . . \<IJj6U kJDHGJ\i " U

x1 = ±π,±3π, . . .

G-D q XM\c<r-W-IJB<lzDT\<cW-Y-I\I-j6U^ C ;IXHSRKACxad6IJIk-DMGJ\<iJc<BmUac<Blzr j-c<iJ\i-DM\ XMc<c\<BDM\<XN\<B q DM\¢dI-G-DMr-G-c<V q c<XHkJIZd-d6[Z\<c

i q 2Z\i6UDMYJ\<c<iJI-I_j-i-DeEXM\PUBad-[- \ XMc<x<c<r RUVY-G q \lzDnY-i-I-B<lzD "Jxad6IJY-IJD'Y c<Y-i q ¡ G-c<XN\R[-V "*c\i6U ¡ U I-Vp2Ui6UaYJ\<DMi V]UPdlCp8jJcUaVjZdIJ\ q ED^Y2cadDMxyUG-I-r "xadIJHY-IZpY_eUi-Vc<GJI-W-DHXMY-I-VzK 9DHi-DNG-D q XM\<cr-W-I-B<lzD\<c<WJY-IjJi-cpycHE[J\ c<XMc<x<lzD9I-GJ\DHMi6UPdfG-lzD Y-i-I-B<lzD

Γ0

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XHDMj6Ui6UP\<i-I-XaKS UPdf<G-DMr DMD q BDNdI-WJDHG-I-D MG-DHi-MI-I j-i-I-B<DFEDM\8Y B<i-U Ua\DNdfG-c<V qEB<I ¡ DHG-IJ EB<I ¡ DHG-IJ[ B<GJDzXHDHj-Ui6Ua\i-I-XeK IZ\ q Uak-IJ[ "2Y2c< EU HG-DMi-HIZ[ V]UP[J\<GJI-Y;UXHcc\<B<DN\<XN\<B q DM\EB<I ¡ DMG-I-Lj-cTXMDHj6Ui-Ua\<i-IJXHD!"JG6UalzB<UDM\X%[_G-DHMi q x<c<rK UPdDMr I-Dc\YZdc<G-DMG-IJ[ HG-DMi-HIJI B \ q I;dI=I-G q XM\<ci-c<G q j-iJI-B<cHE[J\Y=Y;URW-DHXN\<B<DMG-G-cbi6UP dI-WJG-lzV\<IJj6UV EB<I ¡ DMG-IJ[26Y2cad6DHxyUa\DNdfG-c<V q IZdIB<i6U Ua\<Dedf<G-cV q KhnUY1BI;EG-c IJTiJI-XKO<K Q "XHc<XN\<cR[-G-IJD V]Ua[J\G-I-Y;U^c<jJi-DeEDed6[-DN\<X%[ q dc<V¤DMHc c\

YZdcG-DHG-IZ[¢c\nj-cadc ¡ DMG-IJ[¢i6UaB<G-c<B<DMXHIJ[x1I¢XMY;ci-c<XM\f<

x2"G-c E-d6[G6URW-DMG-I-r

x1"

c\dIJW6U TIZpyX%[¢G-U k-Dedc<D]W-I-Xedc2nπ

"MEI-G6UVI-Y;U XHI-XN\<DHVl I2EDHGJ\I-W-G6UZK!9cN\<c<V qjZdcXHY2c<XM\fj-DHiJDHVDHGJG-lCp

x1, x2G-Dn[-Bad-[-DM\X%[ "XM\i-c<Hc^Hc<B<ci2[ "mÙPc<B<crjZd6c<XHY2c

XM\f<XHI-XN\<DMVl"B XMIZd q ca\<X q \XM\<BIJ[bcHEG-caG6URW-G-cXM\<I K 9c<Y;Ui-DMW-f I2EDM\ cTEB<I ¡ D G-IJ[;p "IJHc<x<i6U ¡ U TI-D \i6UDMYJ\<c<iJI-I Y;ca\<c<i-lCp8dD ¡ Ua\bBG q \<i-I XMDHj6Ui-Ua\<i-IJXHG-c<McY2c<GJ\ q i-U "\oK DK*cY2cadDHx<UG-IJ[;p=BcY-i-DHXN\<G-cXM\<I kJDHGJ\i6U "*G-DM[JXHG-cXM\<DMr G-D¢B<cHG-I-Y;U DM\oK ©9cbB1Xed q W6UD!"DHXedI HG-DMi-HIZ[ XHI-XN\<DMVl j-i-DMB<l UaDM\=Y-i-IZ\<I-W-DMXHY2c<DG-UoWJDHG-I-DI EBI ¡ DHGJI-D'XM\PUaG-c<B<IJ\X%[_B<i6U Ua\<Dedf<G-lzV "ZmUac<B<Ua[^j;dc<XHY2c<XN\<fG-DuHcHEIJ\X%[ E6d6[cHEG-cHG6URW-G-c<Mcc<j-I-XHUG-IJ[IbB i6UXHXMVc\<iJDHG-I-DTB<B<cHE[J\ BALXHAB?B;YFDA C1 LS>C mUacB<c<Dj-i-cXM\<i-UG-XM\B<c w K01,;,? <34>- ")2+->J&£+-,;$5uSTI-XMXHI-j6Uak-IJ[ MG-DHiJHI-I " c<x q

XNd6c<BadDMG-G6Ua[ G6UPdI-WJI-DHV"j-ca\<DHiJf"cYZUPlzByUDN\bj-iJI-G-k-I-jJI6UPdf<GJc<D BadIZ[-G-I-D G6Up2U i6UYZ\<DHiEB<I ¡ DHG-IJ[¢XMI-XM\DHVl9KZ©nUI-x<cad6DHD j-i-c<XN\<lzD RUY2c<GJc<VDHi-GJc<XM\I^j-i-cR[JBad6[- \X%[B XMI-XM\DHV]UPpXnj-cad6G-c<r EI-XHXMI-j6UkJI-DHr MG-DHiJHI-I "Y2c< EU XHIZd6l \<i-DMG-IJ[EDHrJXM\<B q \bj-cbB<XMDHV"XM\DHj-DMGJ[-VXHBc<x<cHEl "U1j-c<XN\ q jZdDMG-I-D HGJDHi-MI-I IJB<GJD c\<X q \<XM\B q DN\oK UaXHXHVca\<i-I-V j-i-c<k-DMXHXMl B^dI-GJDHr-G-cV¤EIJXHXHIJj6Ua\<IJB<G-c<VLc<XHk-I;d-d6[J\c<i-D "Y;c EUXHI dU\i-DHGJIJ[ j-i-c<jJc<i-k-I-cG6UPdf<G-U1XHY2c<i-cXM\<I IJHVDHG-DMG-IJ[ Y2c<c<iEIJG6Ua\<l9K niJI-VDHi-cV\PUY2c<r XMI-XM\DHVl XNd q ¡ IJ\bY2cad6DHxyUa\DNdfG-lzr Y2c<GJ\ q i "XHcHEDMi ¡ U I-r UaYJ\<I-BG-c<D^XMcj-i-ca\<I-Bad6DHG-I-D

RK-¨'i6UaB<G-DHGJI-DnY;cGJ\ q i6U

Lq +Rq + q/C = 0O<K Z\Z

d8Q/gT r/YfhYf+hc/YX (`` hY/Yf*d " gk

RUaVDHG-c<rj-DMi-DHVDMG-G-lCp1XHB<cHEIJ\X%[1Yx<DNi6UaHVDHi-GJc<r1mnc<i-VD

x + 2δx+ x = 0, 2δ = R√

L/C, τ = t/√LC.

O<K Z Q 9i-I

δ = 0IJVDHDHVY2c<G-XMDHi-B<Ua\<I-BG-lzr dI-G-DMr-G-lzr c<XMk-IZd-d6[Z\<c<i " i6UXMXHVc\i-DHG

G-lzr_B<l TDK6ACBDeEDMG-I-D9V]UPdcHc \<iJDHG-IJ[Y;URW-DHXN\<B<DMG-G-c VDHGJ[JDM\¢mUaHc<B<lzrj-c<iZ\<i-DN\XHIJXM\<DMVl9K6Sd6[

0 < δ < 1i-DTDHG-IJDHV q i6UB<GJDHG-IJ[ O<K Z Q [-Bad-[-DM\X%[

x = Aexp(−δτ) cos(ωτ + ψ), ω = (1 − δ2)1/2,O<K Za

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DHXN\<f E S A )SD i-IJXKO<K " U eK XNdI Y;c!HmnmnI-k-I-DMGJ\\<iJDHG-IJ[

δ > 1"-j-i-c<kJDHXHXB¢XHI-XN\<DMVD Uj-DMi-I-cHEI-WJDHXHYJI-r2

x = A1exp(λ1τ) + A2exp(λ2τ), λ1,2 = [−δ ± (δ2 − 1)1/2]/2, O<K Z t

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t→ ∞ duxyUP[ IJHc<x<i6U ¡ U Ua[T\c<W-Y;UXM\i-DHVIZ\<X%[ YG6URW6UPd q Y;cc<iEI-G6Ua\ B q XN\<c<r-WJI-B<lzrbmnc<Y q Xnd6I-x<c q DNdK j-I-XHUG-G-c<D XHBc<r-XM\B<c[JBad6[JDM\<X%[ c<x! TI-VE6d6[ EI-G-UVI-W-DMXHY-IZp§XMI-XM\DHVX jJcad

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*$%- -*.$%/.10*! # #$%"!

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x− a(1 − bx2)x + x = 0. O<K Z w

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j-i-DFEXM\PUaBad6[-DN\<X%[ Y;UY

x1 = x2, x2 = a(1 − bx21)x2 − x1

O<K Z j-i-IJW-DHV2

a(1 − bx21) 6≡ 0.

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Γ"-IJcx<i6U ¡ DMG-G-c<Mc G-U¢iJI-XK*O<K t "K

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f+IR*+-`8' (.` X-(XhN+q " fd8X +hR` QI'Dd *+-'( r*d8f*dgq +hNf/gT a = 1, b = 0, 3l6 " f/8+.( R*T i q +f1'*/ M/ hN/Yfd 'DdWr`*/8X./YXh r*+f+q +-'' x1, x2

(*XI`*+-''*/Y+A('*hN+Mf(f/8TNdg'(*+i f*dJTY'*+I'( " R*` WQ-'*d +-'( r*d8fd8q +hNf/8T a = 1, b = 0, 3, B = 1, 0, ϕ0 = 0

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x− a(1 − bx2)x + x = B sin(pτ + ϕ0).O<K Z

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x1, x2

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N = 3"\oK DaK<B<lCp<cE XzjZdcXHY2c<XM\I^B'\i-DNpZVDHiJG-c<DumUaHc<B<c<DCj-i-c<XN\<i6UGJXM\<Bc c<B<XMDHV

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N ≥ 3XHcc\<B<DN\<XN\<B q

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t → ∞ \i6UDMYJ\<c<iJIJ[G-Dj-c<Y-I;EUDM\ UVY-G q \c<r c<xadUXM\I I G-Dj-i-IJ\R[JHI-B<UDM\X%[LY(IJBDHXM\G-lzV \<I-j6UaVUa\a\<i6UYZ\<c<i-cB XMVzKJDHY-k-IJ Q eK

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N ≥ 3K

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x = mx + y − xz, y = −x, z = −gz + gI(x)x2,O<K Q [

I(x) =

1, x > 00, x ≤ 0

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B<j-cPdG-Dzc<j-iJDeEDedDHG-GJlzr^XHVlzXed d6I fnB'\<cV§Xed q W6UD!"Y;c EUG-UV q EUDM\X%[ jJDHi-DMB<D XM\IIZpTG6Uumnc<i-V]UPd6f<G-lzrT[ZlzY V]Ua\DHV]Ua\I-Y-IK XMG-c<B<l XN\<i-cHc<r¢V]Ua\DHV]Ua\I-W-DHXMY2c<r\<DMc<i-I-I q XM\<cr-W-I-B<cXM\<Ix<lCdIRUodc ¡ DHG-lB \i q EUPpYJi q j-G-c<Mc i q XHXMY2c<Hc¢V]Ua\DHV]U \<IJYZU K K;n[Jj q G-c<ByUnc<Y2cadc O [\[ dDM\TG6UaRURE2i6UaBIJ\<I-DuYZURW-DMXM\B<DHG-GJc<r \<DMc<i-I-I I\<DMc<i-I-I§x<I-m q i-Y;Uk-I-rbEI-G-UVI-W-DMXHY-IZp8XMI-XM\DHVXMB[JUG-c_XI-VDMG6UVI=i-c<XMXHI-r-XMY-IZpq W-DHGJlCp K K G;Ei-c<G-c<B<U "6AuK K i-G-cPdfPEU¢I_IZp q W-DHGJI-Y2c<B-K9c<j-l \PUDMVX%[bG6U¢j-i-cXM\<lCpIj-c<GJ[Z\<G-lCpj-i-IJVDHi6UPpjJi-c<IZd-d6uXM\i-I-i-cByUa\<fXMc

EDMi ¡ UG-I-DnIVDM\cHEl i-D DMG-IJ[_RUoEUoW_c<x q XM\c<r-W-IJB<c<XM\IbIx<I-m q i-Y;Uk-IZ[ZpÊIJG6U VI-W-DMXHY-I;p1XMI-XM\DHV Q t " " ] K

,&! , !. )#$%- , -#$+

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dx(t)

dt= x = F (x, µ),

Z K O ED

x(t) j-DHiJDHVDHGJG6Ua[ XMc<XM\co[JG-IJ[ "

F=G-DMY2c\<c<i-Ua[ m q G-Y-k-IZ[ XHc<XN\<cR[-GJIJ[ "p2U

i6UYZ\<DHiJIJ q Ua[8RUY2c<G HBcadukJI-I " µ j6Uai6UVDM\i=XHI-XN\<DHVl9KoXNd6I=UoEUGJcG6U W6UPd6f<G-c<D XHc<XN\<cR[-GJI-Dx(t0)

"\<cX q DMXM\B q DN\=DeEI-GJXM\<BDHG-G-cD i-DTDHG-IJD q i6UB<G-DMG-IJ[ Z K O "<Y2c\<ci-c<D j-i-DeEXHY;UalzB<UDM\Tx q E q DHD XHcXM\<cR[JG-I-D x(t) E6d6[ d6ux<lCp t > t0

KsX dI¢W-I-Xedcnj-DHiJDHVDHGJG-lCp XHc<XN\<cR[-GJIJ[ i6UB<G-c EB q V IZdI¢x<cadDMD "\<cnVcHEDNd6f<¤x q EDM\XHIJXM\<DMV]U¢EB q p IZdIx<cadDMD q i6UB<GJDHG-I-r/2

x1 = F1(x1, x2, µ), Z K Z

x2 = F2(x1, x2, µ).

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XHIJXM\<DMVl G6UV]UPdc<D B<cV q TDHG-I-DDMD XHcXM\<cR[JG-IJ[ "G6U1j-DMi-B<c<V N\PUj-Dc<G-U1Vc ¡ DM\

x<l \f I-XHXedDeEc<ByUG6U Bî6UaVYZUopdI-G-DMr-G-c<Mc j-i-I-xPdI ¡ DMG-IJ[K 9cR[-XHGJI-V M\c-K q XM\fx0(t)

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"-Y2c\<ci6Ua[_RUoEUDM\ V]UPdc<D9ca\<YZdcG-DHG-IJDTc\ W6UXN\<G-cHc i-DTDHG-IJ[/2

y(t) = x(t) − x0(t), Z K Q

eEDHXMfx(t)

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V q DMG-IJ[ y(t) "-Y2c\c<i6Ua[j-cHEWJI-GJ[-DN\<X%[ q i6UB<G-DMG-I- Z K O eK6 UaNdc ¡ IJV(m q G-Y-k-IJFB¢XM\DHj-DMG-G-c<rbiy[2E B¢c<Y-iJDHXM\G-c<XN\<IbW6UaXM\<GJc<Hc i-DTDHG-IJ[

x0(t)2

F (y) =dF

dx|x=x0(t)y(t) +

d2F

dx2|x=x0(t)y

2(t) + . . . Z K

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y(t) = F (y, µ) =dF

dx|x=x0(t)y(t) + Φ(y),

Z K t! ED

Φ(y) =d2F

dx2|x=x0(t)y

2(t) + . . . Z K w

dUeUaDHVlzDΦ(y)

B<YZd W6U \`B<XMD W;dDHG-lXyn (n ≥ 2)

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y(t)DMXM\<fV]Uodc<DTca\<YZdcG-DHG-IJD

c\ W6UXN\<G-cHcîJD DMG-IJ[ K 9cN\<c<V q GJDNdI-GJDHr-G-lzVI1WZdDHG-UVIB q i6UB<GJDHG-I-I Z K tCBj-DMi-B<c<V(j-i-I-xad6I ¡ DHGJI-I1Vc ¡ G-c¢j-i-DHGJDHx<i-DMW-f-K UaY-I-V c<xi6UacV "<E-d6[MB<cad k-I-I V]UPd6c<Hc`B<cV q TDHG-IJ[ Vl¤j-cad q W6UDMV dI-G-DMr-G-c<Dq i6UB<GJDHG-I-D2

y = A(t)y, A(t) =dF

dx|x=x0(t)

Z K UaXHXHVca\<i-I-Vj-i-I-VDMiK q XM\f¢EI-G-UVI-W-DMXHY;Ua[XHI-XN\<DHV]U¢RUoEUG6U q i-UB<G-DMG-I-DHV2

x = a− bx2, a > 0, b > 0. Z K ]

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x = 0IVl j-cad q W-UDHV2

x01,2 = ±

√a

b

Z K

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1

K

y = −(2bx01)y = (−2

√ab)y = λy,

Z K O [ λ =

dF

dx|x0

1= −2

√ab.

D DMG-I-DHV q i6UB<GJDHG-IJ[ Z K O [ x q EDN\ y = exp(λt)KACcHV q DHGJI-D y HY2XHjJc<G-DHGJk-I UPdfG-c8Ua\ q pyUDM\ Bc8Bi-DHVDMG-I

λDHXM\f c\i-I-k6UP\<DNd6f<G-c<DbWJI-XNdc NK \<c§cG-UoW-UDM\ "

WJ\c§XMc<XM\co[JG-I-Dx0

1q XM\c<r-W-IJB<c UY Y;UY B\c<i-c<DXMc<XM\cR[-G-I-D x0

2

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2

GJD q XN\<c<rJW-I-B<c STc<XM\PUP\<c<W-GJc"j-i-c<XN\PUa[ I;EDN[ªj-i-DFEXHY;UaUG-IJ[ q XM\<cr-W-I-B<cXM\<I j-cLd6I-G-DHrJG-c<V q

j-i-IJxadI ¡ DMG-I- c<Y;UaRUodUXHf B<DMXHf<V]U jZdcHEca\<B<c<iJG-c<rK59XHjJcadf q [V]Ua\<DMV]Ua\<IJW-DHXMY-I-rmnc<i-V]UPd6IJV "*Vc ¡ GJccx<c<x! TIJ\<fi-DM q d6fo\PUP\ Z K uG6U_XNd q W-Ur8EB q p8I8x<cadDMD¢j-D i-DMVDHG-G-lCp XHc<XN\<cR[-GJIJ[K ©Uaj-i-I-VDMi "BXNd q W6UaD N = 2 q i6UBG-DHG-IJD Z K 9j-i-I-VDN\B<I;E 2

y1 = a11y1 + a12y2, Z K O<O

y2 = a21y1 + a22y2,

EDaij = ∂Fi

∂xj|xi=x0

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x0(t) j-i-I-VDMG-DHGJI-DHV

XHcB<i-DHVDMG-G-lCpY2c<Vj-f< \<DMi-c<B N\<I_\<i q EGJc<XM\IdDHMY2c^j-i-DMcHEcadI-Vl9K

# ) ( ',# #$%

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c\dIJW6UDM\X%[ c\'IJVG-DHrKoSUaByUrJ\D i6UXMXHVc\i-I-V8B'Y;UoWJDHXM\B<Dzj-i-IJVDHi6U9c<x DHV BcElB cDMi-DKh9cadI-WJDHXM\B<DHGJG-c<rbVDMi-c<r1jJi-I-B<G-cXHI-Vc<rXHcPdG-DHWJG-c<r MG-DHi-MI-I[JBad6[JDM\<X%[\<DMVj-DHi-Ua\ q i6U9B<cHEl \<c<WJG-DHD " MG-DHi-MIJ[ jJi-c<j-c<iJk-I-c<G6Uodf<G6U \<DHVjJDHi6Ua\ q i-DeKDM\<cVB<cHEU¢B caDHiJD\DHjZdUP[1IVc ¡ G-c¢Y q j6UP\<f<X%[ K G6UXM\ q jZdDHGJI-DHVc<XMDHG-I\<DHVjJDHi6Ua\ q i6UB<cHElj-c<XN\<DHjJDHG-G-c q VDHG-f! UDM\X%[Kh q j-Ua\<f<X%[ q ¡ DG-DTp<c<W-DN\<X%[ "*cHEG6UaY;cB<cHEUI jJi-I x<cPdDHD G-IJHY;cr "GJcjZd6uXHcB<c<r§\DHVj-DMi6Ua\ q iJD!" c<XN\PUDM\X%[ B<cHEc<r #2d q x<c<Y2c<rc<XMDHG-f< B<DHi<pZG-I-r¤Xedc<r B<cHEl B§cDMi-Dbc<XN\<lzByUaDM\ Ec§G q dDHB<cr\<DMVj-DHi6UP\ q i-l IBPEi q jJi-DHB<i-U UDN\<X%[ B1dDFE *S UPdDMDI j-i-I −200 dDFE(c<XM\PUDN\<X%[ dfPEc<VzK '\<c¡ D9j-iJc<IJc `dc niJIjJi-cpyc ¡ EDHG-I-I_\<DMVj-DHi-Ua\ q i-lW-DMi-DMG q df¢B<cHEU i-DNY2c IJ VDHGJIZdUXHB<cIXHB<c<rJXM\<B<U 2-c<G6UÌJ ¡ I2EY2c<HcXMc<XM\co[JG-IJ[_j-DMi-DdU¢BT\<B<DMiEc<DK G-DjZdUaB<G-c"UXHY;URW-Y2c<VzK

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00 "-W-DMi-DM

1000 "6Y2c< EU¢B<cHEU j-i-DMB<i6U UDM\X%[bBj6Ui KeXM\f I¢Ei q HI-Dnc<XMc<x<lzD G6URW-DMG-IJ[\<DMVj-DHi-Ua\ q i-lBcEl9K Y;UalzByUaDM\<X%[ "6WJ\<cx<cad6f! I-GJXM\<BcIJGJ\<DMi-DHXMG-lCp§mIZI-W-DMXHY-I;p RUoEUoW jJi-I=IZpbV]UP\<DHV]UP\<I-W-DMXHY2c<Vcj-I XUaG-I-I j-i-IJB<cHE[J\_YEI-mnmDMi-DHGJk-I6UPdfG-lzV q i6UaB<G-DHGJIJ[-V " UB<I-X%[ I-Vc\ cEG-c<HcIZdIG-DMXHY2cadf<YJIZpj6Ui6UVDN\<i-cB-K UaXHXHVca\<i-I-VªB8Y;URW-DHXN\<B<DbjJi-I-VDHi-U q i6UB<GJDHG-IJ[(Y2cadDMxyUG-I-r(c<x<lzY-GJc<B<DHGJG-c<Hc

V]Ua[Z\<G-I-Y;UTI;dI WJ\<c X V]Ua\DHV]Ua\I-W-DHXMY2c<r \<c<WJY-IîJDHG-IJ[^j-cadGJc<XM\f< I;EDHGJ\I-W-G-c j6Ui-UPd-dDedf<G-c<Mc

RLC Y2c<GZ\ q i6U 2

x + αx+ ω20x = 0

Z K O Z ¨'i-UB<G-DMG-I-D Z K O Z *XMcHEDHi ¡ IJ\ EByU j6Ui6UVDN\<i6U 2

αnj6Ui-UVDM\i RUa\ q pyUG-IJ[ "ap2Uai6UY

\<DMi-IJ q I-r8\<i-DMG-I-D "I ω0

_j6Ui6UVDN\<i " c<j-i-DFEDNd6[J TI-r§W-UXM\c\ q Y2cad6DHxyUGJI-rKXNdI1j-c\<DMi-IMG-DHiJHI-Ibca\<X q \XM\<B q \ "j6Ui6UaVDM\<iRUa\ q p2UG-IZ[ α = 0"-\<c i-DTDHG-I

DHV q i6UaB<G-DHGJIJ[ Z K O Z nx q E q \beUi-Vc<G-IJW-DHXMY-I-D G-DNRUa\ q p2U I-D_Y2cadDMxyUG-IJ[ K 9i-IV]UPdcV§\<iJDHG-I-I

0 < α < 1EB<I ¡ DHG-IJD XMI-XM\DHVl¤x q EDN\TY2cad6DHxyUa\DNdfG-lzV X UVjZdI

\ q Ecr "aY2c\c<i6Ua[ q VDHG-f UDN\<X%[B<c B<i-DMVDHG-I`j-cMY;XMj-c<G-DMG-k-I6Uodf<G-c<V q RUaY;cG q KP©nU Y2c<G-DMk "6j-i-IÊc<XN\PUa\<cW-G-c^xcadf!Tc<V \<i-DMG-I-I

α > 1 EB<I ¡ DMG-I-DnV]Ua[Z\<G-I-Y;Ux q EDN\

Uj-DMi-I-cHEI-WJDHXHYJI-V "RUa\ q pyU TI-V Bc B<i-DHVDMG-IK ¨ ¡ D B N\<c<Vj-i-cXM\<cVsj-iJI-VDHi-DB<l EDed6[- \<X%[ EByU`c<XMc<x<lCp¢HG6URW-DHG-IZ[ j6Ui-UVDM\i6U

α = 0Iα = 1

"2c\<Y;dc<G-DMG-IJ[ ca\Y2c\<ci-lCpYZURW-DMXM\B<DHG-GJc VDHGJ[J \ XHBc<r-XM\ByU XMI-XM\DHVl9K

'VDMG-DHG-IJDj6Ui6UVDN\<i6U B q i6UaB<G-DHGJI-I Vc ¡ DM\ B<l HByUa\f j-c\<DMi- q XN\<c<rJW-I-B<cXM\I¢cEG-c<HcnXMc<XM\co[JG-IJ[ IZdIi-D ¡ I-V]U'm q G-Y-kJI-c<G-I-iJc<ByUG-IZ[XMI-XM\DHVlIjJDHi-DepycHEDHD B_Ei q Mc<D!"c\dIJW-G-c<D ca\1j-DMi-B<c<McbXHcXM\<cR[JG-I-DK \<c1[JBadDMG-I-D^G-UalzByUaDM\<X%[ R>A F =&BADC c\¢XNd6c<ByU i6UaeEB<c<DMG-I-D "UTG-UoWJDHG-I-Dj6Ui-UVDM\i6U "-j-iJIY2c\c<i-c<V(c<G-cj-i-cI-X pycHEIJ\ E S LS R>A F =&BABA6K c<XN\<cR[-G-IJD'XHI-XN\<DMVlGJI ¡ D \<c<W-YJI^x<I-m q i Y;Uk-I-I8I=B<l D¢DMDj-i-I8IJVDMG-DHG-IJI§j6Uai6UVDM\i6U_B<XHD \PUY-I8VDHGZ[-DM\X%[K 9XHG-c "WJ\<c

B<cHEU j-iJI=\DHVj-DMi6Ua\ q iJD +30 "I +220 M\ci-UaG-lzD¢XMc<XM\co[JG-IJ[ K*©ncjJi-I M\<cVB<cHEU1cXM\PUDN\<X%[ B<cHEc<r A V]Ua\DHV]Ua\I-Y2D_I mnIJI-Y2D X q DHXN\<B q DM\ j-c<GJ[Z\<I-D Mi q x<cXM\I¤IZdI(XM\<i q YZ\ q i-G-c<r q XN\<c<rJW-I-B<c<XN\<IK q \f M\c<Hc jJc<GJ[J\IJ[ B=\<cV " WJ\c j-i-IV]UPdcV IJVDMG-DHGJI-I j6Ui-UVDM\i6U Hi q x<Ua[XMI-XM\DHV]U§p<c\<f I IJHVDHGJ[JDM\ B EDM\PUPd-[Zpi-D ¡ I-V m q G-Y-k-I-cG-I-i-c<B<UG-IJ[ "zG-c G-D=j-i-I-G-kJI-j-I6UPd6f<G-c-K M\c<r\c<W-Y-IiJDHG-IJ[E6d6[^Hi q x<lCp XMI-XM\DHV j-DHiJDNpycHEW-DMi-DMu\<c<W-Y q xI-m q iJYZUak-I-I cHG6URW6UDN\ XHVDMG q cEG-cHcXM\i q YJ\ q i-GJc q XM\c<r-W-IJB<c<Hc1i-D ¡ I-V]U_G6U Ei q Hc<rK$9i-I M\c<V B_\<c<WJY;D x<I-m q i Y;Uk-I-I XHI-XN\<DHV]UG-D [JBad6[JDM\<X%[ Hi q x<cr2*V]UPdc<D IJVDMG-DHGJI-D^j6Uai6UVDM\i6U1B_\ q IZdII-G q XM\c<i-c<G q jJi-I-B<cHEIJ\ Yi-DMHY-I-VIZVDHGJDHG-IJ[JV XHcXM\<cR[JG-IJ[KS UByUrZ\<DB<DHi-GJDHVX%[(YG6U TDHV q j-i-IJVDHi q X q XM\c<r-W-IJB<c<XM\f< XN\PUk-I-c<G-Ui-G-lCp

XHcXM\<cR[JG-I-r_BTXMI-XM\DHVD Z K ] eK l q XNdcB<IZdI-XMf"JWZ\<cB q i6UaB<G-DHGJI-I Z K ] j6Ui6UVDN\i-l

aIbj-cadc ¡ IJ\<Dedf<G-l9K¨]XM\<cr-W-I-B<cXM\<fcj-i-DeEDNd6[JDM\<X%[=HG6UY2c<Vj-i-c<IZB<cHEG-cr

j-i6UaB<c<r W6UaXM\<I q i6UBG-DHG-IZ[ Z K ] B XM\PUak-I-c<G6Uai-G-c<r\<cW-Y2D!"J\oK DK;G6UY2c<V B<DedI-W-IJG-lλ Z K O [ eK$niJI j-cad6c ¡ IJ\DNdf<GJlCp HG6URW-DHG-IZ[Zp§j6Ui6UaVDM\<iJc<B

aIbN\PU1j-i-cIJB<cHE

G6Ua[(B<XMDH EU c\d6I-W-G6U ca\ G q d6[ K WJ\<c DHXNd6I Vl x q EDHV q VDMG-f!UP\<f G-UoWJDHG-I-Dj6Ui-UVDM\i6U

a hUaY B<I;EGJc IJ Z K O [ ¢jJi-I

a = 0 G-DMUB<I-XMI-Vc c\ B<DNdIJW-I-G-l

b > 0B<DNd6I-W-I-G6U

λc<x<i6U UDM\X%[ B8G q df"]B<cHV q DHGJI-D y G-DG6Ui-UXM\PUDN\ IGJDRUP\ q p2UDN\ UPdc_\<c<Mc"*j-i-I a = 0

BXHI-XN\<DHVD Z XMc<XM\co[JG-IJ[ i6UG-c<BDHXHIZ[ Y;UY8x<lXNd6I-ByU \<X%[B`cEG-c

x = 0 S UPdDHD "ZDMXNdI "

a < 0"y\<cTXHc<XN\<cR[-G-IJri6UB<G-cB<DHXMIJ[^GJDM\

B<c<BXHD S DMr-XM\B<IJ\DNdf<GJc"*B M\<cVLXed q W6UD x01,2 = ±j

√|a|b

"\oK DaKXM\PUG-cB[J\X%[§WJI-XM\cVG-I-VlzVI K9i-I-B<DFEDHVL\<DHjJDHi-fbiJDM q dfR\PUa\l V]Ua\<DMV]Ua\<I-WJDHXHY2c<Mc=UG6UPd6IJRU M\<cr x<I-m q i-Y;U

k-I-I " Y2c\<c<i-Ua[IJBDHXM\G6U YZUaYxI-m q iJYZUak-IJ[@; s FL=WEK?DS<C F =W?DSWCIADCeKzA G-c<B<fi6UXMXHVc\i-I-V q i-UB<G-DMG-I-D Z K ] eK q XM\<f x0(a)

DHXN\<f=Hi q x<cDXHc<XN\<cR[-G-IJDi6UB<G-cB<D XHIZ[ "C\oK DK

λ(a) 6= 0K \c caG6URW6UDN\ "zWJ\<c jJi-IV]UPdc<r¤ByUai-I6Uk-IJI¤j6Ui6UVDN\<i6U

ai6UBG-c<B<DMXHI-Dx0(a)

j-iJcEcad ¡ UDN\ X q TDHXM\B<c<ByUP\<f8YZUaY q XM\c<r-W-IJB<c<DIZdI GJD q XN\<c<r W-I-Bc<DK9i-I G-DMY2c\<c<iJc<VG6URW-DMG-I-I j6Ui-UVDM\i6U

a = a∗XHc<xXM\<BDHG-G-cDW-I-XNd6c

λ(a∗)B

j-cad6c ¡ DHG-I-Ii6UBG-c<B<DMXHIJ[Vc ¡ DM\ c<xi6Ua\<IZ\<f<X%[B G q d6f 2

λ(a) =dF

dx|x0 = 0, a = a∗.

Z K ORQ Sd6[i-DUPd6IJRUkJI-Ibx<I-m q i-Y;Uk-IJI EB q Y-i-Ua\<G-cDTi6UBG-c<B<DMXHI-D G-DHcxRp<cHEI-Vc"WJ\<c<xlB\c<i6Ua[j-i-cIJB<cHEG-Ua[x<lCdU¢c\dI-W-G-U c\ G q d6[

d2F

dx2|x0 6= 0.

Z K O Sd6[ Blzj-cadG-DMG-IJ[ q XNdcB<I-r Z K ORQ*I Z K O B'c<x! TDHV XNd q W-UD G-DMc<xpycHEI-Vc"WJ\<c<xlI-X p<cEG-c<D q i6UB<G-DMG-I-D9Bj-i-UB<c<r W-UXM\IBYZduW-UPdcY;UYVI-G-IJV q V Y-ByUREi6Ua\I-W-G-c<DG-DedI-G-DMr-G-c<D`XNdUeUaDHVc<D "Y;UY_B G6U TDHVjJi-I-VDHiJD Z K ] NK

( i f.1d( lR*T i .f*d8h8'*/Y+f*dJTY'/8TN+-X-(*+J" f( a > 0 TcX-(*X-hN+q + " RT:d,XhYd (/8'Ddgf'X/8XhN/ 1'( x0

1

( x0

2

0 r*f1( a = 0 /g'( X-`(T:d#hNX LT /R*'*/ ( rf( a < 0 XhYd (/8'Ddgf' + X/YX-hN/ 1'( (*X*+QMduhXNdI q Xedc<B<IZ[ Z K ORQ I Z K O B<lzj-cad6G-DHG-l "-\c x0 DHXN\<f EB q Y-i6UP\<G-lzrY;ci-DHG-f

I-X p<cEG-c<Hc q i6UBG-DHG-IZ[ Z K ] NK G6URW-DMG-I-D9j6Uai6UVDM\i6U a∗ "ZjJi-IY2c\<c<iJc<V B<lzj-cadGZ[ DM\X%[ q XNdcB<I-D Z K ORQ "[-BPd6[-DN\<X%[=\<c<WJY;cr x<I-m q i-Y;Uk-IJIKSTc_\c<W-Y-I8x<I-m q i-Y;Uk-IJIa > a∗

VlI-VDHDMV Z XHcXM\<cR[JG-IJ[i6UB<G-cB<DHXMIJ[KJA¤\c<W-Y2Dx<I-m q i-Y;Uk-I-I a = a∗c<G-I

XNd6I-ByU \<X%[(B8cHEGJc"EUPd6DHD_j-i-Ia < a∗

XMc<XM\co[JG-I-r i6UB<GJc<B<DHXMIJ[ B XHI-XN\<DMVD1G-Dx q EDN\ AG-U DMVXNd q W-UD Z K ] a∗ = 0

KDN q dfR\PUa\lVc ¡ G-c¢j-i-DFEXM\PUBIJ\<f Mi6UmnI-W-DMXHY-I XMVzK6i-I-XK Z K O eK

#$+ *&# )+ *$#$ - # !

©9DHXMVc\<iy[ G-U VG-c<McadDM\G-u I-XM\c<i-I- X q DHXN\<B<cByUG-IJ[ Ii6UaHB<IJ\IJ[ YZdUX XHIJW-DHXMY;cr \DHc<iJI-I q XN\<c<r-WJI-B<c<XN\<I I x<I-m q i-Y;Uk-I-r " G6UXN\ q j-IZd(Vc<VDMGJ\ YZUaY M\<cW6UXN\<c x<lzByUDN\ "6Y;c EU Y M\c<r_\<DHci-I-I1xlCdc¢BPEi q nj-i-I-BPdDHW-DMG-c B<XHDMc<x! DMD`BG-I V]UG-IJDK 9i-I-W-I-GJc<r \c<V q j-c<Xed q ¡ I;dI(jJc<j q d6[-iJG-c§IZMdc ¡ DMG-G-lzD1BDHi-XMI-I(i-Ux<c\mni6UG-k q HXHY2c<Mc"V]Ua\<DMV]Ua\<I-Y;UDHG-D c<V]Uj-c\PUY G6UPlzByUDMVc<rs\DHc<iJI-I Y;Ua\PU XM\i-c<m¢Ko DHci-IJ[=Y;Ua\PUXN\<i-c<mgBG6URW6UPdD XHDMVI;EDMX%[J\<lCp HcHEc<BXM\PUodUVcHEG-cr "*j-cGJ[Z\<G-c<r Y;UYI-V Y;UaRUPd6c<XHf E6d6[_G-DMXHj-DMk-I6UPd6I-XM\c<BÎ q G-I-B<DMi-XUPd6f<G-c<XN\<f<XHB<cIZpj-i-DN\<DMGJI-rXN\PUPdU§G-Uj-c<VI-G-Ua\<f§j-XHDMBPEc<G6U q W-G-lzD\<DHci-I-I j-iJc!`dlCp(B<iJDHVDHG K AW-DMV ¡ D^X q \<f1EDedU ncR[JBadDHGJI-D \<DMc<i-I-I Y;Ua\PUXN\<i-c<m KBc<V]U1XMj-DHk-I-UPdI-XN\PUVIx<lCdc§B<c<XMj-i-I-GJ[Z\<c GJc<i-V]UPdfG-c-K [;EiJDM q dfR\PUa\c<B M\c<r\DHc<i-IJIRUXed q ¡ IJByUDM\XUaVc<Hc d q x<c<Y2c<Hc q ByU ¡ DMG-IJ[K*©9c mnIZdc<XMc<mnXHY2c<Mc c\Y-i-l \<IZ[ NEDMXHfG-DM\oK$9c[-XMG-I-V(j-c<W-DMV q K

f1(*q +f q M./ g( i f.1d ((&"P$ h8d(/8'*d8f'/Y+cX/YXhY/ '(+ l6 hN+f 1+h i XhN/ (TN/8Xh ( T 8`(Q-(x'+ M/Br/ 1T8` uhNX BR*TNdc'*/gT i XhN/ (TXh8d(*/g'Dd8f1'WX/8XhN/ 1'( <@

q \<f EDNdU^RUYZd6uW6UDN\<X%[8B \<cV "WJ\<c_i-DMW-f_I;EDN\_B<XHDc \DNp ¡ DTx<I-m q i-Y;Uk-I [Zp "G-c jJi-I N\<c<VªB<lzxI-i6UDN\<X%[ cHEIJG IJ\<I-j-cB \PUYG6UalzB<UDHVlzD , CIE ADCR>A F =&BABA6K6S`d-[1j-cR[JXHG-DMG-IJ[=i6UXMXHVc\i-I-VEByU j-i-c<XN\<lCp jJi-I-VDHi-UJKA jJDHi-B<cVXNd q W-UD XMVzK i-IJXK Z K Z BîJDM q dfR\PUa\D¢x<I-m q i-Y;Uk-I-I I-X p<cHEG-c<DXN\PUk-I-c<G-Ui-G-c<DXMcXM\cR[-G-I-Dn\<DMiy[-DM\ q XN\<c<rJW-I-B<c<XN\<f I_i-c ¡ EU \<X%[ EByU G-cB<lCp q XN\<c<rJW-I-B<lCpXM\PUak-I-cG6UiJG-lCpXMc<XM\co[JG-IJ[ K 9i-IM\c<V B<G-c<B<f`j-cR[-B<IJB! I-DMX%[ÊByUTXN\PUk-I-cG6Ui-G-lCpXHc<XN\<c[-GJIJ[ i-I-XaK Z K Z " B i6UXMj-cadc ¡ DMG-l"B GJDHj-c<XMi-DeEXM\<BDHG-G-cr8xPdIJcXM\<I1c\ I-X p<cHEG-c<McXHcXM\<cR[JG-IJ[ "yY2c\c<i-c<DzjJc\<DMi2[;dc q XN\<c<r-WJI-B<c<XN\<f j-c<VDMW-DHG-c9HB<DMeEc<W-Y2c<reKyCI-m q i Y;Uk-I-I=\PUY2c<Mc\I-j6UG6UaHlzByU \ )+: A "A "I-VDN[=BB<I;E q \<c"WZ\<cB<G-c<B<fiJcEI-B IJr-X%[1iJD ¡ I-V(m q G-Y-kJI-c<G-I-iJc<ByUG-IZ[XMI-XM\DHVl"Y;UYx<lj-cR[-Bad6[JDM\X%[1IJi-D ¡ IJV]U "j-c\DHiy[-B DHMc q XN\<c<rJW-I-B<c<XN\<f"IXHc<X q TDHXN\<B q DM\_iy[;EcVXnG-I-VzKSTi q Hc<r jJi-I-VDHi=x<IJm q i-Y;UkJI-I Y;UoWJDHXM\B<DHGJG-cjJi-DeEXN\PUBadDMG§G-U i-I-XK Z K QJK 9i-I

µ < µ∗ i-IJXK Z K Q " = UiJI-YG6UopycHEIJ\X%[ B q XN\<c<r-WJI-B<c<VªXN\PUk-I-c<G-Ui-G-c<V XHc<XN\<c[-GJI-IK$9i-I N\<c<V X q DMXM\<B q DM\8D D^cHEGJc"G-D q XM\c<r-W-I-Bc<D XHcXM\<cR[JG-I-D j-c<VDMW-DHG-cBDMNEc<W-Y2c<reK

+-XhN.1d r*/YhY+f i XhY/ (TY/YXhY(uXh8d(/8'*d8f' qxX/8XhN/ '(*+-q 0-.1d8h8d8XhNf/ d"_,d *+-XhYTN+I''Dd (`` #XhNf*d ( 8( i f.1d(( lR*T i .f*dghY'*/8+f*dJTY'*/gTN+XI(*+ f(*X8" "

A¤\<c<W-Y2Dx<I-m q i-Y;Uk-I-I µ = µ∗ q XM\<cr-W-I-B<cDTIG-D q XM\<cr-W-I-B<cD XHc<XN\<cR[-GJIJ[1XedI ByU \<X%[_B cHEGJc i-I-XaK Z K Q " RNK2S UPd6DHD'c<GJII-XHW-DNRU \¢IXHI-XN\<DMV]U¢Blzx<I-i6UDN\¢G-c<B<lzri-D ¡ I-V G6Uj-iJI-VDHi "aY;UY M\c'j-c<Y;UaUG-c9G6U i-I-XK Z K Q " "aY2c\c<i-lzrX q TDHXN\<B<DMG-G-cc\dI-W-UDM\X%[1c\¢jJi-DeEl E q TDHMcÎ_G-D9G6UPp<cHEIJ\<X%[BG-DMj-c<XHiJDeEXN\<B<DMG-G-c<rbxPdIJcXM\<Ic\I-X p<cEG-c<HciJD ¡ I-V]UJK UY2c<r_\<IJjbx<I-m q i-Y;Uk-I-r1G6UalzB<U \ , CIE A I1IJVDHG-G-c¡ DMXM\<YJI-DTx<IJm q i-Y;UkJI-I[JB<IZdIJXHf j-i-DeEVDM\<cV¤UG6UPd6IJRU¢B\<DHci-I-I1Y;Ua\PUaXM\<iJc<m¢K UaXHXHVca\<i-DMG-G-lzrB<l TD§jJi-I-VDHix<I-m q i-Y;Uk-IJI EB q YJi6Ua\<GJc<D=i6UB<G-cB<DHXMI-D B

XHIJXM\<DMVD Z K ] j-iJDeEXN\PUBad6[JDM\ XHc<x<cr \<I-j-IJW-G-lzrj-i-I-VDMi ¡ DMXM\Y;crxI-m q iJYZUak-I-I "Y2c\<ci-lzr1Y;URW-DHXN\<B<DMG-G-c jJi-c<IZd-d6uXM\i-I-i-cByUGbG6Ui-IJXK Z K QJK

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dxi

dt= xi = fi(x1, . . . , xN , µ1, . . . , µk)

K O

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n+1 = fi(x1n, x

2n, . . . , x

Nn , µ1, . . . , µk),

i = 1, 2, . . . , N.

EDMXHfxi(t)

IZdIxi

n

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Vl fi(x, µ)

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0

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IZd6Ixi

0

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D = j +

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|λj+1|,

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D = 1¢E6d6[j-i-DFEDNd6f<G-c<Mc^k-I-Y;dU I

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XHBc<r-XM\ByU "-cx<G6Ui q ¡ DMG-G-lzDnI Ec<Y;UaUG-G-lzDuE6d-[ M\c<rXMI-XM\DHVl "6x q E q \ Ec<XN\<c<BDHi G-c j-i-I-VDMG-I-VlYj-c\c<Y q B R3 K DMc<i-DN\<I-W-DMXHY-I q XM\PUGJc<BadDMG-c" WJ\<csB XHIJXM\<DMVD K Q Bc<xadUaXM\<I G6UaW-DHG-IJr1.3 < a < 1.8

X q TDHXN\<B q DM\ DFEI-G-XN\<B<DMG-G-lzr§pyUc\<IJW-DHXMY-I-r UP\\<i-UYJ\<ci "*Y;ca\<c<i-lzrG-DbXMcPEDMi ¡ IZ\ q XM\c<r-W-IJB<lCpG-DMj-cHEB<I ¡ G-lCp \c<W-DMYK \<ca\ Ua\\i6UYJ\c<iLIZB<DMXM\<DMGB dIJ\<DMi6Ua\ q iJD Y;UY Y-B<UaI-MI-j-DMi-x<cadIJW-DHXMY-I-rUa\a\<i6UYZ\<c<i cHI " E6d6[LY2c\<ci-c<HcG6Ui q UDM\X%[ Z D q Xedc<B<I-DHIJj-DHi-xcadI-W-GJc<XM\IK©nU i-I-XaK K Q j-i-I-B<DFEDHGUa\a\<i6UaYJ\<c<icIIc<xPdUXM\f xyUXMXHDMr-GCDHMc¢j-i-IJ\R[ ¡ D

G-IJ[ K \\i6UYJ\c<i cIG0

1DeEI-GJXM\<BDHG-G-cD j-i-IJ\R[JHI-B<U TDHD VG-c ¡ DMXM\<BcBI-GJ\DHi-B<UPdD

1.3 < a < 1.8j-i-I

b = 0.3X`cHEGJc<i-cHEG-lzV¤xyUaXHXHDMr-G-c<V¤jJi-IJ\R[ ¡ DHG-IJ[

G1

K` xyUa[G6URW6UPd6f<G6Ua[\<c<WJYZU x0, y0

"oj-iJI-G6UoE6d6D ¡ U Ua[`xyUXHXMDHr-G q jJi-IJ\R[ ¡ DHG-IJ[G1

"-XHc B<i-DMVDHG-DMV XM\i-DHVIJ\X%[bYUa\\i6UYJ\c<i q cI K¦9Uai6UYJ\DHi-G-cr§[-BPd6[-DN\<X%[8RUB<IJXHI-Vc<XN\<fXM\PUi TDHMcbj-c<Y;UaUa\<Ded6[§n[Jj q G-c<ByUc\

j6Ui-UVDM\i6UaKniJI mnI-Y2XMI-i-c<ByUaG-G-c<V

b = 0.3Ua\a\<i6UYZ\<c<i `caI¢B<cHG-I-Y;UDM\ ¡ DMXM\

Y2c j-i-Ia = 1.3

Ic<XM\PUDN\<X%[ pyUc\<IJW-DHXMY-I-V B<c B<XMDHr c<xPdUXM\I X q TDHXM\B<c<ByUaG-IJ[1.3 < a < 1.8

K UB<I-XMI-Vc<XM\fλ1(a)

GJD'I-VDHDN\¢j-i-c<B<UPdc<BEcG q d6[ Ij-iJDeEXN\PUBad6[

-2 2-2

2

x

y

G 1

G 0

%hNhNfd8.hN/Yf>/YQI( G0( Yd8XX+' + M/ rf(*h 1ao+I'( G1

R*` a = 1.5 r*f1( b = 0.3 hNfd8+.hN/Yf1((W(*QX+-f/ / 8`*d8Xh8(x(*q + #hBT .1d *+X-hYTN+>dghNhNfd8.hY/Yf*d Y+-X./8'*+'*/8Xh

0 2 4 6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

Ψx

k %TYhN/Y./Yff+-` (*/g''Dd i '*.( AR*` a = 1.75 " i '*.hY(*f1'*/ `('(+ xr/Y.1d8Qdg'DdBdgrrf/8.SX-(q d ( .X-r/8'+-'(Ddg`Y'/ i '*.(*+ λ1 = 0.53 y Xh8d8f ,(x` 1r i '/8TYX.(Wr/Y.1d8Qd8hN+-`0X/Y/8hYTN+-hYSXhYT i u)>( rDd8fd8q +hNf i a

DM\^XHc<xc<r1 dUREY q j-cad6c ¡ IJ\DNdf<GJc^c<j-iJDeEDedDHG-G q ªm q G-Y-kJI-TK \<c\ i-DM q d6fo\PUP\c\i6U ¡ UDN\bmUYJ\c\X q \<XN\<B<IJ[ q XM\c<r-W-IJB<lCp GJDHj-cHEB<I ¡ G-lCp8\c<W-DHY c<Y2c<G q XN\<c<r W-I-Bc<XM\I B¢c<xPdUXM\I1X q TDHXN\<B<c<B<UG-IJ[1Ua\\i6UYJ\c<i6U cHIK j-DMYJ\<i8Vc! GJc<XM\I

S(ω)"i6UXHXMW-IJ\PUGJG-lzr j-cY2c<ci<EI-G6Ua\D

xnBc<xadUXN\<I=X q

DMXM\B<c<ByUGJIJ[Ua\\i6UYJ\c<i6U cHI "J[JBad6[-DN\<X%[ dUoEY2c<rm q G-YJk-I-DHr I_G-DB<YZd6 W6UDN\ [-B<G-lCp B<lzx<i-cXHc<B G6U Y;UYJIZp dI-xc pyUi6UYZ\<DHiJG-lCpW6UXN\<c\PUPpK ACXedDeEXN\<B<IJDM\c<HcUB\c<Y2c<i-i-Ded6[-kJI-c<G-G6UP[=m q G-YJk-IJ[ h'~ uj-i-ck-DHXMXU xn

XHj6UREUDM\_jJc_UY2cG q "6xadIJHY;cV q Y HY2XHjJc<G-DHGJk-I6UPdfG-c<V q i-IJXK K NKhnUY q ¡ D c\VDHW6Uodc<XHf "]E6d6[ Hi q x<lCpMI-j-DMi-x<cadIJW-DHXMY-IZpUa\a\<i6UaYJ\<c<iJc<B Ecad

¡ G-c B<lzj-cad6GJ[J\f<X%[ q XNd6c<B<I-D\<i-UG-XHBDHi-XHUPdf<G-cXM\<Ij-DHi-DMXHDMW-DHG-IZ[ q XM\<cr-W-I-B<lCp IG-D q XM\c<r-W-IJB<lCp¤VG-cHc<c<xi6UaIJrXHDFE6dc<Bc<r(\i6UDHYZ\<c<i-IJIK c<j-c<XN\PUBadDMG-I-Di-I-XK K QI K tj-c<Y;UaHlzByUDM\ "WJ\c q XM\<cr-W-I-B<lzDTVG-c<Hcc<x<i6UaHIJ[XHDeE-dc<B<lCpk-IJYZdc<B c<j-i-DFED d6[J \ Mi6UG-IJk-l xyUXMXHDHrJG6U j-i-IZ\R[ ¡ DMG-IJ[¤UP\\<i-UYJ\<ci6U " U=XUaVsp2Uca\<I-W-DMXHY-IJrUa\\<i-UYJ\<cii6UXHjJcadUeUDN\<X%[BoEcadf`G-D q XM\<cr-W-I-B<lCpXHDHj-Ui6Ua\i-I-X!"Zj-c<B\c<iy[J[ IZp mnc<i V q K '9i-I-XaK K t`B<I;EG-c";WJ\<cj-DMi-DHXMDHW-DMG-I-DVG-c<Mc<c<x<i6UPI-r BXH E q \i6UG-XMB<DHi-XHUPdfG-c¢I_j-cR[-Bad6DHG-I-DnMc<Vc<YZdIJG-I-W-DMXHY-IZp_\i6UDMYJ\<c<iJI-r1G-D9BDeEDN\ Y_i-c ¡ EDHGJI- q XN\<c<r W-I-BlCpj-DHiJI-cHEI-W-DMXHY-IZpci-x<IJ\oK #I-jJDHi-x<cPdI-W-DMXHY2c<Dbp2Uca\<I-W-DMXHY2c<D8VG-c ¡ DHXN\<B<c DeEI-G-XM\B<DHGJG-c<Dnj-i-IJ\R[JHI-B<U TDHDnj-i-DFEDNdfG-c<D'VG-c ¡ DMXM\B<cBTmUaHc<B<c<V j-iJc<XM\i6UG XM\B<DXHI-XN\<DHVl K Q NK I-XN\<DHV]U cHI [-Bad6[JDM\X%[cHEG-c<rIJ'j-i-c<XN\<DHr IZp "ZE6d6[ Y;ca\<c<i-lCp q Xedc<B<I-DuHI

j-DMi-x<cadIJW-G-c<XN\<I \<i-UDHYJ\c<i-I-r G6UpyUc\I-W-DHXMY2c<VUa\\i6UYJ\c<i-D Vc ¡ G-c_j-i-c<B<DMi-IJ\fB_W-I-XedDHG-GJc<V HY2XMj-DHi-IJVDHGJ\DK M\c<r§kJDNdf< x<lCdUi6UPi6Uxc\PUG6U_XHj-DMk-I6UPd6f<G6Ua[j-i-cHi6UVV]U "Cj-cHB<cad6[J Ua[ i6UaXHXHWJIJ\PUa\<f B<DMi-cR[J\<GJc<XM\f

P (φ) q dU φ VD ¡ E qG6UjJi6UBadDMG-IJ[JVI q XN\<c<r-WJI-B<c<Mc I GJD q XN\<c<rJW-I-B<c<Mc VG-c<Mc<c<x<i6UPI-r XHDeE-dc<B<c<r

q

Ws

Wu

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-0.50

-0.25

0.00

0.25

0.50

x

y

/8TY+IRD+I'(*+ i XhY/ (TY/ M/A( '+ i XhN/ (TN/ M/ qs'*/ M/Y/ Yf*dgQ-(X+IR`*/gTN/ / X/8XhN/ '( f*dJTY'*/gSTN+-X-( q /8hN/ 8f*d8a,+-'( >/8Q-(xrf( a = 1.7, b = 0.3

\<i-UDHYJ\c<i-I-I(xn, yn)

j-i-In→ ∞ i-I-XK K w " = KZUXHWJDM\<l q dc<B x<lCd6Ij-i-c

B<DFEDHG-l E6d6[O ]\[\[@[ \c<W-DMY^G6UnUa\a\<i6UYZ\<c<i-DaKe' Hi-UmnI-Y;U`B<I;EG-c"yWJ\c`X q TDHXN\<B q DM\G-DMY;ca\<c<i-cD VI-G-IJV]UPdf<G-cD8HG6URW-DHG-IJD!" j-i-I-G-IJV]UDHVc<D q dc<V φ

Ic<G-c c\dI-W-GJcc\ G q d6[ K I-GJI-V]UPdf<GJc<DbG6URW-DMG-I-D φmin

UB<I-XMIJ\ c\ j6Uai6UVDM\i-c<B c\c<x<i6U ¡ D G-IJ[ i-I-XaK K w " RNKACcbB<XMDHVI-GJ\DHi-B<UPdD^HG6URW-DHG-IJr

a" EDX q TDHXN\<B q DM\=p2Uc\I-W-D

XHYJI-rUa\\i6UYJ\c<i "zVI-G-IJV]UPdf<G-lzr q HcPdVD ¡ E q G-Uj-i6UBPdDHG-IZ[-VI q XN\<c<r-WJI-B<c<McI G-D q XN\<c<r-WJI-B<c<Mc8VGJc<Hc<cx<i6UaHI-r mUacB<c<r \i6UDHYZ\<c<i-IJI xcadf!TD 400 I B1G q dfG-D'cx<i6U UDM\X%[K GJc<Hc<cx<i6UaHIJ[^p2Uac\<I-WJDHXHYJIZp_\<i6UaDHYJ\c<i-I-rB<DeE q \¢XMDHxR[ \PUY ¡ D!"Y;UYIVG-cHc<c<xi6UaIZ[_XHDeE-dc<B<c<Mc kJI-YZdU 26cG-IBXHDH EU \<i-UG-XHBDHi-XHUPdf<G-l9K¢>->-+->J 012,3+ K- UaXHXHVca\<i-I-V j-i-IJVDHiY-ByUPI-MI-j-DHiJx<cadI-WJDHXHY2c<Mc Ua\

\<i-UYJ\<ci6UBEI-mnmnDHi-DMG-k-I6Uodf<G-c<r_XMI-XM\DHVD BXHIJXM\<DMVD'c<i-DHGJk6UJK2Sd6[Ua\\i6UY \<ci6Uc<i-DMG-k6U "Y;UY§I E-d6[ UP\\<i-UYJ\<ci6UcI "G6Ui q UaDM\<X%[ cEG-cIZ \i-DHx<cByU G-I-rMI-j-DMi-x<cadIJW-G-c<XN\<I q XNdc<BI-D Z eK \a\<i6UaYJ\<c<iJlL\<IJj6UTc<i-DHGJk6UTc<x<G-Ui q ¡ DHGJlB iy[;ED1XHIJXM\<DMV I[-Bad-[- \ XHc<xc<r(\I-j-I-W-GJlzrjJi-I-VDHiY-ByUaHI-HI-jJDHi-x<cPdI-W-DMXHY-IZpUa\a\<i6UYZ\<c<i-cB-K S cYZUPRUG-c " WJ\c Ua\a\<i6UaYJ\<c<i c<i-DHGJk6U B<YZd W6UDM\=\cadf<Y2c=XHDFE6dcB<lzD \i6UDHYZ\<c<i-IJI "*j-i-I ByUiJI6Uk-I-I=j6Uai6UVDM\i-c<Bx<IJm q i-Y;UkJI-I§BGJDHVc\<X q \<XM\B q \ " q XM\c<r-W-IJB<lzD\c<W-Y-I IZd6I k-I-YZdl G-D BcG-IJYZUa \oK \\i6UYJ\c<i `ci-DHG-k-Up2U i6UYZ\<DHiJIJ q DM\<X%[§Y;URW-DHXN\<B<DMG-G-c \<DMVI ¡ DXHB<c<rJXM\<B<UVI "WJ\<c I Ua\\i6UYJ\c<ibcI "Ii6UXMXHV]Ua\i-I-ByUaDM\<X%[bYZUaYY;dUXHXMI-W-DMXHY-I-r1j-i-I-VDMi1Y-ByUPI-MI-j-DHiJx<cadI-WJDHXHY2c<Mc p2Uac<XUJK¨'i-UB<G-DMG-IJ[ c<iJDHG-k6Ux<lCdI j-cPd q W-DHG-l IJ q i-UB<G-DMG-I-r ©nUB<f<D \c<Y2XU1B1U

EURW-Dc\<DHj;dc<B<c<rY2c<G-B<DMY-k-I-IIIJVDH \ B<I;E

x = −σ(x− y), y = rx− y − xz, z = xy − bz, K

0 20 40 60 800.00

0.05

0.10

1.50 1.55 1.60 1.65 1.70 1.7538.0

40.0

42.0

44.0

46.0

φ a

φ minP(φ)

= R d8XIr*f+ RD+-`+-'(+TN+f/ 1hY'/YXhN+ i I`*d φ q +aoR i 'DdJr*f*dJTY`+-'( qs( i X-hN/ (TY/ M/A( '+ i XhY/ S(TN/ M/ qs'*/ /Y/ 8f*dgQ-( dg/YhY(*+X./ hNfd8+.hN/Yf1(( '*d d8hYhNf*dg.hN/8f+ >/8Q-( R*` a = 1.7, b = 0.3

6 QMdJTY(*XI(*q /YXh q ('(q dJ`8'*/ / i I`*d φmin/YhAr*d8fd8q +hNfdcX-(XhN+q a r*f( b = 0.3

EDσ"bIr q j-i-UBad6[J IJD9j6Ui6UVDN\<i-l9Kh q i6UBG-DHG-IZ[-V \<I-j6U K XMB<cHE[J\X%[

G-DMY;ca\<c<i-lzD`VcHEDNdI_dUaHDHi-c<B "U\PUaY ¡ DnVcHEDNdf EI-XMY2c<B<c<McEIJG6UVc Q " " K \<VDN\<I-V "6WJ\cBXHI-XN\<DMVD K i-D ¡ IJV Y-B<UaI-MI-j-DMi-x<cadIJW-DHXMY;cHc pyUc<XHU i-D

UPdIZ q DN\<X%[§BY2c<G-DHWJG-c<r=c<xadUaXM\<I HG6URW-DHG-IJr§DMD q jJi6UBad6[J TIZp=j6Ui-UVDM\i-c<B-K©nUi-I-XaK K ^j-i-DFEXM\PUBPdDHG6Ux<I-m q i-Y;Uk-I-cG-G6Ua[1EI-UHi6UaVV]U XHI-XN\<DHVl9K q DMXM\B<c<ByU G-I- UP\\<i-UYJ\<ci6Uc<iJDHG-k6U'ca\<B<DMW6UDM\nU T\i-IZpycByUG-G6UP[¢c<xadUXN\<f9B9j6Uai6UVDM\i-I W-DMXHY2c<Vj-i-c<XN\<i6UaG-XM\B<DK*~UaHc<B<lzr j-ciJ\<i-DN\_Ua\a\<i6UYZ\<c<i6U c<i-DMG-k6U^jJi-DeEXN\PUBadDMG

( i f.1d (*/g''*d R*(Dd Mf*dgq q d X-(XhN+q >/Yf+-'*dc'DdBr`*/YX-.1/8XhY( rDd8fd8q +hNf/8T r ( σ R*` b = 8/3

G6U iJI-XK K ] " =aKCACGJD q Y;UaRUGJG-c<r¤c<xadUXM\I¤XHB<c<rJXM\<B<U8p2Uca\<I-W-DMXHY2c<McUa\a\<i6UaYJ\<c<i-Ux q E q \ I-G-lzVI2Ua\a\<i6UYZ\<c<i`ci-DHG-k-U¢\i6UG-XMmci-VI-i q DM\<X%[bB Y-B<UaI6UP\\<i-UYJ\<ciK©nUacB<DHV"\<IJj-I-W-G-lzDXHBc<r-XM\ByU=Ua\\i6UYJ\c<i6U c<i-DHGJk6U 2 XMj-DHYZ\<i(n¦ G-DÎJ

VDHGZ[-DM\X%[ j-i-I ByUi-I-Uk-I-I G6URW6Uodf<G-lCp q XNd6c<B<I-r "\PUY YZUaY Ua\\i6UYJ\c<i c<iJDHG-k6U[-BPd6[-DN\<X%[ DFEI-G-XN\<B<DMG-G-lzV "xyUaXHXHDMr-G-c<V¤jJi-IJ\R[ ¡ DHG-IJ[1Y2c\<c<iJc<Hc Xed q ¡ IZ\B<XHDmU cB<c<D9j-i-cXM\<i-UG-XM\B<c XHjJDHYJ\in¦ j-i6UYZ\<I-W-DMXHY-IG-DnVDHGJ[JDM\X%[ "DHXedI_ByUi-f<IJi-cByUa\f q j-i6UBad-[- TI-D j6Uai6UVDM\i-lXHI-XN\<DHVlsB^c<xPdUXM\I X q DMXM\B<c<ByUGJIJ[§Ua\\i6UY \<ci6Uuc<i-DHGJk6UJK \I¢XHB<c<rJXM\<B<UnG-U d6[;EG-c9IZd-d XM\i-I-i q \THi q x<cXM\<fUa\a\<i6UaYJ\<c<i-Uc<i-DMG-k6U`X \<cW-Y-I Hi-DHGJIJ[ MY2XHj-DMi-I-VDHGZ\PU 2-XM\i q YJ\ q i6U¢Ua\a\<i6UaYJ\<c<i-UTXHcpZi6UGJ[JDM\X%[j-i-I B<Ui-I6UkJI-Ibj6Ui-UVDM\i-c<BIbG6URW6Uodf<G-lCp q XNd6c<B<I-r "x<I-m q i-Y;Uk-I-I8Ua\a\<i6UaYJ\<c<i-Uc\X q \<XN\<B q \oK dDFEXM\B<I-DHV8p2Uc\I-W-DMXHY2c<r¢EI-G6UVI-YJI [JBad6[-DN\<X%[p2Uai6UYJ\DHi-G-lzr^B<I;EÛB\c<Y2c<i

i-Ded6[-k-IJc<G-G-c<r_m q G-Y-k-IJIK Ba\<c<Y2c<i-iJDNd6[Jk-I-c<G-G-Ua[_m q GJY-k-IJ[Ua\a\<i6UaYJ\<c<i-U¢c<i-DMG k6UMY2XHj-c<GJDHG-k-I-UPdf<G-c9XMj6UoEUaDM\'X q B<DNdIJW-DHG-IJDHV=B<i-DMVDHG-Ij-i-UYJ\<IJW-DHXMY-IVc<G-c\c<G G-c"WJ\<c I;d-duXN\<i-IJi q DN\ I-XK K ] " RoK i6UB<G-DMG-I-DMi6UmnI-Y2c<B=UB\c<Y2c<i-i-Ded6[-kJI-c<G G-lCp=m q G-Y-kJI-r8Ua\\i6UYJ\c<i-c<B_c<i-DHGJk6U I `caI XHBI;EDM\DNdfXM\<B q DM\bc Y;UoWJDHXM\B<DHG G-c<r HY-B<IJByUPdDMGJ\<G-cXM\<IEIJG6UVI-W-DMXHY-I;p¤j-i-ck-DHXMXHc<B B§j-c<WJ\I¤HI-j-DMi-x<cad6I-W-DHXMY-IZpXHIJXM\<DMV]UPpK

−30.0 −20.0 −10.0 0.0 10.0 20.0 30.00.0

10.0

20.0

30.0

40.0

50.0

Z

X

= R dgQ/8T r/YfhNf+h l6 ( dJTNhN/8./Yff+-` (/8''Dd i '*.( dghNhNf*dg.hN/8f*d >/8f+-'DdBrf(σ = 10 0 r = 28 ( b = 8/3

*.

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/ *$%$ * $%- &! (',/- ,$

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8kM\I-VXNdc ¡ G-c<XM\f G-D c<Mi6UG-IJW-I-ByUDN\<X%[ K 9i-I Y2c<G-DHWJG-c<r ByUiJI6Uk-I-I j-Ui6UVDN\<i-c<BXHIJXM\<DMVliJDUPdIZ q \X%[LY;UXMY;UoElgi6UaNdI-W-G-lCpx<I-m q i-Y;Uk-IJr¤YZUaY i-DH q d6[-i-GJlCp "\PUY I p2Uac\<I-WJDHXHYJIZp¤UP\\<i-UYJ\<ci-c<B-K c<c\B<DM\XM\<BDHG-G-c cX q DMXM\<BPd6[-DN\<X%[x<I-m q i Y;Uk-I-cG-G6Ua[(j-DHiJDHXM\i-c<r-Y;U IZp x<UXHXMDHr-G-cB j-i-IZ\R[ ¡ DMG-IJ[K 9i-I-W-IJG-c<rXedc ¡ GJc<XM\IY-ByUPI6Ua\a\<i6UaYJ\<c<iJc<B [JBad6[J \<X%[MmnmDMYJ\<lHc<VcYZdI-G-IJW-DHXMY;cHc YZUaXUG-IZ[ q XN\<c<r W-I-BlCpIG-D q XM\c<r-W-IJB<lCpVG-c<Hcc<x<i6UaHI-rXHDFE6dc<BlCp \<i-UDHYJ\c<i-I-rIZdI_B<caG-I-YJG-cB<DMG-I-D j-DM\dI XHDHj-Ui6Ua\i-I-XHlªXMDeE6d6c mnc<Y q XU "*Y2c\c<i-lzDÎJVDH \VDHXN\<cG6UVGJc ¡ D XM\B<DzG6URW-DMG-I-r j6Ui6UVDN\<i-cBGJDHG q dDMB<c<r VDHi-l9KaSTi q HIJVI Xedc<ByUVI "<G6Ui q UDM\X%[q Xedc<B<IJD\i6UG-XMB<DHi-XHUPdf<GJc<XM\I VG-c<Hcc<x<i6UaHI-rKXNdI^j-i-IN\<c<V q W-DMXM\<f "JWJ\c xyUXMXHDHrJG-lj-i-IJ\R[ ¡ DHG-IZ[ XHcX q DMXM\<B q IZpjJi-D EDedf<G-lCp VG-c ¡ DHXN\<B"Vc< q \I-VDN\<fmni6UYZ\PUPdf<G q XN\<i q YJ\ q i q I XMc<XM\PUBPd6[J\fW-i-DNB<lzW-Ur-G-c q Y-I-D8c<xadUXM\IB mUaHc<B<c<V j-i-cXM\<i-UG-XM\B<D!" \<c XN\PUG-c<B<IZ\<X%[Lj-cGJ[Z\<G-c"JG6UXHY2cad6f<Y2c B->J+->- >-+82,;,,+ K- UaXHXHVca\<i-I-V XM\i q YJ\ q i q I

XHBc<r-XM\ByU YJByUaI-Ua\\i6UYJ\c<i6U IZB<DMXM\<GJc<rbVcHEDNd6I¦uDHG-c<G-U 2

xn+1 = 1 − ax2n + yn,

yn+1 = bxn. tZK O

9i-I0 < b < 1

ca\<c<x<i-U ¡ DHGJI-D tZK O'[JBad6[JDM\<X%[8EI-XHXMI-j6Ua\I-B<G-lzVI§pyUi6UYZ\<DHiJI q DM\X%[ G6UPdIJW-I-DHVLY-ByUaHI6Ua\a\<i6UYZ\<c<i6UZK \<cx<i6U ¡ DMG-I-D t;K O uBaRUI-VGJccHEG-caG6URW G-c"6\oK DK[-Bad-[-DM\X%[_EI-mnmDMc<Vc<i-mnIJHVc<VzK \<c<x<i-U ¡ DHGJI-D`¦uDHG-c<G-U c\dIJW6UDM\X%[ c\c\c<x<i6U ¡ DHG-IJ[ cHI \<DMV "WJ\<c=B Y;URW-DHXN\<B<DG-DNd6I-G-DHrJG-c<XM\IB<Y;duW6UaDM\8 dUoEY q Y-ByUREi6Ua\I-W-G q m q G-Y-kJI-TK UaXHXHVca\<i-I-V1mÙPc<B<lzD]j-ciJ\<i-DN\<l j-iJIJ\R[-MI-ByU IZpTVG-c ¡ DHXN\<B XHI-XN\<DMVl tZK O

B<VDMXM\<D X¢xyUXMXHDHrJG6UVI jJi-IJ\R[ ¡ DHG-IJ[§I8IZp HBcadukJI- j-i-I8ByUi-I6Uak-I-I8j6Ui6UVDN\i6U

aE6d6[

b = 0.3K I-X q G-cY t;K O" =1I;d-duXN\<i-IJi q DN\ i-D ¡ I-VªXMc<X q TDHXM\B<c<ByUaG-IJ[

EB q p j-i-IJ\R[JHI-ByUa I;pj-cHEVG-c ¡ DHXN\<B G6U mUaHc<B<c<r¤jZd6c<XHY2c<XN\<IK ACI2EG6U XNd6c ¡ G6Ua[XN\<i q YJ\ q i6UBadc ¡ DMG-G-lCpxyUaXHXHDMr-G-c<BIZpjJi-IJ\R[ ¡ DHG-IJ[ Km]XedIVDHGJ[Z\<fG6U W6UPd6f<G-lzD q XNdcB<IJ[ "\c1G6UxPd EUDN\<X%[ i-DNY2c<D W-DHi-DFEc<ByUaG-I-D EB q p§i-D ¡ I-Vc<B-K*©nUi-I-XaK-t;K Z " = j-i-DeEXM\PUBad6DHG-li-DM q dfR\PUa\<li6UXMW-DM\PU XN\PUi DMHcj-cYZUPRUa\DNd6[ n[Jj q G-cByU E6d-[

a = 1.078B UB<I-XMI-Vc<XN\<I8c\_IJVDMG-DHGJIJ[8G6URW6UPdf<GJc<HcG-UoWJDHG-IJ[8Y2c<c<i

EI-G-Ua\<lxj-i-I¢mnI-Y2XHIJi-c<ByUGJG-c<V

yK UY2XHIJV]UPdf<G-lzr j-c<Y;UaUa\<Dedf

λ1XNd q W6UrJG-lzV

c<x<i-Uac<V XHY;URW-DM\ VD ¡ E q HG6URW-DHG-IZ[-VI λ1 = 0.126Iλ1 = 0.062

"XMB<I;EDN\<DedfXM\B q [ cjJDHi-DepycHEUPp§XHIJXM\<DMVlªX cHEGJc<Hcp2Uca\<I-W-DMXHY2c<Mc Ua\\i6UYJ\c<i6U1G-U Ei q Mc<rKXNdI1XHi6UBG-IJ\<f M\<I1i-DM q d6fo\PUP\<l XB<I;Ec<V XM\i q YJ\ q i-lsxyUXMXHDHrJG-c<B^jJi-IJ\R[ ¡ DHG-IJ[ iJI-XK tZK O" = "\c1XM\PUaG-c<B<IJ\X%[ j-c<GZ[J\<GJlzV "j-cW-DHV q M\c1j-iJc<I-X pycHEIZ\oK 'VDHGJDHG-I-DG6URW6UPd6f<G-lCp q XNdcB<I-r¢j-i-I-BcEIJ\Yj-DHi-DMXHDMW-DHG-IJMi6UG-IJk XMc<c\B<DM\XM\0sf+MdJ`(Q i u)>(*+X T XI(*XhN+-q3+u+-'/8'*d10 ( X-hNf i .h i fd NdgXX+ '*/8T ( rf(*h 1ao+I'( mT d8Q/gTN/8q rf/YX-hNf*dJ'*XhYTY+ (xn, yn) r*f1( b = 0.3 R` a = 1.078

6 (R*` a = 1.32

UXMXHDMr-G¤Ua\a\<i6UYZ\<c<i6U E6d6[α = 1.32

i-I-XK tZK O" R¢B<lz d-[;EIJ\ cHEG-c<i-cHEGJlzV "WJ\c=Ecad ¡ G-c=XMB<I;EDN\<DNd6f<XM\B<c<ByUa\f c=G6UPdI-WJI-I dI f cHEGJc<Hc§Ua\\i6UYJ\c<i6UJK E G6UY2c8IJBDHXM\G-c"]WJ\c B=XHI-XN\<DMVD1XHcX q DMXM\<B q \ q XN\<c<r-WJI-B<lzD1kJI-YZdl|xcadf!TIZpj-DMi-I-cHEc<B=X c<WJDHG-f q YJI-VI(x<UXHXMDHr-G6UaVI j-i-IJ\R[ ¡ DHG-IZ[ "Y2c\<ci-lzDB W-I-XedDHG-GJc<VXHWJDM\<D`G-Dni-DHMI-XM\i-I-i q \<X%[KXNdI VDHGJ[Z\<f q j-i-UBad6[J IJr|j6Ui6UaVDM\<i a

BscxadUXN\<I YJByUaI-Ua\\i6UYJ\c<i6U

1.1 < a < 1.4"\<c_G6Uaxad EUDN\<X%[§W-DHiJDeE q Ua[-X%[§Y;UiJ\I-G6UXHVDMG-l i-DM q d6[Ji-G-lCp

−0.50 −0.25 0.00 0.250.06

0.08

0.10

0.12

0.14

λ1

x 1.10 1.15 1.20 1.25 1.30 1.35−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

λ1

a= R dgT8(*X-(q3/8Xh Xh8d8f ,+ M/` r i '*/gTNX./ /Br*/8.dgQMd8hY+-` XI(*XhN+-q u+I'*/8'*dA/Yh 'Dd *dg`8'Q-'*d +-'( ./Y/8f:R('*d8h x r*f1( y = 0.5 ( b = 0.3 R*` a = 1.078

6 ( /Yh>r*d8f*dgq +hNf*d a r*f1( b = 0.3

Ibp2Uca\<I-W-DMXHY-I;p8Ua\\i6UYJ\c<i-c<BJK*DM q dfR\PUa\<lªi6UaXHW-DN\<c<B_XM\PUai DMHcd6[-j q G-cB<XHY2c<Mcj-c<Y;UaUa\<Ded6[

λ1

BÙB<I-XMI-Vc<XM\Ic\aj-iJDeEXN\PUBadDMG-lLG-UTi-I-XaK-tZK Z " R 2;UB<I-XMI-Vc<XM\f

λ1(a)pyUi6UYJ\DHi-IZ q DN\<X%[G6UodI-W-I-DMV Y;UY j-cad6c ¡ IJ\DNdf<GJlCp " \PUaY(I c\i-I-k6Ua\DNdf

G-lCp G-UoWJDHG-I-rλ1

"2WJ\cTXHBI;EDM\DNdfXM\<B q DM\ c`G-DHi-DM q d6[Ji-G-c<V W-DMi-DeEc<ByUG-IJI p2Uca\<I W-DMXHY-IZpI_j-DMi-I-cHEI-W-DMXHY-I;p Ua\a\<i6UaYJ\<c<iJc<B^B¢XHIJXM\<DMVDTj-iJIByUai-I6Uk-IJIj6Uai6UVDM\i6UJKA8pyUc\<IJW-DHXMY-IZp Ua\a\<i6UYZ\<c<i6UopVcHEDNd6I¢¦uDHG-cG6Uuc\VDHW6U \<X%[ IEi q MI-DCp2Uai6UY

\<DMi-G-lzDc<XMc<x<DMG-G-c<XN\<I2XHj-DMYJ\<iJlsVc! TG-c<XN\<I B¢RUB<IJXHI-Vc<XN\<I c\ \PUYJ\G-c<XM\IbdDHGJ\p2Uac\<I-WJDHXHYJIZp8Ua\a\<i6UYZ\<c<i-cBI-VDH \

δ B<lzx<i-cXHl"G6U^YJi6Ua\<GJlCp W6UXN\<c\PUop " U hu~

Vc ¡ DM\¢Bc<c<x! TD'G-D9XHj-UoEUa\fÈcG q d6[BXed q W6UD!"6DMXNdI\PUYZ\<G-c<XN\<fdDHGZ\<l i-UB<G6U ZIx<cadf!TDKA Ua\a\<i6UaYJ\<c<iJD'¦'DHGJc<G6UTG6Uai q UDN\<X%[ q Xedc<B<IJDu\<i6UaG-XHB<DMi-XUodf<G-c<XN\<Ij-DHiJDHXHDMW-D

G-IJ[ q XM\c<r-W-IJB<lCp=IbG-D q XN\<c<r-WJI-B<lCp8VG-c<Mc<c<x<i-UaI-r XMDeE6d6c<B<lCp1k-I-YZdcBG6U^XHWJDM\G-c<VsVG-c ¡ DHXN\<B<DHG6URW-DHG-IJrj6Ui-UVDM\i-c<B"j-iJI(Y2c\c<i-lCp I-VDMDM\§VDHXM\c§Y;UXHUG-I-DVG-c<Mc<c<x<i-UaI-r K ©nU_i-I-XK tZK QjJi-DeEXN\PUBadDMG-l i6UXHWJDM\<l q XM\<cr-W-I-B<lCp I G-D q XN\<c<r W-I-BlCp¤XHDMj6Ui6UP\<i-I-X XMDeE6d6c<B<c<r \c<W-Y-IK A G-DHY2c\c<i-lCp¤\c<W-Y;UPp q XM\c<r-W-IJByUa[ W s

I=G-D q XM\c<r-W-I-B<Ua[ W u XHDMj6Ui6Ua\i-I-XMl Y;UXU \<X%[bEi q TEi q eU " q Mcad VD ¡ E q G-I-VIi6UBDHG=G q dTK,XedI§jJi-c<XNd6DeEIJ\fU HB<cad6uk-I-DMr q dU_VD ¡ E q VG-c<Hcc<x<i6UaHIJ[-VIBPEcPdf`p2Uac\<I-WJDHXHY2c<r_\<i-UDHYJ\c<i-I-I "-\<cVc ¡ G-c i6UaXHXHWJIJ\PUa\<f i6UXMj-i-DFEDNdDMG-I-DnB<DMi-c[J\G-c<XN\<DHr q dU φ VD ¡ E q VGJc<Hc<cx<i6UaHIJ[-VI P (φ)

K*DM q dfR\PUa\<l j-i-DFEXM\PUBPdDHG-lG6U i-I-XKtZK " =nIXMB<I;EDN\<DNd6f<XM\B q \c \c<V "WJ\<c B<DHiJco[Z\<G-cXM\<f^c<x<i6U TDHG-IZ[HG6URW-D G-IJ[ q dU φ B G q df EcXM\<cB<DHi-GJc'c\dI-WJG6Uuc\ G q d6[ "R\oK DKoY;UXUaG-I-DCIJVDHDM\uc\dI-WJG q c\ G q d6[B<DMi-cR[J\<GJc<XM\f-K

q

W

Wu

s

-2 -1 1-2

-1

0

1

2

y

x0 2

/8TY+IRD+-'(*+ i XhN/ (TN/ /c( '+ i XhN/ (TN/ M/Bqs'/ M/8/ Yfd8Q-( X+IR`*/8TY/ hY/ *.1( q /YhN/ Yfd8a,+-S'( #+-'*/g'Dd,r*f( a = 1.3, b = 0.3

0 20 40 60 800.00

0.01

0.02

1.10 1.15 1.20 1.25 1.30 1.350.00

0.01

0.02

0.03

0.04

P P

a

(φ)

φ

δφ

= R dgX-r*f+IRD+-`+-'(+ TY+f/ 1hY'*/8XhN+ i I`*d φ q +a,R i 'DdJr*fdgTY`+-'( qs( i XhY/ (TY/ M/ ('+ i XhN/ (TN/ M/ qs'/ M/8/ Yfd8Q-( dg/YhY(*+X./ hNfd8+.hN/Yf1(( d8hYhNf*dg.hN/8f*d u+-'/8'Dd R*` a = 1.179rf( b = 0.3 l6 QMdgT8(*X-(q /YXh TY+f/ 1hY'*/8XhY( P δφ r*/8r*dJRDdg'( i I`*d φ q +a,R i qs'/ M/8/ Yfd8Q-( qs( Tq3dg` i /Y.f+X-hY'*/8Xh B' i ` δφ ≤ 10

/Yh i rf*dgT8` u),+ M/orDdgf*d8q +hYf*dBX-(XhN+q a r*f1( b = 0.3

XNdIIJVDMGJ[J\f j6Ui-UVDM\iaXMI-XM\DHVlIB<lzW-I-Xed6[J\f B<DHiJco[Z\<G-cXM\<f

P δφ j-i-I GJ[Z\<IJ[ q d6c<V VD ¡ E q VG-c<Mc<c<x<i6UPIJ[JVIG-UoWJDHG-IJ[ B<xadIJHI G q d6[ δφ ≤ 10 "\<c j-cad q WJI-V i-DN q dfR\PUa\<l "j-i-DFEXM\PUBPdDHG-GJlzD G6Ui-I-XaKtZK " RoKhnUY B<I;EG-cIJ¢Hi-UmnI-Y;U "X q TDHXM\B q DN\bXMW-DM\G-c<D¢VG-c ¡ DHXN\<B<cHG6URW-DHG-IJr§j6Uai6U

VDM\i6U "zj-iJILY2c\c<i-lCpB<DMi-cR[J\G-c<XM\fP δφ c<x<i6U UDM\X%[B G q d6f-K \<IG-UoWJDHG-IJ[

j6Ui-UVDM\i6UaW-DN\<Y2c XMc<c\<BDM\<XN\<B q \LG-UPdI-W-IJj-DMi-I-cHEI-WJDHXHYJIZpUa\a\<i6UaYJ\<c<iJc<B

c\c<x<i6U ¡ DHG-IJ[ XHiKXÊUGJG-lzVI(iJI-XKtZK Z " RNK x<i6U DHGJI-DP δφ B8G q d6f" B<c<cx! D

HcB<c<iy[ "-Vc ¡ DN\¢XHB<I2EDM\DNdf<XN\<B<cByUa\<f c¢i-DUodIJRUak-I-I_i-D ¡ I-V]U MI-j-DHiJx<cadI-WJDHXHY2c<Mcp2Uac<XU "cHEG-UY2cBjJi-c<B<DFEDHG-GJlCp HY2XHj-DMi-I-VDMGJ\PUPp M\c q XNdc<BI-D¢XHc<c\B<DM\XM\B<c<ByUPd6ci-DHUPdIJUk-I-Ii-D ¡ I-Vc<B¢i-DM q d6[-iJG-lCpk-I-YZdcB c\c<x<i6U ¡ DHG-IZ[¦'DHGJc<G6U K©nUY2c<G-DMk " c<x<X q EI-V XHB<c<rJXM\<Bc(Y-B<UaI6UP\\<i-UYJ\<ci6U " B<DHXMf<V]U ByU ¡ G-c<D1B j;dUG-D

UG6UodIJRUzI9\<i-UYJ\<cB<Y-I HY2XHj-DMi-I-VDMGJ\PUPdf<GJlCp i-DN q dfR\PUa\c<B-K x<i6Ua\I-VX%[YiJI-XKot;K O<K \<i q YZ\ q i6U'xyUXMXHDMr-G-c<B j-i-IJ\R[ ¡ DHGJIJ[TXMB<I;EDN\<Dedf<XM\B q DN\nc B<lzXHc<Y2c<rW q B<XN\<B<IJ\DNdf G-c<XN\<I XMI-XM\DHVlY \c<W-G-c<XN\<I UoEUG-IZ[^G6URW6Uodf<G-lCp q Xedc<B<IJrK 9i-cB<DeEDMV XNd6DeE q IJr HY2XHj-DMi-I-VDMGJ\oK*AClzx<DMi-DHV¤G-UoWJDHG-I-DTj6Ui6UVDN\<i6U

a = 1.078"jJi-IbY2c\<ci-c<VB

t;K O CXMc<X q TDHXM\B q \ Z p2Uac\<I-WJDHXHYJIZp UP\\<i-UYJ\<ci6UJK©nU¢iJI-XKtZK t " =nIRuj-i-DeEXM\PUBdDMG-l jZdca\<G-c<XN\<ILi6UXHjJi-DeEDedDHGJIJ[ BDHi-cR[J\G-c<XN\<DHr

p(xn, yn)E6d-[ UP\\<i-UYJ\<ci-c<B-K

STc<xyUBI-V B q i-UB<G-DMG-IJ[ tZK O UoE-EIJ\I-B<G-lzr "JxadIZY-I-r^Y^x<Dedc<V q " q V I-GZ\<DHGJXHI-BG-c<XN\<I

D = 5 · 10−6 KA cNEDMr-XM\B<I-DXNdUax<c<Hc q V]U=I-G;E q kJI-i q DN\ c<x*DeEI-GJDHG-I-DUa\a\<i6UYZ\<c<i-cB(B cHEI-G i-I-XKztZK t" NK DM q d6fo\I-i q TI-ri-D ¡ IJV G-DRUB<IJXHIJ\ c\2 !LX-(` i X-r+ (#(*.1(Wdg`/Yf(hNq3duT (*X-`+-'( TN+f/ 1hY'/YXh >rf(f*dgT8'(T:dg`*d8X ,' i ` c08+X-`( ud8Q/gT:d hNfd8+.hN/Yf1( '*+u(q +-`Ddc'+ i XhY/ (TY/ M/o'Ddgrf*dJTY`*+I'( 5"

-0.50

0.51 -0.2

-0.10

0.10.2

0.30.4

00.5

11.5

22.5

33.5

44.5

x

y

p

-0.50

0.51 -0.2

-0.10

0.10.2

0.30.4

0

0.5

1

1.5

2

2.5

x

y

p

= R

-0.50

0.51 -0.2

-0.10

0.10.2

0.30.4

00.10.20.30.40.50.60.70.80.9

1

x

y

p

`*/8hY'*/8XhY(>fd8X-rf+IR*+-`*+-'( TN+f/ 1hY'/YXhN+ p(x, y) '*d d8/8hY(+X.( #d8hNhNfd8.hN/Yfd 50X/8X i S),+Xh8T i u)>( BT,X-(*XhY+q + #+-'*/g'Ddr*f1( a = 1.08 0:To/YhNX i hNX-hYTY(+ i q3d 680 #+.h,TN/8QIRD+ XhYT8( i q3dB('*hY+-'*X-(TY'/YXhY( D = 5 · 10−6 'Ddcrf(TN+IR*+-'' +f*dgX-r*f+IRD+-`+-'( ATN+f/ hY'/YXhY+ @

\<cHc"6Y;UY2c<r_IJUa\a\<i6UaYJ\<c<iJc<B^x<lCdB<lzxi6UGj-DMi-B<c<G6URW6Uodf<G-c¢j q \<DMVRUREUG-IJ[_XMcc\B<DM\XM\<B q IZpbG6URW6UPdfG-lCp EUG-GJlCp K \<VDN\<I-V "WJ\c§jJi-I

b → 0VcHEDedf ¦'DMG-c<G6U t;K O j-DHi-DepycHEIJ\ B IJBDHXM\G q

cHEG-c<VDMi-G q VcEDNdfdc<MI-XM\I-W-DHXMY2c<Hcc\c<x<i6U ¡ DHG-IZ[2

xn+1 = 1 − ax2n.

tZK Z \<c<xi6U ¡ DMG-I-D tZK Z G-DMc<x<i6UP\<I-VcIEI-mnmDMc<Vc<i-mnIJHVc<VG-D`[JBad6[JDM\<X%[ KuDHV¤G-DVDHGJDHDuXMB<c<r-XN\<ByUTUa\\i6UYJ\c<i-c<BVcHEDedI tZK Z ]Y;URW-DHXN\<B<DMG-G-cx q E q \Tj-c<B\c<iy[J\<fTB<XHDB<l TDHj-DMi-DHW-IJXNdDMG-G-lzDXHBc<r-XM\ByU ca\<c<x<i-U ¡ DHGJIJ[¦'DMG-c<G6UJK+J)2+->->J+->- $92 3+-,,$ a2,2+->- ,;3*#

,,(,y ,;,<>J.15 UXHXMVc\<i-IJV EIJmmnDMi-DHG-kJI6UPdf<G q EI-G6UVIJW-DHXMY q XHIJXM\<DMV q HDMG-DHi6UP\<c<i G-I TDHG-Y2c XN\PUPpyc<B<U Q "2IZdJduXN\<i-I-i q q "VDNpyUG-IJHVi-c ¡ EDHG-IZ[§I=\<IJj-I-W-G-lzD XHBc<r-XM\ByU_Y-ByUaHI6Ua\\i6UYJ\c<i6U1B_XMI-XM\DHV]UPp§X¢j-DM\d6DHr XMD j6Ui-Ua\<i-IJXHl XHDFE6dc mnc<Y q XHU§XMc<XM\co[JG-IJ[(i6UB<GJc<B<DHXMIJ[K \PU8VcHEDNdf8j-i-DFEXM\PUaBad6[ DM\XHc<xc<rL\i-DNpZVDHiJG q EB q j6Ui-UVDM\i-I-W-DMXHY q EI-mnmnDHiJDHG-k-I-UPdf<G q XMI-XM\DHV qq i6UB<GJDHG-I-r/2

x = mx + y − xz,

y = −x, t;K Q

z = −gz + gI(x)x2, I(x) =

1, x > 0,0, x ≤ 0.

UmnI-Y2XHIJi q DMV G6URW-DMG-IJ[=j6Uai6UVDM\i-c<B m = 2.412, g = 0.097I=B<lzW-I-XedI-VRU

B<I-XMI-Vc<XN\<f_XM\PUi DHMc_j-c<Y;UaRUP\<DNd-[=XHj-DMYJ\<i6U n¦ ca\G6URW6UPdfG-lCp q XNdc<BI-rKD q dfR\PUa\l(jJi-DeEXN\PUBadDMG-l G6U i-IJXKtZK w " =ITXMB<I;EDN\<DNd6f<XM\B q \cuXMc<X q DMXM\B<c<ByUGJI-IEB q ppyUc\<IJW-DHXMY-IZpIjJDHi-I-cHEIJW-DHXMY;cHc i-D ¡ IJVc<B¢Y;cPdDHxyUaG-I-rK9i-I1IZVDHGJDHG-I-I=j6Uai6UVDM\i-c<B XMI-XM\DHVlXHc<Bc<Y q j-G-c<XN\<f DMDTj-iJDeEDedf<G-lCpbVG-c

¡ DMXM\<Bj-i-DN\<DMi-j-DHB<UDM\1x<I-m q i-Y;Uk-I-IK,ud-d6uXM\i6Uk-IJDHr N\<c<Mc_mÙaYJ\PUVc ¡ DN\_XNd q ¡ IJ\<f`Hi6UmnI-Y¢RUaB<I-XHIJVc<XM\I XM\PUi DHMcj-c<Y;UaUa\<Ded6[^XMj-DHYZ\<i6U`n¦ c\`j6Ui6UVDN\i6U

m"Zj-i-DeEXM\PUBad6DHG-G-lzrG6UTi-IJXKtZK w " RoK x<i6U DHGJI-D9B G-cad6fj-c<Y;UaRUP\<DNd-[

λ1

XMB<I EDN\<DNd6f<XM\B q DN\¢cì-c ¡ EDHGJI-I cHEGJc<HcIJ VG-c ¡ DHXN\<B`j-i-DeEDNdf<GJlCp^kJI-YZdc<B "yj-i-DM\DHi j-DMByU TIZp Y;UXHY;UoEl x<I-m q i-Y;Uk-IJr q EB<cDHG-IJ[ jJDHi-I-cHEUZK]CI-m q i-Y;Uk-I-IUa\\i6UY \<ci-c<B XMc<j-i-c<Bc ¡ EU \<X%[1IJVDMG-DHG-IJDHVXM\<i q YZ\ q i-l x<UXHXMDHr-G-cB IZpj-i-IJ\R[ ¡ DHGJIJ[ "Y2c\<ci-lzDTj-iJI-c<x<i-DN\PU \ mni6UYJ\PUodf<G-lzD`XHB<c<rJXM\<B<UJK9i-I-X q \XM\B<I-D9BY-ByUaHI6Ua\\i6UYJ\c<i-D q XM\<cr-W-I-B<lCpÎ^XMDeE6d6c<B<lCp^k-I-YZdcBTG6Uiy[2E q

X pyUc\<IJW-DHXMY-I-VI^jJi-DeEDedf<G-lzVI¢VG-c ¡ DMXM\ByUVI¢j-i-cR[-BPd6[-DN\<X%[¢B9XM\i q YJ\ q i-D UB\cY2c<i-i-Ded6[-kJI-c<G-G-cr m q GJY-k-I-I ";Y2c\<ci6Ua[ BTXHiJDeEG-DMV HY2XHjJc<G-DHGJk-I6UPdfG-c XHj-UoEUDN\ B<cB<i-DMVDHG-IÎ BXM\i q YJ\ q i-D9XMj-DHYJ\i6UTVc! TG-c<XN\<I "2[-Bad-[- TDHMc<X%[^XMjZdc!TG-lzVzK EG6U Y2cB_XN\<i q YJ\ q iJD h'~ j-i-I-X q \XM\B q \=j-DMi-I-cHEI-W-DMXHY-IJD^Y2c<Vj-c<GJDHGJ\lªI§B_XHjJDHY \<iJD=UVDM\G-l i-DMHY-I-D=B<lzx<i-cXHl G6U G-DHY2c\c<i-lCp pyUi6UYZ\<DHiJG-lCpW6UXN\<c\PUPpK \Ic<XMc<x<DHGJG-c<XM\I h'~ I XMj-DHYZ\<i6U1p2Uac\<I-WJDHXHY2c<Mc=i-D ¡ I-V]U[JBad6[- \<X%[ \I-j-I-WJG-lzVIIX q DMXM\<BDHG-G-c^c\dI-W6Ua \ Y-ByUaHI6Ua\\i6UYJ\c<i1ca\ÛP\\<i-UYJ\<ci6U c<i-DMG-k6U "Z\PUY ¡ D!"Y;UYY-ByUaHI6Ua\\i6UYJ\c<i1¦uDHG-c<G-U c\ Y-B<UaI-MI-j-DMi-x<cadIJW-DHXMY;cHcUa\a\<i6UaYJ\<c<i-U cIK

−4.0 −3.5 −3.0 −2.50.00

0.01

0.02

0.03

x

λ 1

1.08 1.10 1.12 1.14 1.16 1.180.00

0.01

0.02

0.03

0.04

0.05

m

λ1

= R dJTY(*XI(*q /YXh8(AXh8d8f o+M/ur*/8.1d8QMdghN+-` r i '*/gT:d /Yhu'*d *dg`Y',QI'Dd *+-'(c./Y/YfNR('Ddghx rf( m = 2.412, g = 0.097

6 ( /YhAr*d8fd8q +hNfd m rf( g = 0.3

~ i6UYJ\PUodf<G-c<XN\<f Mi6UG-IJk xyUXMXHDHrJG-c<BTjJi-IJ\R[ ¡ DHG-IJ[Î^XMB<c<r-XN\<B<c`XHI-XN\<DHVl "Zc<x q XNd6c<BadDMG-G-c<D VGJc ¡ DHXM\B<c<V"x<IJm q i-Y;UkJI-r i-D ¡ I-Vc<Bj-i-I V]UPdc<V TDHB<DedDHG-IJI j6U i6UVDN\<i-cB"6j-i-I-BcE6[J\ YBlzXHc<Y2c<rW q B<XM\B<IJ\DNdf<GJc<XM\I XHI-XN\<DHVl"Y_B<GJD GJI-V q Vc<B<lzVB<cHV q TDHG-IZ[-V j-cHEcx<G-csUa\\i6UYJ\c<i q ¦'DMG-c<G6UJK¢ DM q XNdcB<G-c"¢m q G;EU VDHGZ\PUPdf<G-crªj-i-I-W-IJG-c<r XMc<B<c<Y q j-GJc<XM\I q Y;UaUG-G-lCp XHBc<r-XM\BY-ByUPI6Ua\a\<i6UaYJ\<ci6U1[JBad6[-DN\<X%[ XHWJDM\<GJc<D_VG-c ¡ DHXN\<B<c=HcVc<YZdI-GJI-W-DHXMY-IZp Y;UXHUG-I-r q XM\<cr-W-I-B<lCp IG-D q XM\c<r-W-IJB<lCp"VG-cHc<c<xi6UaIJr " B(XMIZd q Y2c\<ci-c<HcXHI-XN\<DHVl \I-j6U tZK Q [-Bad-[ \X%[ G-DMHi q x<lzVIKAsXMI-XM\DHVD^¦uDHG-cG6UHmnmnDHYZ\<lªHcVc<YZdI-GJI-W-DHXMY2c<Hc Y;UXHUG-IJ[Vc ¡ GJc G6Uaxad EUa\f^B W-I-XedDHG-GJc<V HY2XMj-DHi-IJVDHGJ\D!"B¢\<i-DepZVDHi-G-lCp1XHIJXM\<DMV]UPpM\<cj-c<Y;UnG-DHi-Uai-D I-V]UP[ UoEURW6UJK EG6UaY;cuEUaG-G-lzD i-I-XK;t;K w'[-BPd6[- \<X%[ "yxDM q XNdcB<G-c"XNd6DeEXN\<B<I-DMVHmnmnDHYJ\PU Hc<Vc<Y;dI-G-I-WJDHXHY2c<Mc Y;UXUaG-IJ[IVc q \î-UXHXMV]Ua\<i-IJByUa\<fX%[B YZURW-DMXM\B<D\I-j-I-WJG-lCp_p2Ui6UaYJ\<DMi-I-XM\I-Y=Y-ByUaHI6Ua\a\<i6UYZ\<c<i6UZK

$ * ! ' -*$% ' -*$% $ * &! $+$ * $%- &!

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Z `jJi-I XNdc ¡ G-c<r MDHc<VDN\<i-IJW-DHXMY;cr(XN\EKA C1 A ?DC EGFL=@?B? =@EKEGFL=& E SYFDS

"6B\c<i-c<r IEGFL=W?B? ?DC*D=@STEKA C A =@EKEGFL=& E SYFDS : PK+->-:;<1,;<>-+-,,?¢b+->J>-+J>-?

¦9Uac\<I-WJDHXHYJI-DUa\\i6UYJ\c<i-l "G-Du[JBad6[- I-DMX%[Xu\<c<WJY-Ii-DMG-IJ[IZp^HDHcVDM\<iJI-IXM\i6UG-GJlzVI "IJHB<DHXN\<G-lc\<GJc<XHIJ\DNdfG-cÊUB<G-c "cHEG-UY2c IJ q W-DMG-lG-DeEcXM\PUa\c<W-G-c-KA YZURW-DMXM\B<D1j-iJI-VDHi6U1EI-G-UVI-W-DMXHY2c<rXMI-XM\DHVl X¦9© Vc ¡ GJc8jJi-I-B<DMXM\<I(VcEI-mnI-kJI-i-c<ByUaG-G-c<D c\<c<xi6U ¡ DMG-I-D i-G-cPdfPEUJK \c1c\c<x<i6U ¡ DHG-I-D j-i-DFEXM\PUBPd6[-DN\XHcx<c<r IJB<DMXM\G-c<D X Ec<xyUBPdDHG-IJDHV GJDNdI-GJDHr-G-cHcTjJDHi-I-cHEIJW-DHXMY;cHc XedU eUDMVc<Hc 2

xn+1 = xn + yn + δ cos 2πyn, mod 1,

yn+1 = xn + 2yn, mod 1. t;K

9i-I q Xedc<B<IJI "yWJ\<c δ < 1/2π"aca\<c<x<i-U ¡ DHGJI-D t;K DHXM\fuEI-mnmnDHc<Vc<iJmIZV8G6U \<c

i-DaKPSTi q HI-VI¢XNd6c<ByUVI "ac\<cx<i6U ¡ DMG-I-D tZK BUI-VG-c9cHEG-caG6URW-G-c c<x<i6Ua\I-Vc Ij-DMi-DHB<cHEIZ\^DFEI-G-I-WJG-lzrYJByUoEi6UP\¢G6UjZdc<XMY;cXM\<I

(xn, yn)B¢XHDMxR[K \<cx<i6U ¡ DMG-I-D

t;K [-BPd6[-DN\<X%[ÈIJXHXHIJj6Ua\<IJB<G-lzV "\oK DKaj-i-I Y;U ¡ Ec<rÌJ\<DMi6Uk-I-I edDHVDMGJ\njZdc UoEIX ¡ I-V]UDM\X%[K \c XHBc<r-XM\B<c^dDMHY2c Ec<Y;UaUa\<f"DHXedIbB<lzW-IJXNdIJ\f[-Y2c<x<I6UaGbj-i-DMc<xi6UaHc<ByUGJIJ[ tZK 2

J =

∣∣∣∣

1 1 − 2πδ sin 2πyn

1 2

∣∣∣∣6= 0, δ <

1

2π.

tZK t

i-DFEG-DHD1j-c B<i-DMVDHG-I(G6URW-DMG-I-D |J | < 1K niJI M\c<VªXHIJHG6Ua\ q i6U XHj-DMYJ\<i-U§d6[

j q G-c<BXHY-IZpHY2XMj-c<G-DMGJ\j-i-DeEXM\PUBad-[-DM\_XMc<x<c<r +, − "\oK DKmnI-Y2XHI-i q DM\X%[§G-UPdI-W-IJDj-DMi-DHVD I-ByUaG-IJ[KhnUaRUodc<XHfux<l "aVl IJVDHDHV1EDedc'X cx<lzW-G-lzV1pyUc\<IJW-DHXMY-I-V XN\<i6UaG-G-lzV§Ua\\i6UY

\<ci-c<VzK©9c M\c G-D9\PUY K#2dUBG-c<rc\dIJW-IJ\<Dedf<G-crbc<XHcx<DHG-GJc<XM\f<i6UXHXMV]Ua\<iJI-ByU DHVcHc8XNd q W6Ua[ [JBad6[-DN\<X%[ \<c "WJ\<c"GJDHXHVca\<iy[G-U=X ¡ Ua\<I-DjZdc UoEI "EB<I ¡ DMG-I-DIJHc<x<i6U ¡ U TDHr \c<W-Y-Ica\<c<x<i-U ¡ DHGJIJ[ tZK [JBad6[JDM\<X%[ Hi-McHEI-W-DMXHY-I-VzK c<W-Y;Uj-i-I

n → ∞ j-c<XHD UDN\8d6ux<c<r edDHVDMGJ\8DeEIJG-I-W-G-cHc8Y-ByUoEi-Ua\PU " j-iJDeEXN\PUBad6[ TDHMcbj-cadG q i6UaHB<DHiJ\Y q EB q VDHiJG-c<Hc\<c<i6U BI;EDM\DNdfXM\<Bc<V M\<cHcbmUYJ\PU[-BPd6[-DN\<X%[\<c "WJ\cVDN\<i-IJW-DHXMYZUP[ i-UaVDMi-G-c<XN\<fUa\\i6UYJ\c<i6U DMVY;cXM\<fnj-c'hncPdVcHci-c<B q i-UB<G6U Z Kudc\<GJc<XM\f \<c<W-DMYUa\a\<i6UYZ\<c<i6U ";pyca\R[ I G-DMi6UB<GJc<VDHi-G-U`BTDFEI G-I-WJG-c<V YJByUoEi6UP\<D!"GJI- ED G-D¢c<x<i6U UDM\X%[ B_G q dfJK UYJI-VLcx<i6UaHc<V "G-DMXHVc\i2[G6U^X ¡ Ua\<I-D "Ua\a\<i6UaYJ\<c<iJc<VXHI-XN\<DHVl t;K z[-Bad-[-DM\X%[bB<DMXHf DFEI-G-IJW-G-lzr=Y-ByUoEi6Ua\oK

A N\<c<VXHVlzXedD_Ua\a\<i6UYZ\<c<i iJG-cadfPEU1G-D^[JBad6[-DN\<X%[ XN\<i6UaG-G-lzV " \PUY Y;UY DMV qG-DedfH[j-i-IJj-I-XUP\<f^mni6UaYJ\PUPdf<G q HDMc<VDM\i-I-`K©9DHXMVc\<iy[ G6U9\<c"<WZ\<cn\<c<W-YJI^j-i6UaYJ\<I-WJDHXHYJI^j-cad6G-c<XM\f<j-c<Y-iJlzByU \`Y-ByUREi6Ua\

Y;UY B<I;EG-c=IJmÙPc<B<cHc j-c<iZ\<i-DN\PU Ua\\i6UYJ\c<i6U8G6U i-I-XKt;K " jZdca\<G-c<XN\<f§IZpi6UXMj-i-DFEDNdDMG-IJ[[-B<GJcG-DHcHEG-ci-cHEG6U h9cadIJW-DHXN\<B<DMG-G-c<rVDHi-cr N\<c<rG-DMcEG-c<i-cHE G-c<XN\<I [-BPd6[-DN\<X%[ B<DedI-W-I-G-U IJG-mnc<i-V]Uk-IJc<G-G-c<r i-UaVDMi-G-c<XN\<I

1 < DI < 2K

©nUj-i-IJVDHi " E6d6[ G-UoWJDHG-I-rδ = 0.05 DI ' 1.96

"E6d6[δ = 0.10 DI ' 1.84

K9i-I M\c<V "Y;UY q ¡ DHc<Bc<i-IZdcXHf"DHVY2c<XM\f DC = 2.0

N\<cXM\<iJc<HI-r8i-DN q dfR\PUa\ I-G6UP[NKZA(iJDM q dfR\PUa\DnG-DHcHEG-ci-cHEG-c<XN\<IjZdca\<G-c<XN\<I_i6UXMj-i-DFEDNdDMG-IJ[BDHi-cR[J\G-cXM\DHr \c<W-DHY(G6U UP\\<i-UYJ\<ci-D1G-UoWJDHG-IJ[B<XHDep(B<DHi-ca[J\<GJc<XM\G-c VDN\<i-IJW-DHXMY-IZpi6UP VDHiJG-c<XM\DHr¤UP\\<i-UYJ\<ci6U iJG-cadfPEU8x q E q \=dD ¡ Ua\<f§B8IJGJ\<DMi-ByUPdD 1 < D < 2

K \<I(i6UaHVDHi-G-cXM\<I q WJIJ\<lzByUa \ GJD_\cadf<Y2c8HDMc<VDM\i-I-W-DMXHY-I-D " G-c=I EI-G6UVIJW-D XHYJI-D`XMB<c<r-XN\<ByU Ua\a\<i6UYZ\<c<i6UZK¦9© cx<G6Ui q ¡ DMG-lBi2[2ED Ei q MIZp_c\c<x<i6U ¡ DHG-I-rG-UT\<c<i-DaK c ¡ G-cj-i-DFEj-c

dc ¡ IJ\<f" WJ\<c HiJHcHEI-W-DMXHY-IJD_pyUc\<IJW-DHXMY-I-D EB<I ¡ DMG-IJ[ \I-j-I-W-GJl E6d6[ EI-mnmDMcVc<i-mnIJHVc<BnG6U'\<c<iJDK;~UYJ\X q TDHXM\B<c<ByUaG-IJ[^¦'© B'\PUY-IZp c\c<x<i6U ¡ DHG-IZ[Zp¢j-ca B<cad-[-DM\ j-i-DeEj-cadUeUP\<f"-WZ\<c DMXM\<fjJc\<c<Y2c<BlzD EIJmmnDMi-DHG-kJI6UPdf<GJlzDCXHIJXM\<DMVl BRN

N ≥ 4 "RI-VDM IJD i-D ¡ IJVl§¦9© K EG6UY2c Ec G6UXM\cR[ DMHc'Bi-DHVDMG-I¦'©

B EI-mnmnDHi-DMG-k-I6UPd6f<G-lCp=EI-G6UVIJW-DHXMY-IZp§XHI-XN\<DMV]UPp GJD cx<G6Ui q ¡ DMG-l9KAXMB[JHI XM\I-V "BW6UaXM\<GJc<XM\I "Ec XMIZp j-ci1[-Bad-[-DM\X%[=c\<YJi-l \<lzVLB<c<j-i-cX c B<cHVc ¡ G-cXM\<IX q TDHXM\B<c<ByUaG-IJ[=p2Uc\I-W-DMXHY2c<Hc1Ua\\i6UYJ\c<i6UG6U j-c<B<DMi<pJGJc<XM\I§\<iJDNpZVDHi-GJc<Hc\<ci6U "6BPdc ¡ DHG-GJc<Hc B¢mÙPc<B<cD9j-i-c<XN\<i6UGJXM\<Bcî6UaHVDHi-GJc<XM\I

N ≥ 4K

ud8/Yh8(*+-X.( '+XhNfdg'' dghNhNfd8.hY/YfT /YhN/ Yfd8a,+-'(( %#f1'*/g`8RDdorf( δ = 0.15

`>-+-,,?¢b,2£+->-:Z<b+->->-+->-?

hnUY1Vl q ¡ D`Mc<B<c<i-I;dI "6p2Uc\I-W-DMXHY-I-D¢UP\\<i-UYJ\<ci-lc<xadUoEU \ HDMc<VDM\i-I-W-D XHY2c<r XM\i6UG-G-cXM\<f IbjJDHi-DMVD IJByUG-I-DMVzKSTi q HIJVIbXNdcByUVI "6Xedc ¡ G6UP[_EIJG6U VI-Y;U j-DMi-DHVD I-ByUa DMr XMI-XM\DHVl jJc<i-c ¡ EUDM\I MDHc<VDN\<i-IJW-DHXMY q Xedc ¡ G-cXM\<fXHcc\<B<DN\<XN\<B q TDHMc Ua\a\<i6UYZ\<c<i6UZKt DMVbG-D VDHG-DMDCBuXed q W6UDC¦'© Vl B<lzG q ¡ EDMG-li6UaeEDNd6IJ\<f N\<I XMB<c<r-XN\<ByU 2PjJDHi-DMVD IJByUG-I-DCVc ¡ DM\ G-D]j-i-I-B<cHEIZ\<f'YHDMc<VDM\i-I-W-D XHY2c<r XM\<i-UG-G-c<XN\<I Ua\\i6UYJ\c<i6UJK EDMXHf Vl"i-UXHXMVc\<i-IJVLB<caVc ¡ GJc<XM\fî-DHUPdI RUak-I-I j-i-c\I-B<c<j-cPdc ¡ G-cr XMIJ\ q Uk-I-I ";Y2c< EUnXHI-XN\<DMV]U9EDMVc<G-XM\i-I-i q DM\¢Xedc ¡ G-lzrG-DMj-DHi-IJcEI-W-DHXMY-I-r iJD ¡ I-VLY2cad6DHxyUGJI-r "UXMI-VjJ\c\<I-WJDHXHYJI q XN\<c<r-WJI-B<lzr x<DM j-D i-DMVD IJByUG-IJ[ "aU UP\\<i-UYJ\<cij-i-I N\<c<Vb[JB<G-c G-D][-Bad6[JDM\X%[Ti-DM q d6[Ji-G-lzV=X\<c<W-YJIiJDHG-IJ[DHMc MDHc<VDN\<i-IJW-DHXMY;cr XM\<i q YZ\ q i-l9K9i-I-VDMi-lªG-DHMi q x<lCp XM\<i-UG-G-lCp G-DNpyUc\I-W-DHXMY-IZp Ua\\i6UYJ\c<i-c<B © 9j-i-I

B<DMXM\<I GJDM\<i q EG-c-K 9cX q \<I§EDNdUd6ux<c<r§XM\i6UG-G-lzr pyUc\I-W-DHXMY-I-r Ua\a\<i6UaYJ\<c<i BY-i-IZ\<I-W-DMXHY2c<rb\c<W-Y2D`jJDHi-DepycHEU^Yp2UcX q [-BPd6[-DN\^XMc<x<c<r_j-i-IJVDHi © KA¤Y-iJIJ\<I W-DMXHY2c<r \<cW-Y2D d6[-j q GJc<B<XHYJI-r j-c<Y;UaUa\<Dedf i6UB<DMG G q d p2Uac<XUG-DM\ eK 9c1c<jJi-D EDedDHG-IJg\PUaY;crUa\a\<i6UaYJ\<c<i [-BPd6[-DN\<X%[ © K EG6UaY;c cG G-DHMi q x<lzrK \<c<W-YJIiJDHG-IJ[mnIJIJY-IIJGJ\<DMi-DHX`j-i-DeEXM\PUBad-[- \^Hi q x<lzDÙa\a\<i6UYZ\<c<i-l "Y2c\<ci-lzD`X q D XM\B q \G6UVG-c ¡ DHXN\<B<DHG6URW-DHG-IJr§j6Uai6UVDM\i-c<BG-DHG q d6DHB<c<r=VDMi-l I=XHcpZi6UGZ[- \XHBc<"XM\i q YJ\ q i q j-i-I_B<caV q DMG-IJ[ZpKhnUYc<Y;UaUPdc<XMf"2EIJG6UVI-W-DMXHY-IJD`XMI-XM\DHVlXHi q xlzVI © X q DMXM\B q \ Y;UY=B EI-mnmnDHiJDHG-k-I-UPdf<G-lCp "\PUY8I=B EI-XHY-iJDM\G-lCp EIJG6UVI-W-DMXHY-I;p1XMI-XM\DHV]UPp " Oot K © \I-j-I-W-GJlE-d6[EIJG6UVI-W-DMXHY-I;p XHIJXM\<DMVLXY-ByUPI-j-DMi-I-cHEI-WJDHXHYJI-V B<cax q

¡ EDHG-IJDHVzK¨'VDMXM\<GJc q \<c<WJG-IJ\<f "WJ\c1Vlsj-c<G-I-V]UaDHV jJcE Ua\\i6UYJ\c<i-c<V G-DUBa\<cG-c<VGJc<r1XMI-XM\DHVl9K 9i-DFEj-cadc ¡ I-V "WJ\<c UB\<cG-c<VG6Ua[EI-G-UVI-W-DMXHY;Ua[bXMI-XM\DHV]U BRN G6UPpycHEIZ\<X%[j-cHEEDHrJXM\<BI-DHV(j-DHi-IJcEI-W-DHXMY2c<rXMIZdlX'j-DMi-I-cHEc<V T0 = 2π/ω0

K q EDMV|UG6UodIJIJi-c<ByUa\f XHDHWJDHG-I-D q UaG-Y;Ui-D=W-DHiJDM8j-DHiJI-cHE"B<GJD GJDHrXHIZdl9K AXHDMY q DMr j-c<B<DMi<pJGJc<XM\I t = nT0

Vl(YZU ¡ Elzr¢i6UP j-i-Iduxc<Vn x q EDHV§G6Uxad6

EUa\f§GJDHY2c\<ci-c<D_VG-c ¡ DHXN\<B<c=\<cW-DHYK \a\<i6UaYJ\<c<iJc<VªB M\c<V Xed q W6UD_G6UaHlzByU \j-i-cDHY-k-IJ M\c<HcbVGJc ¡ DHXM\ByU\c<W-DHY BXHDMY q IZp " j-cPd q W-DHG-GJc<D E6d6[ j-c<XedDeEcByU \<Dedf<G-cXM\<I

n→ ∞ "6G6UI-X p<cHEG q XMDHY q q sj-c<B<DMiypZG-cXM\<f^j-i-I n = 1K

ACj-DMi-B<lzD © xlCd1cx<G6Ui q ¡ DMGII-XMXNdDFEc<ByUGB c\<c<xi6U ¡ DMG-I-I

xn+1 = λ tan(xn) cos 2πφn,

φn+1 = ω + φn, mod 1. t;K w

9i-i6UkJI-c<G6UPd6f<G-c<DuHG6URW-DHGJI-D9j6Ui6UaVDM\<i-UωW6U DuB<XHDMHcB<lzx<IJi6UDM\X%[_i6UB<GJlzV

\PUYG6UPlzByUDMVc<V q :WSYX STE S C CI?BA 2 ω = 0.5(√

5− 1)KS`d-[^G-UoWJDHG-I-r

λ > 1B8c\<c<xi6U ¡ DMG-I-I tZK w TXM\i-c<Mc=Ec<Y;UaUG-c8X q DMXM\B<c<ByUGJI-D © i-IJXK tZK ] NK©9c © c<x<G6Uai q ¡ DHG-l\PUY ¡ Dj-i-I=B<B<DFEDHG-IJI8YJByUaIJj-DHi-IJcEI-W-DHXMY2c<HcB<cNEDHr-XN\<B<IJ[

Yk

0.0 0.2 0.4 0.6 0.8−4.0

−2.0

0.0

2.0

x

φ $ hNf*dJ''x'*+ d8/8hY(+X.( dghNhNfd8.hY/YfT /YhN/ Yfd8a,+-'(( " R` λ = 1.5

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j-i-ca\<DHY;U DMHc W-DMi-DM`G-DMHc \c<Y;U i-I-XaKwJK Z NK¨'i-UB<G-DMG-I-DuE6d6[_\<cYZU

i(t)B Y2c<GZ\ q i-DI-VDHDN\^B<I2E

d2i

dt2+

[R(T )

L− MS0

LC

]di

dt+

[1

LC+

1

L

∂R(T )

∂T

dT

dt

]

i = 0, wJK

EDS0

YJi q \<IJHG6UnpyUi6UYJ\DHi-IJXM\<IJY-I q XMIZdIJ\DNd6[ "ZY2c\c<i-lzr^jJi-DeEjJcadUeUDN\<X%[ dI G-DMr-G-lzV

M BaRUI-VG-Ua[ I-G2E q YJ\<I-BG-c<XM\fgk-DHj-I cx<i6Ua\G-c<r XHBa[JI R(T )

XHcj-i-c\I-BadDMG-I-D \<DHiJVI-XM\c<i6U "gRUBI-X%[ TDHD c\ \<DMVj-DHi-Ua\ q i-l T

LIC

I-G;E q YZ\<I-B<GJc<XM\fIDMVY2c<XM\f^B Y;cPdDHxyUP\<DNd6f<G-c<V(Y;cGJ\ q i-DaK9cadUeUP[UB<I-XMI-Vc<XM\f

R(T )dI-G-DMr-G-c<r

R(T ) = R0 + LbT IbXMW-IJ\PUa[ " WJ\c

j-i-ck-DHXMX\DHjZdcc<x<VDHG-U j-cHEW-IJGJ[-DN\<X%[1UY2c<G q ©9f< \<c<G-U 2

ρqdT

dt+ kT = R(T )i2,

EDq q EDedf<G6UP[\DHjZd6c<DHVY2c<XN\<f G-IZ\<I\DHi-VIJXM\<ci6U "U ρ DMDnV]UXHXHU "j-cad q W6UDMVRUaVY-G q \ q XMI-XM\DHV q_q i6UB<GJDHG-I-rB<I;EU

õÓÉÌÉÔÅÌØC

R(T)

i(t)

L 1

LM

_`*d8XX-(*+X-.d cX :+q3d +-'*+fd8hN/8f*dXe('*+-f(/8''*/ ,'+-`('*+ S'*/8Xh

d2i

dt2+ ω2

0i = (µ− bT )di

dt− bi

dT

dt,

dT

dt+ γT = α(T )i2,

wJK t

EDµ = ω2

0S0M −R0/L; ω20 = 1/LC; γ = k/ρq; α(T ) = α0 + bLT/ρq;

α0 = R0/ρq.AxDMi-UaVDMi-G-lCpj-DHiJDHVDHGJG-lCp

x = ai, y = −x, z = bT/ω0,τ = ω0t, a =

√

αbρq/ω0kq i-UB<G-DMG-IJ[ wJK t j-iJI-G-I-V]U \ B<I2E

x = mx + y − xz,

y = −x, wZK w

z = −gz + zx2. EDMXHf

m = µ/ω0 = ω0S0M − R0/ω0L, g = γ/ω0, x = dx/dτ.A§\i-DNpZVDHiJG-c<r EB q j6Ui6UaVDM\<iJI-W-DHXMY2c<r XMI-XM\DHVD wJK wj6Ui-UVDM\i m j-i-c<j-ci-k-I c<G6UodDHG8i6UaHG-c<XM\I B<GJc<XHI-Vcr§I=i6UXMXHDMI-ByUDMVc<r MG-DHi-MI-r "g j-Ui6UVDN\<i " pyUi6UY

\<DMi-IJ q I-r ca\<G-c<XMIJ\<Dedf<G-cD`B<iJDHV [_i-DedUY2XUkJI-I\DHi-VI-XN\<c<i-UJKAEUPdf<G-DMr DMVmx q EDHV G6UaHlzByUa\<f j6Ui-UVDM\i-c<V B<cx q ¡ EDMG-IJ[ "-U g j6Uai6UVDM\i-c<V(I-G-DHiJk-I-c<G G-c<XN\<IHDHGJDHi6Ua\c<i6UJK UaXHXHVca\<i-I-V X p<DHV q "IJHc<x<i6U ¡ DHG-G q G6U iJI-XKPwJK QJK EDMXHfuY2cad6DHxyUa\DNdfG-lzrY2c<G

\ q i8Bc\dI-WJI-D¢c\_YZdUXMXHI-WJDHXHY2c<McXed q W6Ua[=G-DXMcHEDHi ¡ IJ\G-DNdIJG-DHr-GJlCp NdDMVDHG \<cB-K¨]XMIZdIJ\DNdf

1 q j-i6UBPd6[-DN\<X%[1Ec<jJcadG-IJ\DNdfG-c<rbk-DMj-f<sc<x<i6Ua\G-c<rbXMB[JHI "XHcEDMi ¡ U DMr§dI-G-DMr-G-lzr q XHI;dIJ\<Dedf 2I8I-G-DHiJk-I-c<G-GJlzr j-i-DMc<x<i6UaHc<ByUa\DNdfJKSTI-m

mnDHi-DMG-k-I6Uodf<G-lzD q i6UBG-DHG-IZ[HDMG-DHi6UP\<c<i6UVc ¡ G-cÙj-I-XHUa\<fBT[JB<G-c<V BI;ED!"ZY2c<G Y-i-DN\<IJHI-i-c<B<UB RUBI-XHI-VcXM\<f

S(x, V ) q XHIZdIZ\<DNd-[ 1I¢RUoEUB q i6UBG-DHG-IZ[ I-G-DMi-k-I

c<G-GJc<Hc j-i-DMc<x<i6UaHc<ByUGJIJ[V (x)

K

1 2

R

C

L L1

M

y(t)

x(t)

z(t)

wx/R*( ((f/gT:dg''Dd X :+q3d M+I'*+fd8hN/8f*dcX('*+f(/8''/ W'*+-`('+ '/YXh

jJj-i-c<Y2XHIJVI-i q DMV j-cPdI-G-c<VcV"m q G-YJk-I- S1(x)"*\oK DaK Y-i q \<IZG q q XHIZd6IJ\<Ded6[

1"6xDM q W-DM\PU B<cNEDHr-XN\<B<IJ[Ec<jJcadG-IJ\DNdfG-c<r1cx<i6Ua\G-c<rXHB[ZI2

S1(x) = S0 − S1x2,

wJK ED

xG6UjJi2[ ¡ DHGJI-DG6U^Bap<cHED q XMIZdIJ\DNd6[ 1

S0IS1

j-c<XM\cR[-G-G-lzD¢j-cPdc ¡ I \<Dedf<G-lzD1Y;c!HmnmnI-k-I-DMGJ\<l9K 9i-DFEj-cad6c ¡ I-V "]WJ\c VDNpyUG-IJHVªB<cNEDHr-XN\<B<IJ[ k-DHj-II-G-DMi-k-I-cG-G-c<rc<x<i6Ua\G-c<rXHB[ZIj-cHEW-I-GZ[-DM\X%[1UY2c<G-c<VDMi-G-c<XN\<I

S = S1(x) − bV = S0 − S1x2 − bV,

wJK ] ED

V = V (x) G6Uaj-iy[ ¡ DMG-I-DnG6U B<lCp<cED9I-G-DHiJk-I-c<G-GJc<Hc j-i-DMc<x<i6UaHc<ByUa\DNd6[

b

j6Ui-UVDM\iK q XM\<f1I-G-DHiJk-I-c<G-GJc<D jJi-DHc<xi6UacByUG-I-D c<X q TDHXN\<Bad6[JDM\X%[ BXMc<c\<BDM\XM\B<I-IX q i6UaB<G-DHGJI-DHV

V = −γV + ϕ(x). wJK

¨'i-UB<G-DMG-I-DuE6d6[_\<cYZU¢B Y;cGJ\ q i-DHDMG-DHi6UP\<c<i6U i-I-XaKwJK Q

Ldi/dt+Ri + C−1

∫

(i−MSdi/dt)dt = 0

XHcB<VDHXN\<G-c_X q i6UBG-DHG-IZ[-VI wJK ] I wJK zEUaDM\ UVY-G q \ q XHI-XN\<DMV q "XHBcE6[ q uX%[B x<DMHi6UaHVDHi-G-lCpj-DMi-DHVDMG-G-lCpYB<I;E q

x = mx + y − xz − dx3, y = −x, z = −gz + gΦ(x), wJK O [

EDd = d(S1)

`j6Ui6UVDN\<i "ac\<BDHW6U I-r¢XN\<DHjJDHG-I BadIJ[JG-IJ[GJDNdI-GJDHr-G-cXM\<I Y-i q \<IZG-l p2Ui-UYJ\<DMi-I-XN\<I-Y-I

Φ(x)nm q G-YJk-IJ[ "ac<j-I-XMlzByU Ua[TXHB<c<rJXM\<B<U'I-G-DHiJk-I-c<G

G-c<Mc j-i-DMc<x<i6UPc<ByUP\<DNd-[K6AMDHG-DMi6Ua\c<i-D EDMr-XM\B q \ EByUTVDep2UGJIJV]U G-DedI-G-DMr-G-cHc c<Mi6UG-I-WJDHG-IJ[TUVjZd6IJ\ q El Y2cadDMxyUG-IJrK nDMi-B<lzr 'xDMlzGJDHi-k-IJc<G-G-lzrIXHB[ZRUG

YkXTGJDNdI-GJDHr-G-cXM\<f p2Ui6UaYJ\<DMi-I-XM\I-Y-I q XHI;dIJ\<Ded6[ " B\<c<iJc<r _IJG-DHi-kJI-c<G-G-lzr "c<x q XNd6c<BadDMG-G-lzrRUBI-XHI-VcXM\<f¤Y-i q \<IJHG-l S c\uG6Uj-iy[ ¡ DHG-IZ[ V K q XM\f q XHIZd6IJ\<Dedfi6Uxc\PUDM\=G-UdI-G-DMr-G-c<V q W6UXN\<Y2D p2Ui6UaYJ\<DMi-I-XM\I-Y-I (S1 = 0)

"U1I-G-DMi-k-I-cG-G-lzrj-i-DMc<x<i6UPc<ByUP\<DNd6fXMc<x<i6UG=j-c_X p<DHVD EB q pZj-cad q j-DMi-I-cHEG-c<Mc1Y-B<UoEi6Ua\I-W-G-cHc ED \<DMYJ\<ci6U^X

RC mnIZdfR\i-c<V I_c<j-IJXHlzByUDN\<X%[ q i-UB<G-DMG-I-DHV

z = −gz + gx2. wJK O<O

nUi6UVDN\<iI-G-DHiJk-I-c<G-GJc<XM\Igi6UaB<DHG_c\G-c! DMG-I-sj-DHi-IJcEU Y2cad6DHxyUGJI-r1Y2c<GJ\ q i6U

T0

Yj-c<XN\<cR[-G-GJc<r1B<iJDHVDHGJImnIZdfR\i6Uτf = RfCf

K9i-IbX EDedUG-GJlCp j-i-DFEj-cad6c ¡ DHG-IJ[;p q i6UBG-DHG-IZ[ wJK O [ zj-DMi-DNp<cE6[J\B q i6UBG-D

G-IJ[§YZdUaXHXHIJW-DHXMY;cHcbMDHG-DMi6Ua\<ci6U wJK wNK G6URW-IJ\ "*DMXNdI q XHI;dIJ\<Dedf 1dI-GJDHr-G-lzr "

U¢I-G-DHiJk-I-c<G-GJlzrbj-i-DMc<x<i6UaHc<ByUa\DNdf q Ec<BadDN\<B<ci2[JDM\ wJK O<O "-\<c V]Ua\<DMV]Ua\<I-WJDHXHYJI-DVcHEDNd6IHDHGJDHi6Ua\c<i-c<B "aX p<DHVl Y;ca\<c<i-lCpTIJHc<x<i6U ¡ DHG-l G6U i-I-XaKwJK Z I wJK Q "PGJDHi6UaNdI W-I-Vl9K pyDMV]UTX EDN\<DMYJ\<c<iJc<V(B HY2XHjJDHi-I-VDMGJ\PUPdfG-c<Vca\<G-c!TDHG-IJIx<cadDMD q Ec<x<G-U "\PUYY;UY_j-cBcad6[-DN\ ByUi-f<IJi-c<ByUa\f¢I-G-DHiJk-I-c<G-GJlzD`XMB<c<r-XN\<ByUHDMG-DHi-Ua\<c<i-U i-DM q dI i-c<BY;cr j-c<XN\<cR[-G-GJc<rB<i-DMVDHG-ImnIZdfR\i6U "WZ\<c9j-i6UaYJ\<I-WJDHXHYJI GJDHc<X q TDHXN\<B<I-Vcj-i-II-XMj-cadfHc<ByUGJI-I\DHi-VIJXM\<ci6UJKACI;E q i6UBG-DHG-IJr wJK O [ G-DIJHVDHG-IZ\<X%[ "*DMXNdI8BY;UoWJDHXM\B<D XHDNd6DHYJ\I-B<G-c<McedD

VDHGZ\PUuI-XHjJcadfHc<ByUa\fRC

k-DMj-c<W-Y q B B<I2ED]Vc<XM\PU'A I-G6UJKMSd6[c<x<DMXHj-DMW-DHGJIJ[ q XNdcB<I-r^HDHGJDHi6UkJI-I_B M\c<V XNd q W6UaD'G q ¡ GJc j-i-I-VDMG-IJ\f EByU`Y;UXMYZUREU q XMIZdDHGJIJ[ "-Y;UYM\c9j-c<Y;UaUG-c'G6U i-I-XKawJK KS`d-[TXMI-VVDM\i-I-W-G-cHcnVc<XM\PU9A I-G6U q j-i6UaBad6[- I-DCj6U i6UVDN\<i-l

mIgB q i6UaB<G-DHGJIJ[Zp wJK O [ j-i-c<XN\<c¢I Xu\c<W-Y-I Hi-DHGJIJ[ HY2XMj-DHi-IJVDHGJ\PU

q Ec<x<GJlzVc<x<i-Uac<V B<lzi6U ¡ U \X%[1WJDHi-DNTj6Uai6UVDM\i-lX p<DHVl12

m = K0 − 3, g = R0C0/τf , wJK O Z

EDK0

^Y2cHmnmnI-k-I-DMGJ\ q XHIZd6DHG-IJ[_EB q pZY;UXHY;UoEGJc<Hc q XHIZd6IJ\<Ded6[ R0C0

Iτf

j-c<XN\<cR[-GJG-lzD9B<i-DMVDHG-IVc<XN\PUACI-G6UÌ mnIZdfR\<i6UEDM\<DMYJ\<ci6UJK6AmnIJHI-W-DMXHY2c<V HY2X j-DMi-I-VDHGZ\<DCj-Ui6UVDN\<i-l

mIgdDMHY2cuVDMGJ[J\f IIJVDMiy[J\<f "PB<Ui-f<I-i q [Y2cHmnmnI-k-I

DHGZ\ q XMIZdDHGJIJ[ITjJc<XM\co[JG-G q B<iJDHVDHGJI mnIZdfR\<i6UZK<hnUYTjJc<Y;UaRUPd6ITI-XMXNdDFEc<ByUGJIJ[ "

1 2

R0 C

0

R0 C

0

x(t)

y(t)

z(t)

$ :+q3d RC S M+I'*+fd8hN/8f*dcX('*+f(/8''/ '*+I`('+ '/YXh

EI-G-UVI-Y;U HDMG-DHi-Ua\<c<i-U " VcHEDedI-i q DHVc<Mc q i6UB<GJDHG-IJ[JVI wZK O [ "j-i-IJG-k-I-j-I-UPdfG-lzV c<x<i6UPc<V UB<I-XMIJ\ c\B<I;EU m q G-YJk-I-I Φ(x)

"J\oK DK-c\ XHB<c<rJXM\<B¢I-GJDHi-k-IJc<G-G-c<Mcj-i-DMc<x<i6UPc<ByUP\<DNd-[KXNd6I

Φ(x)j-iJDeEXN\PUBad6[JDM\ XHc<xc<rXHI-VVDN\<i-IJW-G q m q G-YJk-I- "

\<c q i6UB<GJDHG-IJ[ wJK O [ I-VDM \BY;URW-DHXN\<B<D i-DTDHG-IZ[§\cadf<Y2c_j-i-DFEDNdfG-lzr k-IJYZd*K EG6UY2c DHXedI

Φ(x)G-D[-Bad-[-DM\X%[XHI-VVDN\<i-I-WJG-c<r G6Uj-i-IJVDHi "

Φ(x) = exp(x−1) "

\<c¢VcHEDedf wJK O [ EDHVcG-XM\i-I-i q DM\ Y;UY_j-DMi-I-cHEI-WJDHXHYJI-D!"6\PUY_I^p2Uac\<I-WJDHXHYJI-Dni-D ¡ I-Vl"Y2cadDHx<UG-I-rK CcadDMDEDN\PUPdf<G-lzD¢I-XMXNdDFEc<ByUaG-IJ[ j-cYZUPRUPdI " WJ\cUXHIJVVDM\i-IJ[

Φ(x)[-Bad-[-DM\X%[G-DMc<xpycHEI-VlzV q Xedc<B<I-DMV E6d6[ iJDUPdIZRUk-IJIB VcHEDNdI wJK O [

c<XMc<x<c<r8\<i6UDMYJ\<ci-I-I \<I-j-U1j-DM\d6I XMDHj6Uai6Ua\<iJI-XHl XHDFE6dc mnc<Y q XHUJKA9VDMG-G-cbG6UodI W-I-D8c<XMc<x<c<r\<i6UaDHYJ\c<i-I-I[JBad6[-DN\<X%[Lm q G2EUVDHGZ\PUPdf<G-crj-i-I-WJI-G-c<rLi-c ¡ EDMG-IJ[i-D ¡ I-VcB EDM\<DMi-VI-G-IJi-c<ByUGJG-c<Hc p2Uc<XHUJKALY;URW-DMXM\<BDj-i-I-VDMi6U \<cHc B<I2EU m q G-Y-k-IJI Φ(x)

"Y2c< EU VcEDNdf wJK O [ j-i-I c<x<iJDM\PUDN\ XHBc<r-XM\ByU¢HDHGJDHi6Ua\c<i6Up2Uc<XHU "6Vl Blzx<DHi-DMV

Φ(x) = I(x)x2, I(x) =

1, x > 0,0, x ≤ 0.

wJK ORQ

mnIJI-WJDHXHY2c<r \c<W-Y-I Hi-DHG-IZ[ M\cbXHc<ca\<B<DN\<XM\B q DN\ I-XMj-cadfac<ByUaG-I- cHEGJc<j-cad q j-DMi-I-cHEG-c<Mc¢EDN\<DMYJ\<c<i-U^B¢X pyDHVDÌ-G-DMi-k-I-c<GJG-c<Hc j-i-DMc<x<i6UPc<ByUP\<DNd-[K j-i-DFEDNdIJB¤m q G-YJk-I- Φ(x)

B(XHc<c\B<DM\XM\B<I-IX wZK ORQ " IJ wJK O [ j-cad q W6U DHV q i-UB<G-DMG-IJ[ VcHEIJmIJk-I-i-c<B<UG-G-c<Mc1MDHG-DMi6Ua\<ci6U1X I-G-DHiJk-I-c<G-GJc<r G-DedI-G-DMr-G-cXM\f< "Jj-i-DeEXM\PUBad-[- TI-DTXMc<x<c<r \<i-DepJVDMi-G q s\i-DNpZj6Uai6UVDM\i-I-W-DMXHY q sG-DNdIJG-DHr G q EI-XHXMI-j6Ua\I-B<G q sXHI-XN\<DHV q 2

x = mx + y − xz − dx3, y = −x, z = −gz + gI(x)x2. wJK O

9XHYZd6uW-DMG-I-DHV jJDHi-DMVDHG-G-cry q i6UBG-DHG-IZ[^MDHG-DMi6Ua\<ci6U`X I-G-DMi-k-I-c<GJG-c<r^GJDNdI G-DMr-G-c<XN\<f< wJK O jJi-I-B<cHE[Z\<X%[YB<I2E q wZK Z 2

x− (m− z − 3dx2)x+ [1 − gz + gΦ(x)]x = 0,

z = −gz + gΦ(x). wJK ORt

Ba\<c<V]Ua\I-W-DMXHY-I i-DH q dI-i q DHVlzrGJDNdI-GJDHr-G-lzr(c<XHkJIZd-d6[Z\<c<i wZK Oot!9p2Uai6UYJ\DHi-I q DM\X%[ I-G-DMi-k-I-cG-G-c<r RUaB<I-XHIJVc<XM\f< EI-XHXMI-j6UkJI-I(I W6UaXM\<ca\<l|c\=jJDHi-DMVDHG-G-cr

xK*AXNd q W6UD XHI;df<G-c<r8I-G-DMi-k-I-cG-G-c<XN\<I XMI-XM\DHVl

τf T0 "Y2c< EU

g → 0"XHI

XM\DHV]U B<lzi-c ¡ EUDM\X%[1BTEB q VDMi-G q 2

x− a(1 − bx2)x + x = 0, wJK ORw

a = m− z0, b = 3d/(m− z0), z0 = z(0),

I_G-DMUB<I-XMI-Vc c\¢B<I;EU m q G-Y-kJI-I Φ(x)XHcB<j6UoEUaDM\¢j-c¢mnc<i-VD'Uj-I-XMIX q i6UaB<G-D

G-IJ[JVI HDMG-DHi6UP\<c<i6U=A UaG§EDHi 9cad6[ K nUi-UVDM\i-laIb"YZUaY B<I;EG-c1IJ wZK ORw "

RUaB<I-X%[J\ c\ G6URW6UPd6f<G-lCp q Xedc<B<IJr z(0)K

STi q Hc<r¢UaXHI-VjJ\c\<IJW-DHXMY-I-r^Xed q W6Ur x<DNlzG-DMi-k-I-cG-G-lzr^MDHG-DMi6Ua\c<i "<XMc<c\<BDM\XM\B q TI-r XM\i-DHV dDMG-I- j-Ui6UVDN\<i6U g B xDHXHY2c<GJDHW-G-cXM\<fJKu'\<i-DN\<f<DMHc q i6UaB<G-D G-IJ[XMI-XM\DHVlj-iJI N\<c<V q XNd6c<B<I-IXNdDFE q DN\UPdMDHx<i-UI-W-DMXHY;Ua[BRUIJVc<XHB[Zf¢j-DMi-D VDHGJG-lCp

xIz"6XHBcE6[ UP[IJX pycHEG q XMI-XM\DHV q YB<I;E q

x− [m− Φ(x) − 3dx2]x + x = 0. wJK Oo

9cadG6UP[ UaG6UPdc<MIJ[8X q i6UaB<G-DHGJI-DHV"ACUG EDHi 9cad-[§B M\c<Vj-iJDeEDedf<G-c<V XNd q W6UDEc<XN\<I-eUaDM\<X%[j-iJI q Xedc<B<IJI Φ(x) = x2 K;A(iJDUPdfG-c<V MDHG-DMi6Ua\<ci-D9X IJG-DHi-kJI-c<G-G-crG-DedI-G-DMr-G-c<XN\<f< c<xPdUXM\f1G-UoWJDHG-I-r j6Ui6UVDN\<i6UbIJG-DHi-kJI-c<G-G-cXM\<I

g"*BY;ca\<c<i-cr

XHIJXM\<DMV]U BDeEDN\ XHDHxR[¤jJi-I-G-k-IJj-I6UPdfG-c Y;UY \<i-DepZVDHi-G6UP[ "zc<Mi6UG-I-WJDHG6U G-DMY;ca\<ci-lzVLI-GJ\<DMi-ByUPd6c<V

g1 ≤ g ≤ g2K UDMHc_j-i-DFEDNdUaVI j-i-IJxadI ¡ DMG-G-lzVLc<j-I-XHUG-I

DHV Vc< q \`XNd q ¡ IJ\<fì6UXHXMVc\<iJDHG-G-lzD'UXHI-VjZ\<c\I-W-DHXMY-I-D q i6UaB<G-DHGJIJ[^G-UmUaHc<B<c<rjZdcXHY2c<XM\IK Ua\DHV]Ua\I-W-DMXHY;Ua[VcHEDedfuVcHEI-mnI-kJI-i-c<ByUaG-G-c<McuHDMG-DHi-Ua\<c<i-UuX IJG-DHi-kJI-c<G-G-cr

G-DedI-G-DMr-G-c<XN\<f< wJK O DMXM\<fG-DedI-G-DMr-G6Ua[ \<i-DepZVDHi-G6UP[¢EIJXHXHIJj6Ua\<IJB<G6Ua[ XHI-XN\<DMV]UX9\<i-DMV [1GJDMRUaB<I-XHIJVlzVI1j6Ui-UVDM\i6UVI "6RUREU Ua[j-c\c<YB

R3 2

−∞ < x <∞, −∞ < y <∞, 0 ≤ z <∞, EDj-DHiJDHVDHGJG6Ua[

zc<j-iJDeEDedDHG6U G6U¢j-cadc ¡ IJ\<Dedf<G-c<rj-cad q c<XMI "6\PUY1Y;UYXnmIZI

W-DMXHY2c<r \<cW-Y-I iJDHG-IJ[^j-i-DFEXM\PUBPd6[-DN\ XMc<x<c<r j-i-cHEDN\<DHYZ\<I-i-cByUG-G-cD9G6Uj-iy[ ¡ DHG-I-D

x(t)G-UB<lCp<cHED¢mI;dfR\<i6UJK STI-B<DMi-HDMG-k-IJ[§B<DMYJ\<c<iJG-c<Hcj-cad-[=XHY2c<i-c<XN\<DMr j-c\c<Y;U

wZK O RUB<IJXHIJ\ c\ j6Ui6UaVDM\<iJc<B ImUacB<lCp_Y2c<c<iEI-G-Ua\ 2

divF = m− g − 3dx2 − z. wJK O ]

9XHXedDeEcByUG-IJ[¢B9YJByUaI;dI-G-DMr-G-c<V§j-i-I-xad6I ¡ DHGJI-Im < g 1

XHB<I;EDM\<Dedf<XN\<B q \c\<c<V "WZ\<c_XHI-XN\<DMV]U dc<x<UPdf<G-cÊIJXHXHIJj6Ua\<IJB<G6UI=WJ\cÊ6d6[bd x<lCp=G6URW6UPd6f<G-lCpEUGJG-lCp_IJnc<xPdUXM\I_c<j-i-DFEDNdDMG-IJ[mUacB<lCpj-DMi-DHVDMG-G-lCpB<XMDH EU¢XHjJi6UB<DFE6dI-B<c

divF < 0KAY-ByUaHIZdI-GJDHr-G-cV¤j-i-I-xad6I ¡ DHGJI-I

z ≈ mI1GJDMRUaB<I-XHIJVc c\ Y;cc<iEI

G6Ua\lxEIJB<DHi-MDHG-kJIJ[bc\i-I-k6UP\<DNd6f<G6UJK niJI

m > g"6 ED

gY2c<G-DMW-G-c G6UIJx<cadDMD

I-GJ\DHi-DMXHG6UP[LcxadUXN\<f MDHG-DMi-I-i-c<B<UG-IJ[¤GJDNdI-GJDHr-G-lCp¤Y;cPdDHxyUaG-I-r " G-UY EI-BDHi HDMG-k-I-IRUBI-XHIJ\ c\ Y2c<c<iEI-G-Ua\oK6¨]XNdc<BI-DHV EI-XHXMI-j6Ua\I-B<G-cXM\<Ib[JBad6[JDM\<X%[

m− g < z + 3dx2. wJK OR

Sd6[ UaB\<c<Y2cad6DHxyUGJI-rbj-i-Id 6= 0

N\<c q Xedc<B<IJD B<XHDM EU B<lzj-cPdGJ[-DN\<X%[ KA M\c<VXHVlzXedD j6Uai6UVDM\i

dcj-i-DeEDNd6[JDM\x<DNlzG-DMi-k-I-cG-G q EI-XMXHI-j-Ua\<I-BG q G-DNd6I-G-DHr

G-c<XN\<f XHIJXM\<DMVl9K^]XedI ¡ D q XHIZdIZ\<DNd6f i6Uxc\PUDM\ G-UsdI-G-DMr-G-c<V q W6UXM\Y2Dup2U i6UYZ\<DHiJI-XM\I-Y-I=IGJDNdI-GJDHr-G-cDTc<Mi6UG-I-WJDHG-I-DTUVjZdIZ\ q ElRU¢XHWJDM\ IJG-DHi-kJI-c<G-G-cr

k-DMj-I c<x<i6Ua\G-c<r XHBa[JI G-UXM\ q j-UDM\§i6UG-f D!" W-DHV"G-UoWJDHG-IJ[ j-DMi-DHVDMG-G-c<r x B<l pycHE[Z\_B c<xadUXM\fG-DNdIJG-DHr-GJc<XM\I=pyUi6UYJ\DHi-IJXM\<IJY-IS(x)

"\cB<lzi6U ¡ DHG-I-D wJK OR j-i-IJG-I-V]UDN\ B<I;E

m− g < z(τ). wZK Z@[

9c<XNd6DeEG-DMDGJDHi6UBDHG-XN\<B<c§i6UaNEDNd6[JDM\§mÙPc<B<cDj-i-c<XN\<i6UGJXM\<Bc XHIJXM\<DMVl G6UEB<Dc<xadUXM\I jZdc<XMY;cXM\<f

z = z0 = m − gK Sd6[

z > z0 XHI-XN\<DHV]UEIJXHXHIJj6Ua\<IJB<G6U "E6d6[

z < z0 mUaHc<B<lzr c<x*DMVsBbdc<Y;UPd6f<G-c<r(c<Y-i-DMXM\<GJc<XM\I(duxc<r \i6UDMYJ\<c<iJI-IXHIJXM\<DMVl i6UX I-iy[-DN\<X%[ K \PUak-I-c<G6Uai-G-lzDCiJD ¡ I-Vl UaB\<c<Y2cad6DHxyUGJI-rìJDUPdIZ q \X%[B\c<V"XNd q W-UD!"Y;c EU1j-cHEY;URW-Y;UHGJDHi-MI-I I DHD i6UX p<cE B1XHiJDeEG-DMVj-c1B<iJDHVDHGJIY2c<Vj-DMG-XHI-i q \<X%[ " WJ\c^B<cHVc ¡ G-cjJi-I q XNdcB<I-I

m− g < z, wZK Z O

EDzuXHi-DFEG-DHDj-c B<i-DMVDHG-IG6URW-DMG-I-D]j-DMi-DHVDMG-G-c<r

z(τ)KHSd6[uEc<XN\PUa\<cW-G-c x<cadf

I;pm

m > 1

`G-DMi6UB<DMG-XM\B<c wJK Z O nVc ¡ DM\ G-D^B<lzj-cadGZ[J\<fX%[ I \<i6UDMYJ\<ci-I-IXHIJXM\<DMVlx q E q \ q p<cEIJ\<fnBuxDHXHY2c<GJDHW-G-cXM\<f "yDHXNd6IEI-XMXHI-j6UP\<I-B<G-Ua[¢G-DNd6I-G-DHrJG-c<XM\fc\X q \<XN\<B q DM\ d = 0

eK I-XN\<DHV]U wJK O [ up2Ui6UaYJ\<DMi-IJ q DM\X%[ DeEI-G-XM\B<DHGJG-c<r c<XHcx<c<r§\c<W-Y2c<r B1G6URW6UPdD

Y2c<c<iEI-G-Ua\oKuXedI1m q G-Y-k-IZ[ Φ(x)GJD`XMcEDHi ¡ IZ\ dI-GJDHr-G-lCpj-c

xWZd6DHG-c<B "6d6I-G-D

Ui-IZRUk-IZ[1XMI-XM\DHVlB^c<XMc<x<c<r_\<cW-Y2DTj-i-IJB<cHEIJ\ Y_p2Ui-UYJ\<DMi-I-XN\<I-W-DMXHY2c<V q j-cPdI G-c<V q

(g + s)(s2 −ms+ 1) = 0, wZK Z\Z

XHcx<XM\B<DHG-GJlzD`G-UoWJDHG-IJ[Y2c\<ci-c<Hc DHXN\<f

s1,2 = m/2 ± (i/2)√

4 −m2, s3 = −g. wJK Z Q Ac<xPdUXM\I j6Ui-UVDM\i-c 0, −2 < m < 0EDHrJXM\<BIJ\<Dedf<G-lzDnW6UXN\<IB<XMDNp XHcx

XM\B<DHGJG-lCp HG6URW-DHG-IJrc\<i-IJk6Ua\<Dedf<G-l I cXHc<xyUP[\c<W-Y;U q XM\<cr-W-I-ByUZK mnIJIJW-DHXMY;cr\<cW-Y-I¢i-DMG-IJ[¢j6Ui-UVDM\i

gB<XMDH EUnj-cadc ¡ IJ\<DedDHG¢Y;UY¢c\G-c! DMG-I-D pyUi6UYJ\DHi-GJlCp

B<i-DMVDHGXHI-XN\<DMVl12jJDHi-I-cHEU¢Y2cadDMxyUG-I-r_Y2c¢B<i-DMVDHG-Ii-DNdUY2XUk-IJI1mnIZdfR\<i-UJK nU i6UVDN\<i

mVc ¡ DN\ x<l \f Y;UY VDHGJf! D'G q d-[ MDHG-DMi6Ua\c<iG-DeEc<B<cHx q ¡ EDMG "2\PUY I

x<cad6f! D¢G q d6[ Bi-D ¡ I-V]Uop HDMG-DHi-Uk-I-IeK A"c<xPdUXM\I 0 < m < 2c<XMc<xyUa[b\c<W-Y;U

DHXN\<f_XHDFE6dc mnc<Y q XTXEB q VDHiJG-lzVG-D q XM\<cr-W-I-B<lzVLI cHEG-c<VDMi-G-lzV q XN\<c<r-WJI-B<lzVVG-c<Mc<c<x<i-UaIJ[JVI wJK Z Q eK I-G-IJ[

m = 2x<I-m q i-Y;Uk-I-c<GJG6Ua[ "zI¤cG6U c\<B<DMW6UDN\

XHVDMG-DXHDeE-dc mnc<Y q XU G6UXHDFE6dc q HDNd*KhnUYXNdDFE q DN\9IJ wJK Z Q "RB XMI-XM\DHVD wJK O [ I-VDHDN\<X%[ q G-IJYZUodf<G6Ua[`B<cHVc ¡ G-cXM\<f

G-DNRUB<IJXHI-Vc<Mc q j-i6UaBadDHGJIJ[ XHB<c<rJXM\<B<UVI q XM\c<r-W-I-Bc<HcÌ GJD q XN\<c<rJW-I-B<c<Mc VG-c<Mcc<x<i-UaI-r K-Ai-D ¡ IJVD'HDMG-DHi6Uak-I-I

m > 0]XHc<XN\<cR[-GJI-Dni6UB<G-cB<DHXMIJ[ pyUi6UYJ\DHi-IZ q

DM\X%[ EB q VDHiJG-lzVsGJD q XN\<c<rJW-I-B<lzVsVG-c<Hcc<x<i6UaHI-DHV"I cEG-c<VDHiJG-lzV q XM\c<r-W-I B<lzV " WJ\c cj-i-DeEDNd6[JDM\<X%[ G-DMUB<I-XMI-VlzVI j6Uai6UVDM\i6UVI

mIgK

hnUYbB<I2EG-cIJ wJK Z Q "B x<IJm q i-Y;UkJI-c<G-G-cr=\c<W-Y2D m = 0XHcx<XM\B<DHG-GJlzD¢HG6U

W-DMG-IJ[s1,2

j-DMi-DHXMDHY;U \VG-I-V q c<XHf XnG-DHG q d6DHB<c<rXHY2c<iJc<XM\f< 2

∂Res1,2(m)/∂m|m=0 = 1/2.9i-I N\<c<VL\<i-DN\<f<D XHc<x<XN\<B<DMG-G-c<D G6URW-DMG-I-D

s3 = −g c\HEDedDHG-c1c\1VGJI-Vc<r8c<XHIKDHUPdIJ q DN\<X%[ YZdUXHXMI-W-DHXMY;Ua[8x<I-m q i-Y;Uk-IJ[ G2Ei-c<G-cByU ¦'c<j-mU 2 x<I-m q i-Y;Uk-IJ[i-c ¡ EDHG-IZ[kJI-YZdU¢IJXHDFE6dc mcY q XHUJK

-- . # $ $%- ( ' -*$%#*$%$ *$ - &! # *$ -

#ZcVc<YZdI-GJI-W-DHXMY-I-D¢\<i6UaDHYJ\c<i-I-I \<cW-Y-I'Y;UY8i-DM q d6fo\PUP\bHi q xc<Hcj-DHiJDHXHDMW-D G-IJ[ q XM\c<r-W-I-BlCp I GJD q XN\<c<rJW-I-B<lCp VG-c<Mc<c<x<i-UaI-rXMDeE6d6c<B<lCp¢k-IJYZdc<B q XM\c<r-W-I B<lCpTI G-D q XM\c<r-W-IJB<lCp XHDMj6Ui6Ua\i-I-XzXMDeE6dcB<lCp GJDHj-cHEB<I ¡ G-lCp`\<cW-DHYXHc'B<iJDHVDHGJIc\Y-i-l \<IZ[ IIZ q W-DHGJIJ[ IZp K q UaG-Y;Ui-D!"#K< I-i-Y-Mc<mnc<V=I K VDHrZd6c<V=XNd q ¡ Ua\XHBc<DHMc¤i-cHEUXHIJHG6UPd6c<Vgx<DeEXN\<B<IZ[ "'j-iJDeEB<D U I-V B<cHVc ¡ G-cXM\<f¤Xedc ¡ GJc<HcUj-DMi-I-cHEI-WJDHXHY2c<Mc EB<I ¡ DHG-IJ[XHIJXM\<DMVl9K'X q DMXM\B<c<ByUGJIJ[ HcVc<YZdI-GJI-W-DHXMY-IZp\<i-UDHYJ\c<i-I-r8j-i-I1G-DHY2c\c<i-lCpEc<j-cadGJIJ\<Dedf<G-lCp=jJi-DeEjJcadc ¡ DHGJIJ[Zp1XNdDFE q DN\G6U dI-WJI-D¢BIZp c<YJi-DHXN\<G-c<XN\<I XMW-DM\G-c<McVG-c ¡ DMXM\ByU q XM\c<r-W-IJB<lCp8I=G-D q XM\<cr-W-I-B<lCpj-DMi-I-cHEI-W-DMXHY-I;p \<i-UDHYJ\c<i-I-ri-UaMdIJW-G-lCp j-DHiJI-cHEc<B";B<YZd W6Ua[ Y2c<GJ\I-G q2q V \i6U DHYZ\<c<i-IJr " q XM\c<r-W-I-BlCp1jJc q UXHXMc<G q K EG-I-V"I;p m q G;EUVDMGJ\PUPdfG-lCp i-DN q dfR\PUa\c<B=B1\<DMc<i-I-I§EIJG6UVI-W-DMXHY2c<Mc1p2U

c<XHU^[-Bad6[JDM\X%[ \<DHci-DHV]U IZdfG-I-Y2c<ByU cXMDeE6dc mnc<Y q XHDK,XedI=B EI-G6UVIJW-DHXMY;crXHIJXM\<DMVD¢X q DMXM\<B q DM\EB<cR[JY;c UXMI-VjJ\<ca\<I-W-DMXHY;Ua[ \i6UDHYZ\<c<i-IZ[=B B<I;ED`j-DM\d6I8XMD j6Ui-Ua\<i-IJXHlXMDeE6d6c mnc<Y q XU "a\<c9B'DMD c<Y-i-DMXM\G-c<XM\I^B<caG-I-Y;UDN\nG-DM\i-I-B<I-UPdf<G-cDzHI j-DMi-x<cadIJW-DHXMY;cDj-cHEVG-c ¡ DHXN\<B<c\<i6UDMYJ\<ci-I-rK \cbj-cHEVG-c ¡ DMXM\B<cbVc ¡ DM\1c<Y;U RUP\<f<X%[jJi-IJ\R[-MI-ByU I-V "ZI^\<c EUTB MY2XHj-DMi-I-VDHGZ\<Dnx q EDM\TG6Uxad6 EUa\<fX%[^p2Uca\<I W-DMXHY-I-r1Ua\\i6UYJ\c<ibY;UYc<x<i6UPuEDM\DHi-VI-GJI-i-c<ByUaG-G-c<Mc p2UcXUJK G-cHc<XN\<c<i-cG-G-I-r HY2XMj-DHi-IJVDHGJ\PUPd6f<G-lzr UaG6UPdIJ VDNpyUG-IJHVc<B¤B<cHG-I-Y-G-cB<D

G-IJ[ I \<c<j-cPdc<HIJW-DHXMY;cr XN\<i q YJ\ q i-lp2Uca\<I-W-DMXHY-I;p¢j-i-IJ\R[JHI-B<U TIZp¢VG-c ¡ DMXM\<BnBVcHEI-mnI-k-IJi-c<ByUGJG-c<VHDMG-DHi6UP\<c<i-DXTI-G-DHiJk-I-c<G-GJc<r G-DNd6I-G-DHrJG-c<XM\f< c<x<c<XMG-c<ByUaG G-c j-i-I-B<Ded YVlzXNdIbc X q TDHXM\B<c<ByUaG-I-I B UaB\<c<GJc<VG-c<rEI-G6UVIJW-DHXMY;crbXHI-XN\<DMVDHcVc<YZdI-GJI-W-DHXMY2c<r \<i6UDMYJ\<ci-I-I \<I-j-U ODCIE*XDA CIOB= FL=@EGFBA S5IE S$B?BA FL=<?DS @CIA EKABOB= ICI; X S J)SD MK9DHi-BlzDbj-c<j-l \Y-IG6UrJ\Ij-DM\d XMDHj6Uai6Ua\<iJI-XHlB q i6UB<G-DMG-IJ[;p¤HDMG-DHi-Ua\<c<i-U

wZK O [ Y q XMj-DNp q G-DTj-i-I-B<DedIKA l [-XHGJIZdc<XMf"WJ\c_\PUaY;cHc_i-DTDHG-IZ[ M\<I q i6UBG-D G-IJ[_GJD9I-VDH \oKncYZU ¡ DHV "-WZ\<c M\c \PUY K XHc<xyUP[ \<c<WJYZU¢XHIJXM\<DMVlLp2Uai6UYJ\DHi-IJ q DM\X%[ EB q VDHi-GJlzVsG-D q XN\<c<r-WJI-B<lzV I cHEG-cVDHi-G-lzV q XN\<c<rJW-I-B<lzVsVG-c<Hcc<x<i6UaHI [-VI K UVDMG-I-VB<i-DHV [ B wJK O [ G6U1c<x<i-Ua\<G-cDI RUoEUREI-VG6URW6UPdf<GJlzD q XNd6c<B<IJ[

+hJ` X+IrDd8fd8hNf1(*X X-+IR*`/8S u/Y. i XMdcTBTN/YQ-q i ),+-''*/ XI(*XhN+-q3+ "

x(0) = y(0) = 0"z(0) > 0

G6U cHEG-cVDHi-GJc<V GJD q XN\<c<rJW-I-B<c<V VG-c<Mc<c<x<i6UPI-IK9GJ\<DMHi-IJi-c<ByUGJI-DbXHIJXM\<DMVljJcE6\<B<DMi<EIJ\ q pycHE\<i6UaDHYJ\c<i-I-I¤G6U8x<DHXMY2c<G-DHWJG-c<XM\fBPEcPdfc<XHI

zK>' q i6UB<GJDHG-I-r wJK O [ XedDeE q DM\ "*WJ\c z(τ) = z(0) exp(gτ)

K# i6UaDHY \<ci-IJ[j-i-I

τ → ∞ B cXHc<x q \c<W-Y q G-DnB<caB<i6U UDN\<X%[ ACcHG-I-YZdUTHI-jJc\<DNRU "ZY;ca\<c<i6UP[ c<Y;UaUPdUXMf B<DHXMf<V]U q XHj-D G-c<r K 9DM\d6[^XHDHj-Ui6U

\<iJI-XHl XMDeE6dc mnc<Y q XU X q DMXM\<B q DM\nB G-DMY;ca\<c<i-crTB<cHV q DHGJG-c<rTXMI-XM\DHVDaK GJ[Z\<I-DB<cHV q DHGJIJ[ j-iJI-B<cHEIJ\Y I-XMW-DMHG-c<B<DMG-I- XUVcr j-DN\dI "G-cXN\<i q YJ\ q i6Ubi6UaHx<I DHGJIJ[ mUacB<c<Hc j-i-c<XN\<i6UaG-XM\ByU G6U \i6UDHYZ\<c<i-IJI8cXM\PUDN\<X%[K '\c<x<l"j-cHE\B<DHiEIJ\fM\I_XHc<c<xi6U ¡ DMG-IJ[ ";G q ¡ G-c c<j-iJDeEDedIJ\<f¢B<I2EXNdUax<c B<cHV q DHGJG-c<r_XHI-XN\<DMVl "ZEcY;UaRUP\<fnG6UPdIJW-I-D B'G-DMr jJDM\dI XHDMj6Ui6Ua\i-I-XMl XHDFE6dc mcY q XHU "<B<l [JXHG-IJ\f`XN\<i q YJ\ q i q Ua\\i6UYJ\c<i-c<BIÎJ q W-IJ\fIZp HB<cPduk-IJj-i-I XMGJ[J\I-IB<cHV q DHGJIJ[KJDTDHG-IJDq Y;UaRUGJG-c<rUoEURW-I_G-DMcEG-cG-UoWJG-c"G-c¢B¢XHIZd q XHB<c<rJXM\<B<U Mi q x<c<XM\I1Y2c<G-YJi-DM\G-lzrB<I;EbV]UPdcHc¢B<cHV q TDHG-IZ[1G-DuEcPd ¡ DHGI-VDN\<f^jJi-I-G-k-IJj-I6UPdfG-c<Hc¢G-UoWJDHG-IJ[ KSTc<xyUBI-VB<c B\c<i-c<D q i6UB<G-DMG-I-DÌ-X pycHEG-cr XHI-XN\<DHVl wJK O [ Cj-c<XN\<cR[-G-GJlzr j-c

dc ¡ IJ\<Dedf<G-lzr WZdDMGγIi-UXHXMVc\<i-IJV§B<caV q DMG-G q ¤\PUY-I-V8XHj-cXHc<x<cV8XMI-XM\DHV q "

j-cad6c ¡ I-BTE6d6[j-iJc<XM\c\<ld = 0

2

x = mx + y − xz, y = −x + γ, z = −gz + gI(x)x2. wJK Za

XHc<x<Ua[ \<c<WJYZU_j-c\c<Y;U wJK Za uj-c j-i-D ¡ G-DMV q DeEI-G-XM\B<DHGJG6Ua[ "*XedDHMYZUXHVD D G6U c\<G-cXHIJ\DNdf<GJc G6URW6UPdU Y2c<c<iEI-G6UP\ Ij-i-DeEXM\PUBad-[-DM\_XMc<x<c<rXHDFE6dc mnc<Y q XaKuDY2c<c<iEI-G-Ua\<l12

x0 = γ"y0 = γ(γ2 − m)

"z0 = γ2 K c<XM\cR[-G-I-D1i6UB<G-cB<DHXMIJ[ B

B<cHV q DHGJG-c<r8XHI-XN\<DHVDTE6d-[ m > 0pyUi6UYZ\<DHiJIJ q DM\<X%[8EB q VDHi-GJlzVLG-D q XN\<c<r-WJI

B<lzV I¤cHEG-c<VDMi-G-lzV q XN\<c<r-WJI-B<lzV|VG-cHc<c<xi6UaIZ[-VIKS`d-[ G6UPp<c ¡ EDMG-IJ[j-DN\dI

Γ 10

B q i6UaB<G-DHGJIJ[Zp XMI-XM\DHVl j-i-cIJB<DFEDHVRUVDMG q B<i-DMVDHG-I G6U c<x<i-Ua\<G-cDIXG6URW6UPd6f<G-lzVI q Xedc<B<IJ[JVI=G6U cHEG-c<VDMi-G-c<VG-D q XM\c<r-W-I-Bc<V VG-cHc<c<xi6UaIJI i-D IJV(VG-cHc<Y-i-Ua\<G-cUoEURW q h9c!TI E6d6[_mnI-Y2XHI-iJc<ByUG-GJc<Hc g = 0.3

I_i6UaNdI-W-G-lCp

mIγKAClzx<i6UaB V]UPdc<D^HG6URW-DHGJI-D

γ = 0.1" G6Ur2EDHVx<I-m q i-Y;Uk-I-cG-G q \<c<W-Y q

m∗ = 1.176 . . ."B Y2c\c<i-c<r i-DUodIJ q DM\X%[¤cHEG-c<c<xp<cEG6Ua[ j-DM\d-[(XMDeE6d6c mnc<Y q XU

Γ 10

KP i-DNpZVDHiJG-c<DzIJHc<x<i6U ¡ DHG-I-D]EB<cR[-Y2c UXHI-VjZ\<c\I-W-DHXMY2c<r \i6UDHYZ\<c<i-IJIΓ 1

0

j-i-I B<DFEDHG-cG-UTi-I-XaKJwJK t ";I j-i-I^c\<YZd6c<G-DHGJI-I dux<cHc IJ q j-i6UBad-[- TIZpj-Ui6UVDN\<i-c<BXHIJXM\<DMVl|c<G6U " DHXN\<DMXM\<BDHG-G-c "i-Uai q UDM\X%[K S DN\PUPdf<GJlzDi-UXHW-DN\<l x<I-m q i-Y;Uk-I c<G-GJlCp EI-UHi6UaVVsE6d6[ XMI-XM\DHV wZK O [ ¢I wJK Za j-cHE\<BDHiEIZdIIZp(j-cPdG q Y;U W-DMXM\<BDHG-G q HYJB<I-ByUPd6DHGJ\G-c<XM\f-K©nU`c<XMG-c<ByUaG-I-I M\c<HcVc ¡ G-c q \B<DHi ¡ EUa\<f "-WJ\<cXM\i q YJ\ q i6UIXHB<cr-XM\ByU p2UcXU'B'XMI-XM\DHVD wJK O [ *j-cadGJc<XM\f< c<j-iJDeEDed6[- \<X%[ mUY \<cVX q TDHXM\B<c<ByUaG-IJ[bjJDM\dIXHDHj-Ui6Ua\i-I-XHl"XMDeE6d6c mnc<Y q XUJK Y2XHj-DMi-I-VDMGJ\PUPdf<GJlzD1I W-I-XedDHGJG-lzD_I-XHXedDeEcByUG-IJ[ q x<DFEIJ\DNdf<GJc EcYZUPRUPdI

B<cHVc ¡ G-cXM\<fnMDHG-DMi6Uk-I-I¢p2Uca\<I-W-DMXHY-I;p UaB\<c<Y2cad6DHxyUGJI-r i-UaMdIJW-G-c<r XM\i q YJ\ q i-lI=BRUaI-Vc<XHBa[Jf MmmnDMYJ\PU EDM\<DMi-VI-G-IJi-c<ByUGJG-c<Hc_p2UcXU_Xj-DM\dDMr XMDHj6Ui-Ua\<i-IJXHlXHDFE6dc mnc<Y q XHUJK AsYZURW-DMXM\B<Dj-i-I-VDMi6U1G6Ui-I-XaK wJK w " = j-iJI-B<DeEDHG-l j-i-c<DMY-k-I-I8p2U c\I-W-DHXMY-IZp§\<i6UaDHYJ\c<i-I-r G6U_jZdcXHY2c<XM\f

(x, y)"c\<BDHW6U I-D \PUY G6UPlzByUDMVc<V qIOBA@F = XDJ ?DS EKABO 9Ua\a\<i6UaYJ\<c<i-UJK 9i-I-VDMi WAB?BE S@S+ S =WEKEGFL=t E S FL= j-i-I-BDeEDMG

G6U i-IJXK6wJK w " RoK6©nU M\<cV ¡ Dui-I-X q G-Y2Dnj-i-DFEXM\PUaBadDHG-U j-DN\d6[_XHDMj6Ui6Ua\i-I-XMlXHDFE6dcmnc<Y q XU Γ 1

0

K ACI;EGJc"WZ\<c1p2Uac\<I-WJDHXHYJI-D\<i6UDMYJ\<ci-I-I XHj-IJi6UPdf<GJc<Hc=Ua\a\<i6UaYJ\<c<i-Uj-i-DFEXM\PUaBad6[- \ XHc<x<cr Y;UY x<l i6UaVl \lzr _EB q p;\PUYJ\<GJlzrLk-IJYZd i-IJXK wZK w " = "U BI-GJ\<cB<c<HcÛa\\i6UYJ\c<i6U i6UPVl \ q TjJDM\d XMDHj6Uai6Ua\<iJI-XHl

Γ 10

K6¨'Y;UaUG-G-lzDi-DN q dfR\PUa\<l x<lCd6I§j-cPd q W-DHG-l W-I-XNd6DHG-G-cJK EG-UY2cB<XMD¢c<G-I8B<c<XHjJi-c<IJHB<cHE[J\X%[ BmnIJIJW-DHXMY;cV HY2XMj-DHi-IJVDHGJ\DK niJI M\<cV§I-VDMDM\VDHXN\<c q EI-B<IZ\<DNd6f<G-c'pyc<iJc! DMD XMcc\B<DM\XM\<BI-D MY2XHj-DMi-I-VDHGZ\PUPdf<G-lCp1IW-I-XedDHGJG-lCpi-DM q dfR\PUa\<c<BJKA YZURW-DMXM\B<Dnj-i-I VDHi-U`G6Ui-I-XK;wJK 9j-i-I-B<DFEDHGJl¤mca\<c<Mi6UmnI-IUa\a\<i6UaYJ\<c<i-U "ZjJcad q W-DHGJG-c<HcTB MY;XMj-D i-I-VDMGJ\<DaKA RUY;duW-DMG-I-D c\VDM\<IJV XNdDFE q TI-r^ByU ¡ G-lzrmUYZ\oKyCcadf I-G-XN\<B<ci-DN q df

\PUa\c<B"Rj-cad q WJDHG-G-lCpX]I-XMj-cadfac<ByUaG-I-DHVHDMG-DHi-Ua\<c<i-U "oc\<G-cXHIJ\X%[TY9VcHEDedI wJK O X B<I;EcV G-DedI-G-DMr-G-c<r m q G-YJk-I-I wJK ORQ j-i-I d = 0

K \<c<W-YJI iJDHG-IJ[ mnIJHI-W-DHXMY2cHc I1W-IJXNdDMG-G-c<McHY2XHjJDHi-I-VDMGJ\<cB N\<c cj-i6UBPEUG-c-K EG6UaY;c B^V]Ua\DHV]Ua\I-W-DMXHY2c<VjZdUaG-D DMXM\fG-DMY2c\<c<i-Ua[ cXHc<x<DMG-G-c<XN\<f-K~ q G-Y-k-IZ[ wJK ORQ uB_G q dD¢\<DHiJj-IJ\bi-Uai-lzBj-i-cIJB<cHEGJc<r "6\oK DaK6[-Bad6[JDM\X%[1G-DM dUoEY2c<r K \c^c<x<XN\<cR[J\DNdfXM\<Bc j-i-I-BcEIJ\ Yiy[ E q V]Ua\DHV]Ua\I-W-DMXHY-IZpTc<Xedc ¡ G-DMG-I-rKPClCd6c q XM\PUG-cBadDHGJc"aWZ\<cucXHG-c<BG-lzD XMB<c<r-XN\<ByUHDMG-DHi-Ua\<c<i-UXNdUx<c RUaB<I-X%[J\c\\PUYJIZp=V]Ua\DHV]Ua\I-W-DHXMY-IZp§\<c<GJY;cXM\<DMrKA"W6UXM\G-cXM\I "oHDMG-DHi6UP\<c<iEDHVc<G-XN\<i-IJi q DN\9B<DHXMf XMj-DHYZ\<i XMB<c<r-XN\<B"RDMXNdI`j-i-DeEXM\PUB<IZ\<f'm q G-Y k-I-

Φ(x)B BI;ED HY2XHjJc<G-DHGZ\<l

Φ(x) = exp(x) − 1. wJK Z t

= R f/Y+.( :d8/Yh8(*+-X./ M/d8hNhYf*d8.hN/Yfd X-r(f*dg`Y'/ M/ hY(r*d 'Dd r`*/8X./YXh (x, y)

6 M0dg/YhY(*+X-.(xd8hYhNf*dg.hN/8fWT8('*hY/8TN/ M/AhY(rDd,(xr*+-hg` WX-+-rDdgf*d8hYf(*X X+IR*`/8S u/8. i XMd

\PU m q GJY-k-IJ[ [JBad6[-DN\<X%[ dUoEY2c<r "UaG6UPdIJ\I-W-DMXHY2c<r I Vc ¡ DM\ X q XHj-Depyc<Vsx<l \<fI-XMj-cadfHc<ByUG-U j-iJI\DHc<i-DN\<I-WJDHXHY2c<VUG-UPdIJHDXMB<c<r-XN\<B XHI-XN\<DHVl wJK O NK

9XHXedDeEcByUG-IJ[j-cYZUPRUPdI "zWJ\<c B(BcV q DMG-G-c<rLXHI-XN\<DMVD wZK Za X Φ(x) =exp(x) − 1

\PUaY ¡ D¢i-DUodIJ q DM\X%[ c<XHcx<c<D¢i-DTDHG-IJD^B_B<I;ED¢j-DN\dI XMDHj6Ui-Ua\<i-IJXHl9KDN q dfR\PUa\l i6UXHWJDM\<cB jJi-DeEXN\PUBadDMG-l G6U i-IJXKzwJK ] KW UYJI-V c<x<i6UaHc<V "CMDHG-DMi6U \<ci G-I TDHG-Y2c XM\PUopyc<ByUi-DHUPdIJ q DN\iJD ¡ I-V(EDM\DHi-VIJG-I-i-c<B<UG-G-c<Mc p2Uc<XHUÎbBXNd q W-UDUoEUGJIJ[G-DNdIJG-DHr-GJc<HcEDM\DHYJ\c<i6U^B¢XHcc\<B<DN\<XN\<B<I-I1X wJK Z tNK

f/Y+.((dgQ/8TY/ LhNfd8+.hN/Yf1(( X-r(*fdg`8'*/ M/ dghNhNf*dg.hN/8f*dA'Ddr`/YX./YX-hY( r*+f+q +-''(x, z) l6 ( (x, y) #(*Q-(*+X-.( .X-r+f(q3+I'*h80 m = 1.5 0 g = 0.2

+-hg` X+-r*d8f*dghNf(X X-+IR*`/8S u/Y. i XMd0&f+Mdg`(*Q i #)cd X TWTN/YQ-q i ),+-''*/ X-(*X-hN+q + " XΦ(x) = exp(x) − 1

*.

cHEDedf MDHG-DMi6Ua\<ci6U GJI DMG-Y2c XN\PUPpycByU [-Bad-[-DM\X%[ jJi-c<XM\DHr TDHr EIJG6UVI W-DMXHY2c<rXHI-XN\<DMVc<rXO" t(XM\DHj-DMGJ[-VIXMB<c<x<cHEl " j-i-DMY-i6UXMG-c¤IZd-d XM\i-I-i q TDHrXM\i q YJ\ q i q I¤XMB<c<r-XN\<ByU \PUaY¤G6UalzB<UDHVc<Mc XHjJI-i6UPdfG-c<Hc pyUc<XHUJK j-IJi6UPdf<GJlzrp2Uac<X B<cHG-I-Y;UDN\B¢EI-mnmnDHi-DMG-DHk-I-UPdf<G-lCp XHI-XN\<DHV]Uop "iJDUPdIZ q TIZp=c<XMc<x<c<Dì-D DMG-I-D9B BI;ED'j-DN\dI_XHDMj6Ui6Ua\i-I-XMl XHDFE6dc mnc<Y q XHU IZdI XMDeE6d6c q MdUJK;Sd6[^\PUaY-IZpXHIJXM\<DMV B<cVc ¡ G-c<XN\<fpyUc\I-W-DHXMY2c<rÊI-G6UVIJY-I^XedDeE q DM\¢IJuIJBDHXM\G-c<r \DHc<i-DMVl IZd6f<G-I-Y2c<ByUZK #on©i-DUPd6IJ q DM\`B<XHD IJHB<DHXN\<G-lzD VDNpyUG-IJHVl B<cHG-I-Y-G-cB<DHG-IZ[TED \<DMi-VI-G-IJi-c<ByUGJG-c<Hcp2Uc<XHU 2yY;UXHY;UoE~DMr-HDMG-xyU q V]U "2j-DHiJDHVD ¡ UDHVc<XN\<f i-UaMdIJW-G-cHc \I-j6U " BYZduW-Ua[ j-DMi-DHVD ¡ UDHVcXM\<f pyUc<X <p2Uc<X "j-c<iy[;EcY Ui-Y2c<B<XMY;cHc-KAG-DHUB\<cG-c<VG-c<V"iJD ¡ I-VD#,n© EDMVc<G-XM\i-I-i q DM\§B<XHD¢\<IJj-I-W-G-lzD MmnmDMYJ\<l "c<x q XNd6c<BadDMG-G-lzDj-DHi-DepycHEc<VY_pyUc<X q W-DHi-DNTYJByUaIJj-DHi-IJcEI-W-DHXMY-I-D`Y;cPdDHxyUaG-IJ[ i6UP i q DMG-I-D EB q VDHiJG-c<Hc \c<i6U " q EBc<DHG-IJD_\c<i-c<B I Ei K eKA U ¡ G-c<r cXHc<x<DMG-G-c<XN\<f<#,n© [-Bad-[-DM\X%[§\c"*WJ\<cB<XMD j6Uai6UVDM\i-lªXHIJXM\<DMVlªI-VDH \[JXHG-lzr§mnIJI-WJDHXHYJI-rXHVlzXed IbEc<j q XHY;U \bIJHVDHi-DMG-IJ[§BiJDUPdfG-c<V HY2XHj-DMi-I-VDMGJ\<DaK 9XHXNd6DeEc<B<UG-IJ[JVEI-G-UVI-Y-I #on©YZUaY BnW-I-XedDHGJG-c<V "y\PUYÎ BnG6Ua\ q iJG-c<V MY;XMj-DHiJI-VDHGJ\D'j-c<XMB[ TD G6U j-i6UYZ\<I-W-DMXHY-IjJcadG-c<XN\<f< Vc<G-cHi6UmnIJ[ Q "RY2c\<c;>ABE * FL=t USTEGFHC ?BADC8=<E ST?DS *?B= ;>A ABOB=@EKAW?B= ;>AB?B=*A C1 = A IE C "= WAB;<C ?DC XDAB?DCB? STR ?DS@CI?B? ;>A ,)CPFHC ? BAB= XDJ ? FL=&W?DCI?BA CIE.@CQFDE S+ S ODS F0B;5 =sW SD =<: <=\C E M) E S A IE C *= *S , CIE A *CIEKJ FHC# C$?BADC !WAB;<C E S A ; = E STEK? LSYX C R =@?BA M LSTE SYF S>E.WC =\C E; *CPFD? E S F# FDAWS>;>ABE *'R>A F = BADS>?B?B= ;<AB=&+ FL= *= IA E C A ALXX IEGFDA@FLC E @S : *S , ?DS5IEKJ FHC = XDA :<=&BABA = 1 =@;>= R>A F = BA ; WS>CI?BAE S FL=s

Mk8k

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x(t) = x[φ1(t), φ2(t), . . . , φp(t)], ZK O

EDφi(t) = ωit

"i = 1, 2, . . . , p

KA i-DM q dfR\PUa\<D x(t) ZK O I-VDHDN\_j-DHi-IJcE 2πj-c

Y;U ¡ Ec<V q IZUi- q VDMGJ\<c<B φi(t)"6U XUaV(Y-B<UaI-jJDHi-I-cHEIJW-DHXMY-I-rbj-iJc<k-DHXMXnBc<x! DMV

XNd q W-UD9G-Du[JBad6[-DN\<X%[_j-DMi-I-cHEI-WJDHXHYJI-VzK-\oK DaK x(t) 6= x(t+ T0)K j-DMYJ\<iVc! GJc<XM\I

Y2cadDMxyUG-I-rx(t)

B c<x DHV¤Xed q W6UD`B<YZd W6UDM\^B<XHD`G-DMUB<I-XMI-VlzD W6UaXM\<ca\<l ωi

"IZpeUi-VcG-I-Y-I

nωi

n = 2, 3, . . .

I Y2c<Vx<I-G6Uak-I-c<G-GJlzDW6UXM\c\<l\<IJj6Unωi ±mωk

" ED

nImkJDNdlzDj-cadc ¡ IJ\<Dedf<G-lzDW-I-XedU "U

i, k = 1, 2, . . . , pK

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Y2cadDMxyUG-I-XnGJDMRUaB<I-XHIJVlzVI W6UXN\<c\PUaVIx q EDM\(c\B<DHW6UP\<f n VDHiJG-lzrL\c<iK G-UPdIJ9jJi-c<xadDMV q XN\<c<r-WJI-B<c<XN\<I "6x<IJm q i-Y;UkJI-rI j-DMi-DNp<cEc<BY p2Uac<X q j-i-I-VDMG-I

\<Dedf<G-cuYTY-ByUPI-j-DMi-I-cHEI-WJDHXHYJI-V=Y;cPdDHxyUaG-IJ[-V1XHB[ZRUG`XIZ q W-DHGJI-DHV=x<I-m q i-Y;Uk-IJr

n VDHi-GJlCp \<ci-c<B-K \<IRUREURW-IG6UTj-iJc\R[ ¡ DMG-I-I_VG-c<MIZp dDM\cXM\PU \<X%[j-i-DFEVD \<cVI-XHXedDeEc<ByUG-IJr XHjJDHk-I6UodI-XM\c<Bbj-cG-DedI-G-DMr-G-c<r EI-G6UVIJY;D I=\ q i-x q dDMGJ\<G-c XM\IK$9c<XNdD¢j q xad6I-Y;Uk-I-I§i6Ux<c\ed6[8I UY2DHGJXU_I-GJ\<DMi-DHX Y=I-XHXedDeEcByUG-IJ[JVY-ByUPI-j-DMi-I-cHEI-WJDHXHYJIZp_Y2cadDMxyUG-I-r^X W-IJXNdc<V

ωi

"i− 1, 2, 3, . . . , n

"n ≥ 4

"2Bc<j-i-D EDedDHG-GJc<r_XM\<DMj-DHGJI q VDMG-f! I;dX%[K \cXHB[ZRUG-cTX Ec<Y;UaUa\<Dedf<XN\<B<c<V G-D q XM\c<r-W-I B<c<XN\<I EB<I ¡ DHG-I-r G6Uu\i-DNp I¢W-DN\<lzi-DepJVDMi-G-lCp^\<c<i6Uop K<A \c ¡ DzB<i-DHV [¢iy[2E i-Ux<c\XHBI;EDM\DNdfXM\<B q DM\`cuX q DHXN\<B<cByUG-I-I q XM\<cr-W-I-B<lCp Y2cadDHx<UG-I-r X V [Ì w G-DNRU B<I-XMI-VlzVIW6UXN\<c\PUaVIK \c\9mUYJ\uG-DCj-caB<cad-[-DM\uI-XHY;duW6UP\<f9IJCi6UXHXMVc\<iJDHG-IJ[HIJj-c\<DN q ÙG;EU q c_j-DHi-DepycHED¢Ybp2UcX q W-DMi-DM i-D ¡ IJV\c<i-c<B_B<lzXMc<Y2c<r=i6UaHVDHi G-c<XN\<IK cadDHD \<c<Mc"-iy[;EG-DMi-DTDHG-GJlCp EcY2c<G-k6U j-iJc<xadDMV cXM\PUDN\<X%[Ec XHI;pj-c<iI j-iJI-VDHG-IZ\<DNd6f<G-c Y EB q VDMi-G-c<V q \<c<i q K \VDM\<IJV " G6Uj-iJI-VDHi " x<I-m q i-Y;Uk-I-

Mk

q EB<c<DMG-IJ[(EB q VDHi-G-cHc \<c<i-U "CB<jJDHi-B<lzD G-Uxad EUaB! q uX%[UaB\<c<iJc<VªI¤j-ceEG-DHDVG-c<MI-VI¢I-XHXedDeEc<ByUa\DNd6[JVIKaSTc'XHIZp j-c<iB<cj-i-c<X c9xI-m q iJYZUak-I-c<G-GJc<V=VDNp2UaG-IJ VD q EB<c<DHGJIJ[8cEG-c<HcIJ j-DHiJI-cHEc<B_EB q VDHi-G-cHc\c<i6Uc<XM\PUDN\<X%[ GJD jJcadG-c<XN\<f<[-XMG-lzVzK c ¡ G-c jJi-I-B<DMXM\<II Ei q Hc<r¤j-iJI-VDHiKSTc XHI;p j-ci c<XM\PUaDM\<X%[c\Y-i-l \<lzV i2[2E j-i-c<xad6DHV "XHBa[JRUaG-G-lCp=X HmnmnDHYZ\<c<VLXHI-GZpZi-cG-IJRUak-I-I=EB q pJW-UXM\c\<G-lCpY2cadDMxyUG-I-r KSd6[8i-DTDHG-IZ[ q YZUPRUG-GJlCp I8iy[;EU Ei q HIZp B<cHVc ¡ G-lCp=UoEURW GJDHc<xpycHEIJVc

I-VDN\<f=B i6UXMj-c<iy[ ¡ DMG-I-I j-iJc<XM\ q gxyUPc<B q |VcHEDedf HDMG-DHi-Ua\<c<i-U8UaB\<c<GJc<VG-lCpEB q pZW6UXN\<c\G-lCp Y2cad6DHxyUGJI-rK*hnUY M\<cj-i-IJGJ[J\c1B\<DMc<i-I-I§Y;cPdDHxyUaG-I-r "*G-DMc<xpycEI-Vc B<B<DMXM\<I1Bî6UXMXHVc\i-DHGJI-DEI-G-UVI-W-DMXHY q XMI-XM\DHV q "I-VDH q i-D DMG-I-D BB<I;ED q XM\c<r-W-I-BlCpbEB q pZW6UXM\c\<GJlCp Y2cadDMxyUG-IJrK G6U^[-Bad-[-DM\X%[=\PUY ¡ D¢G-DMc<xpycEI-Vcr "6Y;UY "6G6UjJi-I-VDHi " q i6UaB<G-DHGJIJ[1MDHG-DMi6Ua\c<i6UÂCUGÊDHincPd6[Ê6d6[IJ q WJDHG-IJ[j-i-DFEDNd6f<G-lCpTk-I-YZdcB-KPA8G6UaXM\<cR[ DHrTd6DHY-k-IJIEUDN\<X%[ mnIJHI-W-DMXHY2c<DCcx<c<XHGJc<ByUG-IJDIB<B<cHEIZ\<X%[=B i6UaXHXHVca\<i-DMG-I-D¢xyUaHc<ByUa[1VcEDNdfMDHG-DMi6Ua\<ci6UY-ByUaHI-j-DHiJI-cHEI-W-DMXHY-IZpY2cadDMxyUG-I-rXuEB q V [_G-DMUB<I-XMI-VlzVIbW6UaXM\<ca\PUVIK

- )+. - * ) * , *$ -

UaXHXHVca\<i-I-V1BzYZURW-DMXM\B<D IJX pycHEG-c<rÌJBDHXM\G q (VcHEDedf HDMG-DHi6UP\<c<i6U GJI DMG-Y2c XN\PUPpycByU "JX p<DHV]UTY2c\c<i-c<Mc j-i-DFEXM\PUaBadDHG-U¢G-Uì-I-X q G-Y2D9wJK Z XMVzKy`DMY-k-I- w NK#;D G-DMi6Ua\<ci8BYZduW-UDM\_YZdUXHXMI-W-DHXMY-I-r8HDMG-DHi6UP\<c<i8A UG_EDHi 9cad6[ "B Y2c\c<i-c<VB<BD EDMG6UnEc<jJcadG-IJ\DNdfG6Ua[ IJG-DHi-kJI-c<G-G6UP[ c<x<i-Ua\<G6UP[^XMB[JHf-Ky¨'i6UBG-DHG-IZ[ HDMG-DHi-Ua\<c<i-Uj-i-DFEXM\PUaBad6[- \¢XHc<xc<r \i-DNpZVDMi-G q LEI-G6UaVI-W-DHXMY q XMI-XM\DHV q X \i-DHV [^j6Ui6UVDN\i6UVI/2

x = mx + y − xz − dx3, y = −x, z = −gz + gΦ(x). ZK Z

9DHi-BlzD EByU q i6UB<GJDHG-IJ[ XHI-XN\<DHVl ZK Z cj-I-XHlzB<U \¢MDHG-DMi6Ua\<ciACUG EDHi 9cad6[ KAM\<cV dDMHY2c q x<DFEIJ\<fX%[ "j-cadc ¡ I-B z = 0

"-II-XHjJcadf q [ Φ(x) = x2 K6hnUY_x<lCdcj-c<Y;UaUG-c" XHI-XN\<DHV]U ;K Z ni-DUodIJ q DM\§j-DHiJDNpycHE Y p2Uc<X q B XMc<c\<BDM\<XN\<B<IJIX^\DHci-DMVc<r IZd6f<G-I-Y2c<ByU j-i-I q XNdc<BI-I " WZ\<c G-DedI-G-DMr-G6Ua[1m q GJY-k-IJ[ Φ(x)

[-Bad6[JDM\X%[UXMI-VVDM\i-I-W-G-crbc\<GJc<XHIJ\DNdfG-c^j-DMi-DHVDMG-G-c<r

xIRUoEUDM\X%[1B B<I;ED2

Φ(x) = I(x)x2, I(x) =

1, x > 0,0, x 6 0,

Φ(x) = exp x− 1.

ZK Q

¨]XNd6c<B<IJ[UXMI-VVDM\i-I-I ZK Q Cc<x<DMXHj-DMW-I-ByU \^X q DHXN\<B<cByUG-I-DTB XMI-XM\DHVD ZK Z c<XMc<x<c<Mc9i-DTDHG-IJ[ B9B<I;EDzj-DM\dI XHDMj6Ui6UP\<i-I-XMl¤XHDFE6dc mcY q XHUnI "<Y;UY¢XNdDFEXM\B<I-D!"i-DHUPdIJUk-I-|iJD ¡ I-V]U1XHjJI-i6UPdfG-c<Hcp2Uc<XHUJK mIZI-W-DMXHY2c<r \<cW-Y-I iJDHG-IJ[ M\<c

Mk

Ec<XN\<I-eUaDM\<X%[(RUbXMW-DM\8XUaVc<XHc< dUXHc<B<UG-G-c<Mc=B<cNEDHr-XN\<B<IJ[ G6UbcXHG-c<BG-c<r q XMIZdI \<Dedf XMc XN\<c<i-cG-l"c<x<i6UP\<G-c<rXHBa[JI "UoEUGJG-c<r_\<i-DN\<f<I-V q i6UBG-DHG-IJDHV B XHI-XN\<DMVD ;K Z eK 9i-I V]UPdlCp UVjZd6IJ\ q EUPp XMI-HG-UPdU x(t) M\c`BcNEDMr-XM\B<I-D G-DMHG6URW-IJ\<Dedf<G-c "IXHI-XN\<DHV]U ZK Z MDHG-DMi-I-i q DM\¤jJi-DeEDedf<G-lzrLk-I-YZdK i-c<XN\<c<V j6Ui6UaVDM\<i-U B<ca x q ¡ EDHG-IZ[ m I-GJ\<DMG-XHIJB<G-c<XN\<f8Y;cPdDHxyUaG-I-r x(t) i6UXM\DM\ "XMI-HG6Uod c<x<i6Ua\G-c<r XMB[ I

z(t)G-Ui6UXN\PUDM\ \<c ¡ DIx<cPdDHDUYJ\I-B<G-c G6URW-I-G-UDM\ BadIJ[Z\<f G6UY;c!HmnmnI-k-I-DMGJ\

q XMIZdDMG-IJ[ c<XHGJc<B<G-c<Mc q XMIZdIJ\DNd6[ K I-XM\DHV]U9i-DHUPdIJ q DM\j-cXNdDFEc<ByUa\DNdfG-c<XM\fnx<I m q i-Y;Uk-I-r q EB<c<DHGJIJ[j-DHi-IJcEU k-IJYZdc<B IjJDHi-DepycHE=Yp2UcX q K k-Dedf< cx<DHXMj-DHW-DMG-IJ[GJDMRUP\ q p2U IZp EB q pZW6UXM\c\<GJlCp_Y2cadDHx<UG-I-rB`XHI-XN\<D

V q ;K Z CGJDHc<xpycHEIJVc^B<B<DMXM\INd6DHVDHGZ\ "pyUi6UYJ\DHi-IZ q DMVlzr=XHc<x<XN\<B<DMG-G-c<rW6UXN\<c\<cr "c\dI-W-U TDHr-X%[¤ca\=i-DMHc<G6UG-XMG-c<r(W6UXM\c\<l Y;cGJ\ q i6U8HDHGJDHi6Ua\c<i6UJK EG-IJVIJ^B<cVc ¡ G-lCp8XHjJc<XHc<xc<B[JBad6[-DN\<X%[ I-XHj-cPdfcByUG-I-D Y2cadDHx<Ua\<Dedf<G-c<McbY2c<GJ\ q i-UB8k-DHj-I Ec<jJcadG-IJ\DNdfG-c<rcx<i6Ua\G-c<r XHB[ZIK]©9DHcxRp<cHEI-Vc8X EDNdUP\<f§\PUaY "]WJ\<c<xlXHIJHG6UPd1c<x<i-Ua\<G-crXMB[JHI

z(t)jJi-DeEXN\PUBad6[;dbXMc<x<c<r_Y2cadDMxyUG-IZ[G-DMUB<I-XMI-Vc<rW6U

XM\c\<l " Y2c\<c<iJlzDx q E q \ VcE q d6I-i-c<ByUP\<f8Y;c!HmnmnI-k-I-DMGJ\ q XMIZdDMG-IJ[ I c<x<DMXHj-DMW-I ByUa\f YJByUaIJj-DHi-IJcEI-W-DHXMY-I-DUaB\<c<Y2cad6DHxyUGJIJ[K UaXHXHVca\<i-I-VX p<DHVl " jJi-DeEXN\PUBadDMG-G-lzDG6U i-I-XaKZK O<K©nU iJI-XK ZK O" =nj-cYZUPRUG6U

X pyDMV]U8IJG-DHi-kJI-c<G-G-cHc=Y;UXHY;UoEUEc<j-cadGJIJ\<Dedf<G-c<r c<xi6Ua\<GJc<r XHB[ZI HDMG-DHi-Ua\<c<i-U GJI DMG-Y2c XN\PUPpycByUJK9hnUXMYZUREj-i-DFEXM\PUaBad6[-DN\XHc<xc<r

RC kJDHj-c<WJY q " c<j-IJXHlzByU

DHV q cHEGJc<VDHi-GJlzVªEIJmmnDMi-DHG-kJI6UPdf<GJlzV q i6UBG-DHG-IJDHV \i-DM\f<D q i6UB<G-DMG-I-D BXHIJXM\<DMVD ZK Z eKJ©nUTi-IJXKZK O" R jJi-DeEXN\PUBadDMG6U¢X pyDHV]U¢G-c<Bc<Hc IJG-DHi-kJI-c<G-G-cHc¢Y;UX Y;UoEU "oY2c\c<i-lzrTBYZduW-UDM\uY;cPdDHxyUP\<DNd6f<G-lzr Y2c<GJ\ q i G-DHY2c\c<i-c<rTi-DMHc<G6UG-XMG-c<r`W6U XM\c\<l9K¨'i-UB<G-DMG-IJ[ "6cj-I-XHlzB<U TI-D`X p<DHV q G6U¢i-I-XaKZK O!" R "-I-VDM \ B<I;E+2

z = φ, φ = −γφ+ γΦ(x) − gz, ZK

EDγ8j6Uai6UVDM\i RUa\ q pyUG-IJ[ "U g 8j6Uai6UVDM\i " j-iJDeEXN\PUBad6[J IJrG-c<iJVI-i-cByUGJG q i-DMHc<G6UGJXHG q W6UXN\<c\ q G-cB<c<Hc§mnIZdfR\<i6UZK ©9DM\i q EG-c q x<DeEIZ\<f<X%[ " WJ\<c

C00RNonlinearConverterx(t) z(t)

ϕ(t)NonlinearConverterx(t)

CL 11

1R R $ N+q3dm('+f(*/g''*/ M/L.1d8X.1dJRDdLR*/8r*/g`'(*hN+-`Y'/ / 8f*d8h8'*/ X-T 1Q-( l6 T +-'*+fd8hN/8f+%'(*),+-'.1/gSF%#Xh8d :/gT:d0 T M+I'*+fd8hN/8f+.1T:d8QI(r*+-f(*/R(*+-X.( ./8`*+ Ndg'(

Mk

q i6UB<GJDHG-IJ[ ZK cj-I-XHlzB<U \§EI-XHXMI-j6Ua\I-B<G-lzrY2cadDHx<Ua\<Dedf<G-lzrY2c<GJ\ q iB§i-D ¡ I-VDnB<lzG q ¡ EDHG-G-lCpY2cad6DHxyUGJI-r2

z + γz + gz = γΦ(x). ZK t

ACB<DFEDHV G-c<B q "cx<i6Ua\G q "XHB[ZfB¢HDMG-DHi6UP\<c<i i-IJXK6wJK Z \PUaY "-WJ\c<x<l q j-i-UBd6[J IJrXMI-HG6Uodbc<xi6Ua\<GJc<rXMB[JHI_j-i-DeEXM\PUBad-[Zd XMc<x<c<r

z(t) = φ(t)KZ¨'i6UBG-DHG-IZ[

G-c<Bc<Hc¢HDMG-DHi-Ua\<c<i-U^x q E q \ I-VDM\f^B<I;E 2

x = mx+ y − xz − dx3,

y = −x,z = φ.

ZK w

φ = −γφ+ γΦ(x) − gz.

I-XN\<DHV]U ZK w [JBad6[JDM\<X%[LG-DedI-G-DMr-G-c<r EI-XMXHI-j6UP\<I-B<GJc<rEI-G-UVI-W-DMXHY2c<rXHI XM\DHVc<r i6UPVDHiJG-c<XM\I

N = 4I§p2Ui-UYJ\<DMi-IJ q DM\X%[ W-DN\<lzi-fV [ q j-i6UBad-[- TI-VI

j6Ui-UVDM\i6UVI2mj-Ui6UVDN\<i=B<cHx q ¡ EDMG-IJ[ " d j6Uai6UVDM\i=G-DNdIJG-DHr-GJc<r1EI-X XHIJj6Uk-I-I "

γ j6Uai6UVDM\i RUa\ q p2UG-IZ[_I g ¢j6Uai6UVDM\iI-G-DMi-k-I-c<GJG-c<XM\I_mI;dfR\<i6UJK q DMXM\<BDHG-G-lzVI8j6Ui-UVDM\i6UVI1XMI-XM\DHVl ;K w C[JBad6[- \<X%[EByU 26j-Ui6UVDN\<ibB<ca

x q ¡ EDHG-IZ[ MDHG-DMi6Ua\<ci6U m I8j6Ui6UaVDM\<i I-G-DHiJk-I-c<G-GJc<XM\I g "p2Ui-UYJ\<DMi-IJ q I-ri-DNc<G6UaG-XHG q sW6UXM\c\ q mnIZd6fo\i6UJK9i-I RUREUG-I-I

Φ(x)BXHcc\<B<DN\<XN\<B<I-IX ;K Q ]XHI-XN\<DMV]U ZK wI-VDMDM\ i-DTDHG-IZ[B

B<I;ED q XN\<c<rJW-I-B<lCp EB q pJW-UXM\c\<G-lCp Y;cPdDHxyUaG-I-rK niJI-VDHi q Y;UaUG-G-c<Mc i-D ¡ I-V]UIZd-d6uXM\i-I-i q DM\_i-IJXKZK Z K

# ) -* ! 1 && # #$%"!

©nU i-I-XaKZK Qj-iJDeEXN\PUBadDMG6U x<IJm q i-Y;UkJI-c<G-G6UP[EI6UHi-UVV]UXHI-XN\<DHVl ZK w G6UjZdcXHY2c<XM\IcXHG-c<BG-lCp q j-i-UBad6[J I;pj6Ui-UVDM\i-c<B m I g E6d6[ mIJY;XMI-i-c<B<UG-G-lCpG-UoWJDHG-I-r

γ = 0.2Id = 0.001

KC~ q G-Y-kJIJ[ Φ(x)B ZK w TRUoEUByUPdUaXHf B§B<I;ED

I(x)x2 ;K Q eK©nU8dI-GJI-I

m = 0B XHc<ca\<B<DN\<XM\B<I-IXbV [-MY2c<rx<I-m q i-Y;Uk-I-DMr G;Ei-cG-c<ByU

¦'cj-mU i-c ¡ EUaDM\<X%[ q XM\c<r-W-I-Blzr§j-iJDeEDedf<G-lzr8k-I-YZd T0" Y2c\c<i-lzr=j-i-I jJDHi-DMXHD

W-DMG-I-I x<IJm q i-Y;UkJI-c<G-G-cr§dI-G-IJI l1jJi-DM\DHi-j-DMByUDN\=x<I-m q i-Y;Uk-I- q EB<c<DMG-IJ[§j-D

i-I-cHEU ";G-U9dIJG-I-Il2x<I-m q i-Y;Uk-IJ q EB<c<DMG-IJ[ j-DMi-I-cHEUTj-iJDM\<DMi-j-DMByUDM\ k-I-YZd "2B<ca

G-I-Y I-r¢G6U d6I-G-I-Il1 i-I-XaKyZK Q NKaCI-m q i-Y;Uk-IJc<G-G6Ua[ d6I-G-IJ[ lt

c\<BDHW6UDN\ q Xedc<B<I-B<lCp<cEUbG6UDFEI-G-I-WJG q |c<Y-i q ¡ G-c<XN\<f j6UiJlªY;cVjZdDHY2XMG-c XMc<j-iy[ ¡ DHG-G-lCp V q df\<IJjZdI-Y;Ua\c<i-c<Bk-I-Y;dU

T0

I V [-HY2c<V q i-c ¡ EDHG-IJvEB q VDMi-G-c<Mc\c<i6U µ1,2 =

exp(±jθ) "xI-m q iJYZUak-IJ[§©9DHr-V]UiJYZUNK>XM\DHXN\<B<DMG-G-c"EB<I-eUP[-XHf_BPEcadf dI-G-IJI lt "

Mk

0 30 60 90 120 150t

-3

-2

-1

0

1

2

3

4

X

-2 -1 0 1 2 3X

-2

-1

0

1

2

3

4

Y

R

0 0,05 0,1 0,15 0,2 0,25f

-75

-60

-45

-30

-15

0

S(f)

, dB

01

11

f

f

f +f01 11f +f01 11

+-ac(*q .1T:d8QI(r*+-f(*/R(*+-X.( WRT i Dd8X-hN/YhY' ./8`+NdJ'( l6 TNf+q +-''*d f+MdJ`(QMd( 50 r*f/Y+.( dgQ/8TY/ M/r/YfhYf+h8d10<@ X-r*+.hNf>q /Y)>'*/8XhY(&" 'Dd +-'( #rDdgf*dgq3+-hNf/gT m = 0.06 0 g = 0.5 0γ = 0.2 0 d = 0.001

q Hcad θ x q EDM\ j-iJc<x<DHeUP\<f=VG-c ¡ DHXN\<B<c i-Uk-I-c<G-UPdf<G-lCp G6URW-DMG-I-r " c\<B<DMW6U IZpi-DNc<G6UaG-XUVG6U \c<i-DaKAY;URW-DHXN\<B<Dj-iJI-VDHi6U G-Uî-I-XaKZK Q^G6UGJDHXHDMG6Uc<xadUaXM\<fî-D cG6UG-XHU

θ = 1 : 4"c<Mi6UG-IJW-DHG-G-Ua[ dI-G-IJ[JVI XMDeE6dc q MdcB<lCp8x<I-m q i-Y;Uk-I-r8i-D

cG6UG-XMG-c<Hc1k-I-YZdUG6U\c<i-Dlr"*c<jJI-i6U Ua[JX%[ G6U_\<c<W-Y q A Y2c<i6UaHVDHi-GJc<XM\I Z K

ACl TD dI-G-IJIi-c ¡ EDMG-IJ[`\<c<i6Ultj-c<Y;UaUG6U dIJG-IJ[

lu"aj-iJI j-DHiJDHXHDMW-DHGJI-I Y2c\c<i-c<r

XHGJIJ q BB<DHi<pTG6Uxad EUDM\X%[ jJDHi-DepycHEYTpyUc<X q W-DMi-DM i6UPi q DMG-I-D Y-ByUPI-j-DMi-I-cHEI W-DMXHY-IZpbY2cadDHx<UG-I-rK©UdI-G-IJI

lcI-VDHDN\_VDHXM\cY-i-IJHI-X i-Uai q TDHG-I-D 'B<cGJI-Y

DMHc'G-UzdIJG-I-Ilup2Uac\<I-WJDHXHY2c<McnUa\\i6UYJ\c<i6UJKoI-G-IJ[

ldc

c\<B<DMW6UDN\'x<I-m q i-Y;Uk-IJIXNd6IJ[-G-IZ[1I_j-c<XedDeE q DMHc I-XMW-DMHG-c<B<DMG-IJ[1j6Ui-lLXHDeE-dc<B<lCpk-IJYZdc<B-K

Mk

0 0,15 0,3 0,45 0,6 0,75g

0

0,05

0,1

0,15

0,2

0,25

m

l

ll

ll

l

1

2t

cr

dc

Aul

( i f.1d(*/g''Dd WR(Dd f*d8q q3d f+ao(*q /8T M+I'*+fd8hN/8f*d+ γ = 0.2 0 d = 0.001 " l1,2

y`('((8( i f.1d( i R*TY/Y+-'( r+f(/R5d(*.`/8T0 lt ym`('( f/ga,RD+-'( phY/Yf*d0 lu ym`('( mf*dgQf i ,+-'( hN/8f*d10 lc yx`('( fd8Qf i ,+-'( dg/YhY(*+X./ M/ dghNhNf*dg.hN/8f*d0 lr yx`('((&05/ Mfdg'((T:d#)>(*+A/ 8`DdgXh f+-Q/8'*dg'*Xdc'DdBhY/Yf+ 1 : 4 0 ldc

y `('((x.f*d8h8' (.`/8T0 A y hN/*.1dc./Yfd8Qq +f1'*/8XhY( 0/8hYTN+Dd#)od i X-`/8T8( ψ = 1 : 4

# ) )- )&- - $%-

UmnI-Y2XHIJi q DMVLG-UoWJDHG-IJ[8j6Uai6UVDM\i-c<B g = 0.5"d = 0.001

Iγ = 0.2

I=i6UX XHVca\<i-I-V MB<cad k-I-|i-D ¡ IJV]U\<c<i-U1B1c<xPdUXM\I G6URW-DMG-I-r j6Uai6UVDM\i6U

mVD ¡

E q q YZUPRUG-GJlzVIdI-G-IJ[JVI ltIluKC©nU8i-I-XKCZK " =& +j-i-DFEXM\PUBPdDHG-l j-i-cDHY-k-IJI

Ua\a\<i6UYZ\<c<i-cB1G6UjZdcXHY2c<XM\fjJi-I§jJi-cpyc ¡ EDHG-I-I=\<cW-DHY8x<I-m q i-Y;Uk-I-r q EB<c<DMG-IJ[EB q VDMi-G-c<Mc1\<c<i-UJK x q EB<c<DMG-I-I WJDM\<Y2c XMB<I;EDN\<DNd6f<XM\B q DN\§XN\<i q YJ\ q i6U XHDHWJDHG-IJ[ q UGJYZUai-D!" U\PUY ¡ D UG6UodIJ Bi-DHVDMG-G-lCp iJDUPdIZRUk-IJr I IZp§XHjJDHYJ\i-c<B Vc! TG-cXM\IKCI-m q i-Y;Uk-IJI q EBc<DHG-IZ[_j-DHi-IJcEU\c<i6UTeEDHXMf XMc<c\B<DM\XM\<B q DM\ x<I-m q i-Y;Uk-IJ[q EB<c<DMG-IJ[_j-DMi-I-cHEUVcHE q d6[-kJI-I IZdIx<IJm q i-Y;UkJIJ[ q EB<c<DHGJIJ[j-DMi-I-cHEU¢k-I-YZdUBc\c<x<i6U ¡ DHG-I-I q UGJYZUai-DeK \<cW-Y-I iJDHG-IJ[\DHc<iJI-Ix<I-m q i-Y;Uk-I-rB<U ¡ GJlzV [-Bad6[JDM\X%[c\<BDM\G6U`B<c<j-i-cX!"

q EByUIJByUDM\X%[ dI Hi-McHEI-W-DMXHY-I-r8\<c<i=IZd6I8BxadIJHI=\c<W-Y-I=x<IJm q i-Y;UkJI-I§XMG6URW6UPdUI-VDMDM\VDHXN\<ci-DMHc<G6UG-XG-U\c<i-D " q EB<c<DMG-I-Di-DNc<G6UaG-XHG-cHck-I-YZdU "IJ¢Y2c\<ci-c<HcRUP\<DHV c<x<i6Ua q DN\<X%[ q EB<c<DHGJG-lzr \c<izSd6[¢ca\<B<DN\PU`G6U M\c\`B<c<j-i-cXzj-i-c<IJHB<cHEIZdX%[i6UXMW-DM\j-cadGJc<Hc(XHj-DMYJ\<i6U(j-c<Y;UaRUP\<DNd6DHrn[-j q G-cByU jJi-I j-iJcRp<c ¡ EDMG-I-I \c<W-DMYx<I-m q i-Y;Uk-IJr q EB<c<DMG-IJ[_\<ci6UJKhnUY B<I;EG-c=IJi-I-XaK ZK t" B8\c<W-Y;UPp x<IJm q i-Y;UkJI-r \<c<W-YJI" " B8G-cadf

c<x<i-U U \<X%[XHi6UP q \i-IXN\PUi TIZpj-c<Y;UaRUP\<DNd-[n[-j q GJc<ByU Λ1 = Λ2 = Λ3 = 0NK

CI-m q i-Y;Uk-IJc<G-G-lzr j-DHi-DepycHE p2Ui6UaYJ\<DMi-IJ q DM\X%[_XNdDFE q DMV IJHVDHG-DMG-I-DHV§XHI-MG6U \ q i-l"XMj-DHYZ\<i6U¢[Jj q G-c<B<XMY-IZppyUi6UYJ\DHi-IJXM\<IJW-DHXMY-IZp jJc<Y;UaRUa\DNdDMr2

Mk

f/8+.((Bd8hNhYf*d8.hN/Yf/8T>XI(*XhN+-q " (BX/Y/8hYTN+-hNXhYT i #)>( cX+ *+I'( i dg'.1d8f+ 'Dd#r`*/YXIS./YXh 0td hYdg.a,+A( X-r+.hNf q /Y)>'*/8XhY(t0&r*f1( (*Q-q3+I'*+-'(( r*d8fd8q +hNfd m R` Q-'Dd *+I'( d = 0.001 0γ = 0.2 0 g = 0.5

Mk

0.064 0.066 0.068 0.07m

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

ΛΛ

Λ

Λ

1,2

3

4

i

B C D

VPf*d #(*.LQMdJTY(*XI(*q /YXh8( XIr*+.hNf*d r*/8.1d8QMdghN+-`*+ 1r i '/8TNd /8h r*d8fd8q +hNfd m d = 0.001 0γ = 0.2 0 g = 0.5

T / 8`*d8Xh8(LQ-'Dd *+I'( r*d8fd8q +hNf/8T q +aoR i `('( 1qs( lt ( lu " 0#0 y hY/ *.1(8( i f.1d( i RTN/Y+I'( hN/8f*d

0, 0,−,− =⇒ 0, 0, 0,− =⇒ 0, 0,−,−torus2 torus3 torus2

UaXHW-DN\<l j-iJc<B<cHEIZdIJXHfnXzc<W-DMG-fnV]UPdlzV UMc<V§jJcj-Ui6UVDN\<i q m ∆m = 3×

10−6 IbXHBI;EDM\DNdfXM\<B q \bc^\c<V " WZ\<c_j-i-I j-iJcRp<c ¡ EDMG-I-I \<cW-Y-I x<I-m q i-Y;Uk-IJIi-c ¡ EDHG-IJDjJi-DeEDedf<G-cHc¢k-I-YZdU XHj-DMYJ\<ij-c<Y;UaUa\<DedDHr_n[-j q G-c<B<U 2 [ " " " G-DG6UxPd EUDN\<X%[ CI-m q i-Y;Uk-IJ q EB<cDHG-IJ[8j-DMi-I-cHEU_j-i-DN\<DHiJj-DHByUaDM\ Mi-HcHEI-WJDHXHYJI-r\<ci i-DMHc<G6UG-XMG-lCpk-I-Y;dc<B B WJI-XNdDMG-G-c<V HY2XHj-DMi-I-VDMGJ\<DG-DnG6Uaxad EUDN\<X%[ K9i-I-B<DFEDHGJG-lzD B<l TDEUaG-G-lzD^BlzW-I-XNd6DHG-I-r q x<DFEIJ\DNdf<GJcEc<Y;UalzB<U \ "WJ\<c

B<B<DFEDHGJG6Ua[Bì6UXHXMVc\<iJDHG-I-DnXMI-XM\DHV]U ;K w BTUB<I-XMI-Vc<XM\I_c\THG6URW-DHG-IJr q j-i-UBd6[J I;p j6Ui6UVDN\<i-cB`XMj-c<XHcx<G6U9BUB\<cG-c<VG-c<V8i-D ¡ I-VD i-DHUPdIJHc<ByUa\f q XM\c<r-W-I B<lzDCEB q pZW6UXN\<c\G-lzDuY2cad6DHxyUGJIJ[¢Xzi6UaNdI-W-G-lzVI G6URW-DMG-IJ[JVI W-IJXNdUnB<i6U TDHG-IZ[ "EDMVc<G-XM\i-I-i-cByUa\<f Y;UXMY;UoE x<I-m q i-Y;Uk-IJr q EB<c<DMG-IJ[ EB q VDHi-GJlCp \c<i-c<BI j-DMi-D pycHE Y_pyUc<X q j-i-Ii6Uai q TDHG-IJI1\<c<i-UJK

*.

A|dDHY-kJI-I B<BDeEDMG6U B i6UaXHXHVca\<i-DMG-I-D VcHEDedfMDHG-DMi6Ua\c<i6U EB q pJW-UXM\c\<G-lCpY-ByUPI-j-DMi-I-cHEI-WJDHXHYJIZp"UaB\<c<Y2cad6DHxyUGJI-r "zY2c\<ci q Vc ¡ G-c i6UXMXHV]Ua\i-I-ByUP\<fY;UYcHEG q IJ8xyUaHc<B<lCpVcHEDedDHr \DHc<iJI-IG-DedI-G-DMr-G-lCp Y;cPdDHxyUaG-I-rK cHEDedfj-iJDeE XM\PUaBad6[-DN\UB\c<G-c<VG q XMI-XM\DHV q WJDM\<lziJDNp¢c<xlzY-G-c<B<DMG-G-lCpnEI-mnmnDHi-DMG-k-I6Uodf<G-lCp

Mk

q i6UB<GJDHG-I-r XW-DN\<lzi-f<V [bj6Ui6UVDN\<i6UaVIK IJXM\<DMV]U i-DUodIJ q DM\ q XM\<cr-W-I-B<lzDYJByU IJj-DHi-IJcEI-W-DHXMY-I-DY2cadDHx<UG-IJ[ "ZY2c\c<i-lzD9XuIZVDHGJDHG-I-DMV(j-Ui6UVDN\<i6UEDMVc<G-XN\<i-I i q \Lx<I-m q i-Y;Uk-IJI q EB<cDHG-IJ["\c<i6U¤Ij-DHiJDNpycHElYpyUc<X q j-i-Ii6UaHi q DHGJI-IEB q VDMi-G-c<Mc\<c<i-UJK niJDeE6dc ¡ DMG-G6Ua[ VcHEDNdf(Vc ¡ DM\(x<l \<fI-XHj-cPdfcByUG6U E6d6[EDN\PUPdf<G-cHc1IZ q W-DHGJIJ[8G-DNdIJG-DHr-GJlCp XMB<c<r-XN\RP$, ;>=\C E ,)C E ? , ;<C ? ?DS AB?\FHST?BA :<= BABA ODCPFDADST;>A C A LSYXHCIR =W?BA )=t *=WEGFDAW=@CIE * XDAD?BADC Xs E" =&BA A ,)C EKA<?B= W?DC$#K?0 IAB?@FHST?BA :<=&BA R5K?DS5IEKJUA ?B;>= *CI?BEK= XDJ ?DS5IEKJ FHC : >XDJ EK=@E SALX X IEGFDA@F LCIE XHS , ? O@FBA UCQFHS AB? @FHST?BA :<=&BABAFDABE"= ICPFD;B= CPX S@C1 =s

k

I-GZpZi-cG-IJRUak-IJ[UB\c<Y2cadDMxyUG-I-r (cHEG-c IZ m q G2EUVDHGZ\PUPdf<G-lCp¤G-DNdIJG-DHr G-lCp [-BPdDHG-IJr j-i-I-iJcEl9KeHc`Vc ¡ G-cì6UXHXMV]Ua\<iJI-ByUa\f Y;UY j-i-cXM\<DMr IJrj-i-I-VDMiXUaVc<c<i-eUGJIJRUkJI-I BRUaI-VcHEDHr-XN\<B q IZpbXHI-XN\<DHVzK 9cE XHI-G;pJiJc<G-IJUk-I-DMr8cx<lzW G-c jJc<G-I-V]U \ q XN\PUG-c<Bad6DHG-I-DnG-DMY2c\<c<iJlCpXHcc\<G-c DHGJI-r_VD ¡ E q pyUi6UYZ\<DHiJG-lzVIB<i-DMVDHG6UaVI " W6UaXM\<ca\PUVI IZdI mUaUVI Y2cadDMxyUG-I-r j-Ui-k-I6Uodf<G-lCp XHI-XN\<DMVsBi-D q dfR\PUa\D I;p BUI-VcHEDHrJXM\<BIJ[K mnmnDHYZ\XMI-GZpZi-c<GJIJRUkJI-I "c\Y-i-l \<lzr #ur-MDHG XHcV D TB B<DHY2D "j-iJI-BadMYbY1XMDHx<Dc<XHcx<lzr1I-GZ\<DHiJDHX q WHGJlCp1BEByUoEk6Ua\<cVXM\cadDN\<I-I B9XMB[JHI¢X i-UaB<IZ\<I-DMV G6U q Y-I I¢\DNpZG-I-YJIK;A G-UXM\co[ DMD B<iJDHV [TEDN\PUPdfG-cbi-Uai6Uax<c\PUG6UXM\PUB! Ua[ YZdUXHXMI-W-DHXMY2c<r \<DHci-IJ[ XMI-GZpZi-c<GJIJRUkJI-I j-DHi-IJcEI-W-D XHYJIZp UB\c<Y2cadDMxyUG-I-r "yi6UXHXMVc\<iJDHG-l¤Xed q W6UI XHI-GZpZi-cG-IJRUak-I-I^YJByUaIJj-DHi-IJcEI-W-D XHYJIZp Y2cadDMxyUG-IJr I Y;cPdDHxyUaG-I-r Bbj-i-IJX q \<XN\<B<I-I(XNd q W6Uar-G-lCp m'd q YJ\ q Uk-IJr ORw KA¤i6UVY;UPp_Y;dUXHXMI-W-DMXHY2c<r\DHc<iJI-Ii-UaMdIJW6U \ B<lzG q ¡ EDMG-G q XHI-GZpZi-cG-IJRUak-I- "\oK DKCXMI-GZpZi-c<GJIJRUkJI-vUB\c<Y2cadDHx<UG-I-r¤B<GJD GJI-V XHI-MG6UPdc<V " I BUI-VG q XMI-G pZi-c<G-IZRUk-IJ " G6Uaxad EU q uX%[ jJi-I BRUaI-VcHEDHr-XN\<B<IJI EB q p UB\c<Y2cadDMxyUa\<DedfG-lCp^XHI-XN\<DMVzK-Ac<xc<IZp XNd q W-Ua[Zp^j-i-cR[-BPd6[- \<X%[ cHEGJI I^\<D ¡ DMmnmDMYJ\<lXMI-GZpZi-cG-IJUk-I-I " XHB[ZRUGJG-lzD X¢EB q V [ YZdUXMXHI-WJDHXHYJI-VI VDNp2UaG-IJV]UaVI2 RUopJB<Ua\<c<VXMc<xXM\B<DHGJG-lCpbW-UXM\c\ I "XHc<c\B<DM\XM\B<DHG-GJc" mUa Y2cadDMxyUG-IJr1IZdI ¡ Dnj-cHEUBad6DHG-I-DMVXHcx<XM\B<DHG-GJc<r1W6UaXM\<ca\<l"cHEG-c<rIJBUI-VcHEDHrJXM\<B q IZp1XHIJXM\<DMVzK

' -* -/ ',-!.1* , ./#! $ -

i-DFEIbG6UIJx<cadDMD\I-j-I-W-GJlCpbmnIJHI-W-DHXMY-IZp[-Bad6DHG-I-r "G-Uxad EUa I;pyX%[ BEI G6UVIJW-DHXMY-IZpXMI-XM\DHV]UPpj-i-I B<GJD GJDHV eUi-Vc<G-IJW-DHXMY;cV(B<cax q ¡ EDHG-IJI "Jc\VDM\I-Vi-DNc<G6UaG-X!"'XMI-GZpZi-c<GJIJRUkJI-IB<cHx q ¡ EDHG-I-D§i-D ¡ I-V]U EI-G6UVI-WJDHXHY2c<Mc p2Uc<XHUJK mnmnDHYJ\1i-DMHc<G6UG-XHU_p<c\R[ I8c<W-DMG-f1ByU ¡ G-lzr8BDHXM\DHXN\<B<cHG6UG-IJI " cHEG6UY2c_[-Bad-[ DM\X%[1G-UI-x<cad6DHDnj-i-c<XN\<lzV¤XMi-DeEIG6UaHByUG-GJlCpB<l DaK UaXHXHVca\<i-I-V dI-G-DMr-G-lzrEI-XHXMI-j6Ua\I-B<G-lzrc<XHkJIZd-d6[Z\<c<ij-i-I_B<G-D G-DMVB<cax q

¡ EDHG-IJIeUiJVc<G-I-W-DMXHY2c<rbXMIZdc<r/2

x + γx+ ω20x = a sinω1t

] K O XNdIUmnI-Y2XHI-i-cByUa\<f UVj;dIJ\ q E q B<ceEDHr-XN\<B<IZ[ a = const

Ij-i-c<G6Uaxad EUa\fc\YZdI-YXHIJXM\<DMVl j-iJI ByUi-I-Uk-I-I¤W6UaXM\<ca\<l

ω1

IY;c!HmnmnI-k-I-DMGJ\PU EI-XHXMI-j6Uk-IJIγ"\c Vl j-cad q W-I-Vi-DN q dfR\PUa\ "j-i-DFEXM\PUaBadDHGJG-lzrG6UbiJI-XK ] K OK 9BadDMG-I-D i-DNcG6UGJXU UYZd W6UDN\<X%[ B i-DMHY;cV¤B<ci-UXM\PUGJI-I UVjZd6IJ\ q El"B<lzG q ¡ EDMG-G-lCp1Y;cPdD xyUGJI-r j-iJI§XMc<B<j6UoEDHG-I-I§W6UXN\<c\l BG-DTG-DHr§XHIZdl

ω1

X¢XHc<x<XN\<B<DMG-G-c<r W6UXM\c\<cr

0.0 0.5 1.0 1.5 2.0ω

0.0

1.0

2.0

3.0

4.0

A

γ = 0.05γ = 0.1γ = 0.3

1

dJTY(*XI(*q /YXh Adgq r`(h i R T ' i aoR*+-''.1/g`*+Ydg'(WTBX-(XhN+q + " /8hDdgXhN/8hpTY'+ >S'*+M/uTN/8QIRD+*XhYT8( 50R*` ofd8Q-`(',Q-'*d +-'(o.1/ #((+-'*h8d R(*XX-(rDd ((&" uXh8dg`8' +3rDd8fd8q +hNf ω0 = 1.0, a = 0.2, γ = 0.3.

UB\c<G-c<VGJc<Hc c<XHk-I;d-d6[J\c<i6Uω =

√

ω0 − γ2/2 ≈ ω0

KDMHc<G6UGJX`p2Ui-UYJ\<DMi-IJ q DM\ X%[¢\<I-jJI-W-G-lzVI RUB<IJXHI-Vc<XN\R[-VI

A(ω1)"Y2c\c<i-lzD G-c<X%[J\G6UPByUGJI-Dzi-DMHc<G6UGJXHG-lCp

Y-i-IJB<lCpKaDMHc<G6UG-XMG-lzDXHB<c<rJXM\<B<U XMI-XM\DHVTI-i-c<Y2c i6UXHjJi-c<XM\i6UG-DMG-l B j-i-I-iJcEDII-XMj-cadf q \<X%[^B9\<DNpZG-IJY;DaK-A W6UXN\<G-cXM\<I " MmmnDMYJ\i-DMHc<G6UG-XHUXed q ¡ IZ\TcHEG-IJV IJc<XMG-c<B<G-lCpj-i-IXHcaNEUGJI-IÊI-G6UVIJW-DHXMY-IZpXHIJXM\<DMV "6XHj-cXHc<x<GJlCp_XHc<BDHi Ua\<f¢G-DNRU \ q pyU TI-D j-DMi-I-cHEI-W-DMXHY-IJDTY2cadDMxyUG-IJ[B UB\c<G-c<VG-cV(i-D ¡ IJVDK9i-I-VDMi-c<VVc< q \8XNd q ¡ IJ\f=\PUaY G-UalzByUaDHVlzDHDMG-DHi-Ua\<c<iJl \<cVXHc<G-cB<XHY2c<Mc

\<IJj6U "xadIZY-I-D_Y dI-GJDHr-G-lzV Y;cG-XHDMi-ByUa\I-B<G-lzV cXHk-IZdJd6[J\c<i6UVzKXNd6I(Y i-DNcG6UGJXHG-c<r"XMI-XM\DHVD X V]UPd6c<rEI-XMXHI-j-Uk-I-DMrjJcEYZduWJIJ\<f q XMIZdIJ\DNdfIB<B<DMXM\Ic<x<i-Ua\<G q XHB[Zf" q Ec<Bad6DM\<Bc<iy[- q G-DHYJI-V XHjJDHk-I6Uodf<G-lzV UVjZd6IJ\ q EG-lzVªImUacB<lzV q Xedc<B<IZ[-V "]\<c Vl j-cad q W-I-VªMDHG-DMi6Ua\c<i¤G-DMUa\ q pyU TIZp¤xad6IJY-I;pYeUi-VcG-I-W-DMXHY-I-V¤Y2cad6DHxyUGJI-rK2,2+->J +J,82 4K2©nUI-xcadDHD j-iJc<XM\lzV I YZdUXHXMI-W-DHXMY-I-V j-iJI-VD

i-c<V8HDMG-DHi-Ua\<c<i-Un\c<VXHc<GJc<B<XHY2c<Mc\I-j6UnXNd q ¡ IJ\IJBDHXM\G6Ua[ VcHEDNd6fÀCUGTEDHi 9cd6[ w " Ow 2

x− ε(1 − x2)x + ω20x = 0,

] K Z ED

ε9V]UPdlzr j-cPdc ¡ IJ\DNdfG-lzrj6Ui6UaVDM\<i "Pp2Uai6UYJ\DHi-IJ q IJr^XM\DHj-DMG-fnc<x<i6UP\

G-c<r_XMB[JHI "ω0

W6UXM\c\PU jJDHi-I-cHEIJW-DHXMY-IZpY2cadDMxyUG-I-r K Ua\<DMV]Ua\<IJW-DHXMY-I-V c<xi6U cV jJDHi-I-cHEIJW-DHXMY-IZpY2cadDMxyUG-IJrG-U mUac<Bc<r jZdcXHY2c<XM\I

(x, x)[-Bad6[JDM\X%[¤RU

VY-G q \PUa[mUacByUa[\<i-UDHYJ\c<i-IJ[ "G-UaByUaG-G6Ua[_j-iJDeEDedf<G-lzV kJI-YZdc<VzK mnIJIJW-DHXMY;cr \<c<W-YJI i-DMG-IJ[ j-c<GJ[Z\<I- j-i-DeEDNdf<GJc<Hc k-IJYZdU1c\B<DHW-UDM\1i-D

¡ I-V \PUY G6UalzB<UDHVlCp UBa\<c<Y2cadDMxyUG-IJr "yY2c< EUnp2Ui6UaYJ\<DMi-I-XM\I-Y-I q XN\<c<r-WJI-B<c<Mc

j-DMi-I-cHEI-W-DMXHY2c<Mcì-D ¡ IJV]U'G-DCRUaB<I-X%[J\'c\9G6URW6UPd6f<G-lCp q XNdcB<I-r "yU c<j-i-DFEDNd6[J \X%[I-XMYZduWJIJ\<Dedf<G-c`XHB<c<rJXM\<B<UVIXUVcrXHI-XN\<DHVl9K I-XN\<DMV]U ] K Z *p2Uai6UYJ\DHi-IJ q DN\<X%[DeEI-G-XM\B<DHGJG-lzV XMc<XM\co[JG-I-DMVsi6UaB<G-c<B<DMXHIJ[ \<I-j6Ubmnc<Y q X G6UmÙPc<B<cr jZdcXHY2c<XM\I

(x, x)B G q dDCY2c<ci<EI-G6Ua\oK c<x<XM\B<DHGJG-lzDCG-UoWJDHG-IJ[TXHc<XN\<cR[-GJIJ[ i6UaB<G-c<B<DMXHIJ[ndDMHY2c

B<lzW-IJXNdIJ\f Q 2

s1,2 =ε

2± i

√

1 − ε2

4.

] K Q hnUY=XNdDFE q DN\bIJ ] K Q "j-i-I8j-DHi-DepycHED B_c<xadUXN\<f_j-cad6c ¡ IJ\DNdf<GJlCp G6URW-DMG-I-r εXHcXM\<cR[JG-I-D i6UB<G-cB<DHXMIJ[\<DMi2[JDM\ q XM\c<r-W-IJB<c<XM\f-K niJI ε = 0

I-VDMDHVs1,2 = ± i

K cXM\cR[-G-I-Dni6UaB<G-c<B<DMXHIJ[^\<I-j6U mnc<Y q X9j-i-DN\<DMi-j-DHB<UDM\ x<I-m q i-Y;Uk-IJ"i-c ¡ EDMG-IJ[jJi-D EDedf<G-c<Mc1k-IJYZdU G;Ei-cG-c<ByU y¦'c<j-mUJK VjZdIJ\ q EUk-IJYZdUi-UXM\DM\ j-iJc<j-c<i-kJI-cG6UPd6f<G-cTY2c<iJG- Y-B<UoEi6Ua\G-c<V q IJuG6UoEY-iJIJ\<I-WJG-c<XM\I √ε ";U`j-DMi-I-cHEk-I-Y;dUTj-i-I0 < ε < 1

c<j-i-DFEDNd6[JDM\X%[bXHcc\<G-c DHGJIJ[-VI

T ∼= 2 π

ω(ε), ω(ε) = ω0

√

1 − ε2

4,

] K \<c8DHXN\<f8W6UXN\<c\PU8HDHGJDHi6UkJI-I(jJi-I

0 < ε < 1xadIJHYZU=Y XHc<xXM\<BDHG-G-crW6UaXM\<ca\<D

i-DNc<G6UaG-XHG-cHc^Y2c<GZ\ q i6U ω0

K UaXHXHVca\<i-I-V EI-G-UVI-Y q HDHGJDHi6Ua\c<i6U ] K Z zj-i-I BcNEDMr-XM\B<I-I=G6U G-DMHc_UoEJEI

\<IJB<G-c<reUi-Vc<G-IJW-DHXMY;cr1XHI;dl' ORw 2

x− ε(1 − x2)x+ ω20x = a cos(ω1t+ φ0).

] K t9cad q W-IJ\f i-DTDHG-IJD q i-UB<G-DMG-IJ[ ] K t!nX j-c<Vc! Tf< Y2c<Vj-f \<DMi6UbG-D j-i-DeEXM\PUBd6[JDM\\<i q EUJK EG-UY2c"aE6d6[¢j-cG-I-V]UG-IZ[ X q \<I¢EI-G6UVI-WJDHXHYJIZp [-BadDMG-I-r j-i-c<BDeEDMVj-i-IJxadI ¡ DMG-G-c<DuUG6UPd6IJ\<I-WJDHXHY2c<DuI-XHXedDeEcByUG-I-D "JY2c\c<i-c<D j-cBcadIJ\`G6UV d q x ¡ Dj-c<GZ[J\<f mnIJIJW-DHXMY-I-rXMVlzXNd MmnmDMYJ\PU(XMI-GZpZi-c<GJIJRUkJI-IKCSd6[ M\<cHc c<x<X q EIJVB<G6URW6UodD'j-c<GZ[J\<IJD'mUal¤Y2cadDMxyUG-IJrK DHi-VI-G mUaRUY2cadDHx<UG-I-r uj-DMi-B<c<G6URW6UodfG-c xlCdB<B<DFEDHG E-d6[ eUiJVc<G-I-W-DMXHY-I;p¤j-i-ck-DHXMXHc<B \I-j6U

x(t) = A exp (iωt) =A (cosωt + i sinωt)

KALj-cad6[Ji-G-c<rXHI-XN\<DHVDTY;cc<iEI-G6Ua\M\<c^Y;cPdDHxyUaG-I-DTIZc<x<i-U ¡ UDN\<X%[Y;UY1B<i-U DMG-I-DB<DHYJ\c<i6U

AXj-c<XN\<cR[-GJG-c<r q dcB<c<r1XMY2c<i-c<XN\<f< ω

K~UaDY2cadDMxyUG-I-r¢c\B<DHW-UDM\ q Hcad j-cB<c<i-c\PU'BDHYJ\c<i6U A B<c'Bi-DHVDMG-I φ = ωt

K G-UoWJDHG-I-q dU BbG6URW6UPd6f<G-lzr Vc<VDMGJ\ B<i-DMVDHG-I φ0 = ωt0

ca\<B<DMW6UDM\=G-UoW-UPdf<G6UP[ mUaRU "XY2c\<ci-c<r1iJDHHIJXM\<iJI-i q DN\<X%[=Y2cadDMxyUa\DNdf<GJlzrbj-i-c<kJDHXHXB<c¢B<i-DMVDHG-I t > t0

KAgEIJXHXHIJj6Ua\<IJB<G-lCp"GJDNdI-GJDHr-G-lCp"XHI-XN\<DHV]Uop \PUY2c<D c<j-i-DFEDNdDMG-I-D mUaHl I-X

j-cad6fc<B<Ua\<f G-DNdfaH[ "\PUYbY;UYY2cadDHx<UG-IJ[1B G-I;p1GJDTVc< q \^x<l \<f XM\i-c<HceUai-VcG-I-WJDHXHYJI-VIKhUaY ¡ DjJc<XM\ q jJIJ\<f B\PUY2c<r§XHIJ\ q Uk-IJI \B<DM\RUBI-XHIJ\c\\<c<Mc"Y;UY q EI-G6UVIJW-DHXMY q XHIJXM\<DMV q IY;UY2c<r_\<IJjbUB\c<Y2cadDMxyUG-I-rVl IJXHXNd6DeE q DHVzK

3 ! i f*dJTY'+-'(( " Y+-Qf*dgQq +f'*d r*+f+q +-''*d '*/8fqs(*f/8T:dJ'Dd 'Dd √ε 05r*/ hN/Yq i TY` +hYX r/YXhN/gS ''*/ x(xf*dgT8'*/ "

ACDMi-G-DHVX%[ Y XHI-XN\<DMVD ] K tNK9i-I V]UPdlCp0 < ε 1

Ia = 0 q i6UB<G-DMG-I-D ] K tc<j-IJXHlzByUDN\ UaB\<c<Y2cad6DHxyUGJIJ[ BMDHG-DMi6Ua\c<i-D \c<VXHcG-c<B<XMY;cHcn\<I-j6UZKZA N\<c<V§XNd q W6UD

i-D DHGJI-Dx(t)

Vc ¡ G-c¢I-XMYZUP\<f^B¢B<I2ED

x(t) = A(t) cos [ω1t+ φ(t)], x(t) = −ω1A(t) sin [ω1t + φ(t)], ] K w

EDA(t)

8VDFE6dDMG-G-c=VDHGJ[J Ua[-X%[ B<c Bi-DHVDMG-I UVjZdIJ\ q EU Y;cPdDHxyUaG-I-r " φ(t)1VDeE6d6DHG-G-cVDHGJ[J Ua[-X%[8B<c_B<i-DMVDHG-I8mUaRUY2cadDMxyUG-IJr "ω1

1W6UaXM\<ca\PUB<G-D G-DMHc XMI-HG6UodU B ] K tNK9G6URW-DHc<B<ci2[ "]Vl B<B<cHEI-V B§i6UXMXHVc\i-DHG-IJDbj-c<GJ[Z\<I-DVHGJc<B<DHGJG-c<rbUaVjZdIJ\ q El A(t)

IVMG-c<B<DMG-G-c<rmUalY2cadDHx<UG-I-r2

Φ(t) = ω1t+ φ(t). ] K

D DMG-I-D q i6UaB<G-DHGJI-r ] K t!Cx q EDHV(I-XHY;Ua\f^B¢B<I;ED2x(t) = A(t) cosΦ(t).

] K ] A M\<cV XNd q W-UDRUB<IJX%[ UP[§ca\Bi-DHVDMG-I Y2c<Vj-cG-DHGJ\PUmUal φ(t) = Φ(t) − ω1tj-i-DFEXM\PUaBad6[-DN\"XHcx<c<rVHG-cB<DHG-G q i6UaHG-c<XN\<f mUa VD ¡ E q i-DM q dfR\<I-i q I-VxadIZY-I-V Y eUi-VcG-I-W-DMXHY2c<V q j-i-c<k-DMXHXMc<V x(t) I^BG-DTG-I-V eUi-VcG-I-W-DMXHY-I-V XMI- G6UPd6c<VzK<¨]XNdcB<I-D VDeE6dDMG-G-c<McTIJHVDHG-DMG-IJ[^mÙPl

φ(t)B<cnB<iJDHVDHGJI^cHG6URW6UDM\ ";WJ\c

φ(t) ω1

K j q XHY;Ua[`j-i-c<k-DFE q i q G-DHXedc ¡ GJlCp "PG-c Hi-cVcNEYJIZpj-i-DMc<x<i6UaHc<ByUGJI-r "c<j-IJXUG-G q B<cVG-cHIZp q W-DMx<G-I-Y;UPp "RUaj-I DMVL[JB<G-lzr8B<I;E q i6UB<G-DMG-I-r8j-DHiJB<c<Hcj-i-IJxadI ¡ DMG-IJ[E6d6[_VMG-c<B<DMG-G-c<r1UVjZdIJ\ q El A(t)

I_VHG-c<BDHG-G-cr1mUalφ(t)

A =εA

2

(

1 − A2

4

)

− µ sinφ ,

φ = ∆− µ

Acosφ ,

] K ED

µ = a/2ω1

nj6Ui-UVDM\iG-DNdIJG-DHr-GJc<XM\I XMI-XM\DHVl ] K "∆ = (ω2

0−ω21)/2ω1

∼=ω0 −ω1

i6UaXHXM\i-c<r-Y;U j-c`W6UXN\<c\DuVD ¡ E q XHcx<XM\B<DHG-GJc<r W6UXN\<c\c<rUBa\<c<G-cVG-c<HcHDMG-DHi-Ua\<c<i-U'ITW6UXN\<c\c<rB<G-D G-DMHc'XMI-HG-UPdUuB ] K t "

εnj6Ui-UVDM\i B<cHx q ¡ EDMG-IJ[

HDMG-DHi-Ua\<c<i-UJK¨'i-UB<G-DMG-IJ[ ] K j-c'XMVlzXNd q j-i-DFEXM\PUBPd6[-DN\`XMc<x<c<r q XHiJDeEG-DMG-G-lzD j-c'B<iJDHVDHGJI

q i6UB<GJDHG-IJ[ "aY2c\c<i-lzD W6UXM\cuG6UPlzByU \<X%[TXHI-XN\<DHVcr q Y2c<i-c<WJDHG-G-lCp q i6UB<GJDHG-I-r FK©9DHj-cHEBI ¡ GJc<r\c<W-Y2DXHI-XN\<DHVl ] K

A = 0"φ = 0

Cx q EDM\ c\<BDHW6Ua\f^j-DMi-I-cHEI W-DMXHY2c<D¢i-DTDHG-IJD IJX pycHEG-c<r=XMI-XM\DHVl ] K t! "U j-DHi-IJcEI-W-DHXMY2c<V q i-D DHGJI- ] K EB q pZW6UXM\c\<GJc<DTY-B<UaI-jJDHi-I-cHEIJW-DHXMY;cDTi-D DHGJI-D q i6UBG-DHG-IZ[ ] K tNK \<VDN\<I-V "WJ\c q Y2c<iJc<W-DHGJG-lzD q i-UB<G-DMG-IJ[ j-cad q WJDHG-l B j-i-I-xad6I ¡ DHGJI-I VDeE

dDMG-G-c¢VDHGJ[J TIZpyX%[1UVjZdIZ\ q El ImUaHlLI "JY;UY_j-c<Y;UaHlzByU \ i6UaXHW-DN\<l "6Vc< q \c<j-IJXHlzByUa\f^j-iJc<k-DHXMX

x(t)B ] K tCj-i-I q Xedc<B<IJI "WJ\<c µ ≤ 0.05

K

−0.04 −0.02 0.00 0.020.00

0.02

0.04

0.06

I

II II

ll aa

∆

µ

g`Dd8X-h X-(' 1f/g'(QMd((t0 /8hYTN+Dd#)od i X-hN/ (TY/YXh8( '+-r*/R*T8(*ao'*/ phN/ .( X-(*XhY+q "

9i-DeEj-cadc ¡ I-V "WJ\<cXHIJXM\<DMV]U q i6UBG-DHG-IJr ] K uI-VDHDN\1B_Y;URW-DHXN\<B<D i-DTDHG-IJ[G-DMj-cHEB<I ¡ G q L\c<W-Y q I;dI XHcXM\<cR[JG-I-Dui-UB<G-c<BDHXHIZ[ A = 0, φ = 0

I c<G6U q XN\<c<r W-I-B<UJK¨]Xedc<B<I-D

A = 0cG6URW6UaDM\=j-c<XN\<cR[-GJXM\<Bc8BcbB<i-DMVDHG-I UVjZd6IJ\ q El Y2cadD

xyUGJI-r "U q Xedc<B<IJD φ = 0cHG6URW6UDM\ "WJ\c

Φ = ω1

XMVzK ] K ! "\cbDHXN\<f W6UaXM\<ca\PUB<lzG q ¡ EDHGJG-lCp Y2cadDHx<UG-I-r BnXHI-XN\<DHVD ] K t x q EDM\XHc<Bj6UoEUa\fnX W-UXM\c\<c<r B<G-D G-DMrbXHIZd6l9KuXNd6IM\c\i-D ¡ IJVY2cadDMxyUG-I-r1B<cVc ¡ DMGI[-Bad-[-DM\X%[ q XM\<cr-W-I-B<lzV "\<c W-UXM\c\PU Y;cPdDHxyUaG-I-r BbG-DHUB\c<G-c<VG-cV"HDHGJDHi6Ua\c<i-D ] K t!IJHVDHG-IJ\X%[ I c<Y;U ¡ DN\<X%[ i-UB<G-c<r§W6UXN\<c\D B<G-D G-DMr XHIZd6l

ω1

K$9i-I M\<cVUVjZdIJ\ q EU1Y2cadDMxyUG-I-rB<c B<i-DHVDMG-I VDHGZ[J\<fX%[bG-D`x q EDM\oK #ZDHGJDHi6Ua\c<i j-cHEXM\i-c<IJ\X%[ jJc^W6UXN\<c\D Ibi-DHU dIJ q DN\<X%[ MmmnDMYJ\^BlzG q ¡ EDMG-G-c<rXHIJGZpZi-c<G-IJUk-I-I K©nUrJ\I Y2c<c<iEIJG6Ua\<l¤GJDHj-cHEB<I ¡ G-cr \c<W-Y-I

A0(µ, ∆)Iφ0(µ, ∆)

Vc ¡ G-cUG6U dIJ\I-W-DMXHY-I " i-D Ua[ ] K j-i-I q XNdc<BI-Ibi6UBDHG-XN\<ByU G q d j-i-UB<lCpW6UXN\<DMr q i-UBG-DMG-I-rK c ¡ GJc \PUY ¡ DCUaG6UPdIJ\I-W-DMXHY-II-XMXNdDFEc<ByUP\<f q XM\<cr-W-I-B<cXM\<f A0 I φ0 KoUX W-DN\<l j-c<Y;UalzB<U \ "<WZ\<c9G6Uuj;dc<XHY2c<XN\<I j6Ui6UaVDM\<iJc<B

µ, ∆*X q DMXM\<B q DM\Tc<xadUaXM\<f

IZpG-UoWJDHG-I-r "aB Y;ca\<c<i-crTXHcXM\<cR[JG-I-D i6UaB<G-c<B<DMXHIJ[ q XN\<c<rJW-I-B<c-K \PUuc<xPdUXM\fuj-iJDeE XM\PUaBadDHG-U8G6UbiJI-XK ] K Z K ©nUbHi6UGJI-k6UPp c<xadUXN\<I d6I-G-I-I

laì6UBG-c<B<DMXHI-D \DHiy[-DN\

q XN\<c<rJW-I-B<c<XN\<f"IJì6UB<G-cB<DHXMIJ[1iJc ¡ EUDN\<X%[j-i-DFEDNdfG-lzr k-I-YZd j-DMi-DNp<cHE8IJ`c<xdUXN\<I CB¢c<xadUaXM\<I eKhnUY q ¡ D`c\VDHW6Uodc<XHf "B j-cadGJc<r1XMI-XM\DHVD ] K t!zXMc<XM\cR[-G-I- i6UB<G-cB<DHXMIJ[c\

B<DMW6UDM\(j-i-DFEDNd6f<G-lzrk-IJYZd " U kJI-YZd q EB q VDMi-G-lzr \c<iK \<c cG-UoW-UDM\ " WJ\<cB c<xadUXM\I ¢Vl I-VDHDMV q XM\<cr-W-I-B<lzD8Y2cadDMxyUG-IJ[X W-UXM\c\<c<r ω1

" U BG-D DHD Y-ByUPI-j-DMi-I-cHEI-WJDHXHYJI-DTY2cadDMxyUG-IZ[X EB q V [G-DNRUBI-XHI-VlzVIW6UXN\<c\PUaVI ω1

Iω0

K

−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 ∆−0.25

−0.15

−0.05

0.05

0.15

<ω>−

ωD = 0.0D = 0.02D = 0.07

1

1

2

3

dJTY(*XI(*q /YXh f*d8QI'*/YX-hY( Xf+IR*'+ +DdgXhN/8h .1/g`*+Ydg'( TX-(XhN+q + " ( *d8XhY/Yh TY'+ >S'*+M/oTY/YQIR*+ *X-hYTY( /YhcTY+-`((' r*d8fd8q +hNfd,fd8XX-hNf/ *.(AR` fd8Q-`('AQ-'*d *+-'(('hN+-'*XI(TY'/YXh8( i q d xadUXN\<f "BY2c\<c<iJc<r W6UaXM\<ca\PUbY;cPdDHxyUaG-I-r MDHG-DMi6Ua\c<i6U

ω0 = ω1"G-UalzByUaDM\<X%[

c<xadUXM\f<XHI-GZpZi-cG-IJRUak-I-IbG6Uc<XMG-c<B<GJc<V(\c<G-DK mnmnDHYJ\`XHI-G;pJiJc<G-IJUk-I-I IZd-d XM\<iJI-i q DN\\PUY ¡ DzUB<I-XMI-Vc<XM\fTi6UPG-c<XN\<I W6U

XM\c\<l Y;cPdDHxyUaG-I-r XHI-XN\<DMVl ] K tnI W-UXM\c\<l B<G-DTG-DHMc§BcNEDMr-XM\B<IJ[ c\=i6UaX XM\i-c<r-YJI

∆ Y-i-I-ByUP[1O G6UTi-I-XaK ] K Q eKJhnUY B<I;EGJc IJ'iJI-X q G-Y;U "ZB Y2c<G-DHWJG-c<r c<xPdU

XM\I`HG6URW-DHG-IJr j6Ui6UaVDM\<i-U i6UaXHXM\i-c<r-YJI W6UXN\<c\PU XHc<x<XN\<B<DMG-G-lCpTY2cadDHx<UG-I-rXHc<Bj6U EUDN\ X W6UXM\c\<cr B<G-D G-DMHcB<ceEDHrJXM\<BIJ[KJACG-D c<xadUXN\<I XMI-GZpZi-c<GJIJRUkJI-IW6UXN\<c\PUnY;cPdDHxyUaG-I-r HDHGJDHi6Ua\c<i6Uc\dI-W6UaDM\<X%[^c\W6UXN\<c\l B<G-D G-DMr^XHI;dlI¢BnXHI-XN\<DMVDi-DHUPdIJ q DM\X%[ i-D ¡ I-V EB q pZW6UXN\<c\G-lCpbYJByUaIJj-DHi-IJcEI-W-DHXMY-IZp1Y2cadDHx<UG-I-rK9GJ\<DMi-DHXMG-lzV j-i-DFEXM\PUaBad6[-DN\<X%[_XedDeE q I-r Bc<j-i-c<X2yWJ\<c`x q EDM\`j-i-cI-X pycHEIJ\fj-i-IEUPdfG-DHr TDHV q B<DNd6I-W-DHGJI-I i6UXMXM\i-c<r-Y-I "Z\oK DK2j-i-IEUPdf<GJDHr TDHV q B<DNd6I-W-DHGJI-IW6UXN\<c\l¤B<G-D G-DMr XHIZd6l Bnc<xadUXN\<I Y-ByUaHI-j-DHiJI-cHEI-W-DMXHY-IZp^Y2cadDMxyUG-I-r S`d-[¢c\B<DN\PU G6UTEUG-GJlzrB<cj-i-c<XnB<B<DFEDHV(B i-UXHXMVc\<i-DMG-I-D\PUYG-UalzByUaDHVc<DW-I-Xedc Bi6U DMG-IJ[ q UG-Y;Ui-D "aY;UY`c\<GJc! DMG-I-D W6UaXM\<ca\ θ = ω1/ω

K EDHXHfω1nW-UXM\c\PUuB<G-D

G-DMrXHIZdl "aUωW6UXN\<c\PU'Y2cadDMxyUG-IJr HDMG-DHi6UP\<c<i6UZK<A i6UaXHXHVca\<i-DMG-G-c<V=B<l TD XNd q

W6UDXHIJGZpZi-c<G-IJUk-I-I1G6U c<XMG-c<B<GJc<V(\c<G-DVlI-VDHDMVω = ω1

I1W-I-Xedc^Bi6U DMG-IJ[i6UBG-c§DFEI-G-IJk-DK \<ca\ Xed q W6Urca\<B<DMW6UDM\ i-DMHc<G6UGJX q O 2 O_G6UbEB q VDHi-G-cVs\<ci-DKhnUY8B<I;EG-cIJ i-I-XaK ] K Q Y-i-I-ByUP[ O "j-iJI§B<lCp<cHEDÎJ c<xPdUXM\I§XMI-GZpZi-c<GJIJRUkJI-Iω 6= ω1

"U IZp i6UPG-c<XN\<f8Vc<G-c\c<G-G-c=i6UaXM\<DN\ X q B<DedI-W-DMG-I-DMV ω1

K \<c8cHG6URW6U DM\ " WJ\<c W-IJXNdc B<i-U DMG-IJ[ \PUY ¡ D x q EDM\=IZVDHGZ[J\<fX%[ j-c B<DNd6I-W-I-G-D " j-i-IJG-I-V]Ua[j-c<XedDeEc<ByUa\DNdf<GJc \<c I-i-i-Uk-I-c<G-UPdf<G-lzD "6\<c i6Uk-I-cG6UPdf<GJlzDHG6URW-DHG-IZ[K Uak-I-c<G6Uodf<G-lzVG6URW-DMG-IJ[JVsW-IJXNdU B<i6U DHGJIJ[

θ = m : nca\<B<DMW6U \=xcadDHD

XNd6c ¡ G-lzD B XMi6UB<GJDHG-I-IX1i-DNc<G6UaG-XHc<V1 : 1

¢i-DNc<G-UG-XHGJlzDbk-I-YZd6l G6U8\c<i-DK

xadUXN\<I q XM\<cr-W-I-B<cXM\<I M\IZpkJI-YZdc<BTc<j-i-DFEDNd6[J \¢XMc<c\<BDM\<XN\<B q I-Dc<xadUaXM\<IXHIJGZpZi-c<G-IJUk-I-I IZdI8c<xadUXN\<I=RUPpZByUP\PUW6UXN\<c\l I8mÙPl9K q B<DedI-W-DMG-I-DMV"j-ciy[;EY;Ui-DNc<G6UaG-XHc<B X'i-c<XN\<c<V

mIn c<xPdUXM\I_XHI-GZpZi-cG-IJRUak-I-IXM\PUG-cB[J\X%[U

VDM\G-c q ¡ D "6G-c XHcpZi6UGJ[J \ YZURW-DMXM\B<DHG-GJc mnc<i-V q c<xPdUXM\IXMI-GZpZi-c<GJIJRUkJI-Ibj-i-Ii-DNc<G6UaG-XHD

1 : 1 XMVzKyi-I-XaK ] K Z eK ]XedI^jJi-c<IJHB<DHXN\<Ii6UXMW-DM\l RUBI-XHI-VcXM\ |∆| > 1" I-XHj-cPdf q [IJX pycHEG-lzD q i-UB<G-D

G-IJ[ ] K t "*\cbVl j-cad q W-IJV"Hi6UmnI-Y X^x<DMXHY2c<G-DMW-G-lzVW-I-XNd6c<V j-cPdc<W-DMY \<I-j6Ui-I-XaK ] K Q " I-i-IJG6U'Y2c\<c<iJlCp q VDHG-f UDN\<X%[ X i-c<XN\<c<V m I n K \<ca\nHi6UmnI-YIJVDHDM\mni6UYJ\PUodf<G q XM\<i q YZ\ q i q I_G-cXHIJ\ G6UaHByUG-I-D Ef[-Bcadf<XMY;crd6DHXM\G-I-k-l %K UaXHXHVca\<i-DMG-G-lzr B<l TD Xed q W6UrbEI-G6UaVI-Y-I MDHG-DMi6Ua\c<i6U1ACUG1EDHi 9cad6[=j-i-I

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ω01

Iω0

2

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ω01

"ω0

2

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1 < ωp < ω02

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"Y2c\c<i-c<D`XHcpZi6UGJ[JDM\<X%[=B G-DHY2c\<ci-c<rY;cG-DHW-GJc<rcxadUXN\<I_G6URW-DMG-I-r1j-Ui6UVDN\<i-c<B¢XMI-XM\DHVl "G6UalzB<UDHVc<r_c<xPdU XM\f< XMI-GZpZi-c<G-IZRUk-IJIK xadUaXM\<f^XHI-GZpZi-cG-IJRUak-I-I1p2Uai6UYJ\DHi-IJ q DN\<X%[ HmnmnDHYJ\c<V

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B<XM E q B c<xadUaXM\<IXHI-G;pJiJc<G-IJUk-I-IK UPpZByUa\ q mÙP`ca\<B<DMW6UDM\^j-c<XN\<cR[-G-XN\<B<cî6UaGJc<XM\ImUa Y;cPdDHxyUaG-I-r BUI-VcHEDMr-XM\B q I;pHDMG-DHi-Ua\<c<iJc<B BTcxadUXN\<I^XMI-GZpZi-c<GJIJRUkJI-Iφ = 0, φst = const

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XM\cp2UXN\<I-W-DMXHY-IJV q i6UBG-DHG-IJDHV

x− ε(1 − x2)x + ω20x = a cos(ω1t + φ0) +

√2D ξ(t),

] K O [ B9Y2c\<ci-c<V8B'XHi6UaB<G-DHGJI-I¢X ] K tEc<xyUBad6DHG6U'Xed q W6Ur-G6UP[¢XHIZdU ÌJXM\<cW-G-I-Y¢x<Dedc<Hc q V]U ξ(t) I-GJ\DHG-XMI-B<G-c<XN\' ] K O [ uK K \i6Ua\c<G-c<B<IJW-DHVxlCdI§jJcad q W-D

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A =εA

2

(

1 − A2

4

)

− µ sinφ+√

2Dξ1(t),

φ = ∆− µ

Acosφ+

√2D

Aξ2(t),

] K O<O ED

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U(φ) = −∆ · φ − µA

sinφKACDNd6I-W-I-G6Ui6UXHXN\<i-cr-Y-I

∆c<j-i-DFEDNd-[-DM\=G-UYZdc<G j-c

\<DMG-k-I6UodU "zU8c\<G-c DHGJI-Dµ/A

BlzXHc\ q j-ca\<DHGJk-I6UPdfG-lCpxyUiJf<DHi-cB-K A XNd q W6UaD

∆ < µ/AVIJG-I-V q Vlj-c\<DMG-k-I6UodU φk = arccos (∆ · A/µ) + 2πk

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U(φ)2φ(t)

m'd q YJ\ q I-i q DM\B<xad6IJIVI-G-I-V q Vc<B jJc\<DMG-k-I6UPdU φk

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2πK

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t > T0" ED

T0j-DHi-IJcE B<GJD GJDHHcj-DMi-I-c

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'B<DMXM\G-cTG-DMXHY2cadfY;cc<j-iJDeEDedDHG-IJrHmnmnDHYJ\PUXHI-G;pJiJc<G-IJUk-I-I¢p2Uc<XHUJK EG-c<rIJjJDHi-B<lCpbx<lCdU Y2c<G-kJDHj-k-IZ[ "XMc< dUXMG-cY2c\<c<iJc<r XHI-G;pJiJc<G-IJUk-IJ[ p2Uac<XU j-cG-I V]UPdUaXHf1Y;UY§[-BadDMG-I-D q XM\PUaG-c<BadDMG-IJ[ j-DMi-I-cHEI-WJDHXHY2c<Mc i-D ¡ I-V]Uj-cHE BadIJ[JG-I-DMVB<G-D G-DMHceUi-Vc<G-IJW-DHXMY;cHc1BcNEDMr-XM\B<IJ[§G6UXHI-XN\<DMV q B_i-D ¡ IJVDTp2Uc\I-W-DMXHY-IZpUB\c<Y2cadDMxyUG-I-r K9DHiJDNpycHEca\ pyUc\I-W-DHXMY-IZpYLi-DM q d6[Ji-G-lzV Y2cadDMxyUG-IJ[JV|j-i-IM\c<V G6Uaxad EUDN\<X%[^j-i-I¢Ec<XM\PUa\c<W-G-cTB<lzXHc<Y2c<r I-GJ\DHG-XMI-B<G-c<XN\<IB<caNEDMr-XM\B<IJ[ "2\<cDHXN\<fI-VDHDN\<X%[ G-DHYJI-r jJc<i-c<zXMI-GZpZi-c<G-IZRUk-IJIK DNpyUG-IJHV XHI-G;pJiJc<G-IJUk-I-I B'EUG G-c<V§XNd q W6UaD c<x q XNd6c<BadDMG^j-cHEUBPdDHG-IJDHV i-D ¡ I-V]UupyUc<XUI BcG-IJY-G-c<B<DMG-I-DMV BnXHI XM\DHVD`B<lzG q ¡ EDHG-G-lCpjJDHi-I-cHEIJW-DHXMY-IZp1Y;cPdDHxyUaG-I-rKSTi q Hc<r\I-j¢XHI-GZpZi-cG-IJRUak-I-I pyUc<XHUnIJVDHDM\VDMXM\<cnj-iJI BaRUI-VcHEDMr-XM\B<I-IEB q p

cHEI-G6UaY;cB<lCppyUc\I-W-DHXMY-IZp=XMI-XM\DHVzK q BDNdI-WJDHG-I-DMVLY2cMmnmIJk-I-DHGZ\PU_XMB[JHIbY2cdDMxyUG-IJ[ B'EB q p¢BUI-VcHEDMr-XM\B q I;p XMI-XM\DHV]UPp XM\PUaG-c<B[Z\<X%[ j-cadG-cXM\<fLI;EDHG \<IJW-G-lzVI29B<iJDHVDHGJG-lzD i-DUPd6IJRUkJI-IXHc<c\B<DM\XM\B q I;pEI-G6UVI-WJDHXHYJIZpj-DMi-D VDHGJG-lCp cx<c<IZp`j-cHEXMI-XM\DHV8j-cadG-cXM\<fj-c<Ba\<c<iy[- \ Ei q Ei q eU'x<DM Y;UY2c<Hc dI-x<cX EB<IJeUB<c_B<i-DMVDHG-I K \<c\\<I-j8XHI-G;pJiJc<G-IJUk-I-I G-UalzByUa \ODSYXD?DS IAB?@FHST?BA :<=&BADC 6K \<c<WJY-Ii-DMG-IJ[¢\<DMc<i-I-I¢Y2cadDMxyUG-I-r¢j-iJDeEXN\PUBad6[JDM\<X%[ I-GJ\DHi-DMXHG-lzV c\<B<DN\<IJ\f

G6Uiy[;EXedDeE q IZpB<cj-i-c<XMc<B-K c ¡ G-cTdI_c<xc<x! IZ\<fYZdUaXHXHIJW-DHXMY-I-Dj-i-DeEXM\PUBdDMG-IJ[c`XHI-GZpZi-cG-IJRUak-I-I_Y;UY c<x MmnmDMYJ\PUPp RUPpZByUP\PU IZdI j-cHEUaBadDHGJIJ[ W6UaXM\<ca\<lG6U8XNd q W6Ur p2Uc\I-W-DMXHY-IZp Y2cadDHx<UG-I-r XNd6I EU "\<c YY;UY2c<V q \<I-j q pyUc\I-W-D XHYJIZpUB\c<Y2cadDMxyUG-I-r N\<c Vc ¡ GJc j-i-I-VDMG-IJ\f hnUY-I-V c<xi6UacV Vc ¡ G-c B<BD XM\IUoEDMY-ByUa\G-c<D j-c<GJ[Z\<I-D mUal E6d6[p2Uac\<I-WJDHXHYJIZpUaB\<c<Y2cad6DHxyUGJI-rÎ Vc ¡ G-c'dIM\c X EDedUa\f n9XHXNd6DeEc<B<UG-IJ[ j-c<Y;UaUPdI "WZ\<c G-U=BXHD M\IB<c<jJi-c<XHl X q DMXM\B q DM\ W-DN\<Y-IJrc\B<DM\ "6DMXNdIi6UXMXHV]Ua\i-I-ByUP\<f i-D ¡ I-V \PUYG6UalzB<UDHVc<McÎOBA@FL= XHJ ?DS+ SIZdI -=>:WSM LS+ CPFDCI?BEK?DS+ S9pyUc<XUZK j-IJi6UPdf<GJlzr pyUc<Xnp2Ui6UaYJ\<DMi-IJ q DM\X%[ G6UPd6I-W-I DHV [Ji-Y2cB<lzi6U ¡ DMG-G-c<r_XMj-DHYJ\i6UPdfG-c<r dI-G-IJIx<Uac<Bc<r W6UXN\<c\l "6i-DUodIJ q DM\X%[1B IJi-c<Y2c<V YZdUXHXMD EI-G6UVI-WJDHXHYJIZp XHI-XN\<DHV X jJDM\dDMr XHDMj6Ui6Ua\i-I-XMl XHDFE6dc mcY q XHUI j-cHB<cad6[JDM\ q x<DFEIJ\DNdf<GJcca\<B<DN\<IJ\fG-U^B<XHDTB<l D jJc<XM\PUBPdDHG-GJlzD Bc<j-i-c<XMl9K AG6UXN\<cR[ TDHr1dDMY-k-I-I Vl c<x<X q EIJV Y2c<G-k-DMj-k-I- W-UXM\c\<G-c mUacB<c<r1XMI-GZpZi-c<GJIJRU

k-I-Ij-iJI-VDHG-IZ\<DNd6f<G-cY pyUc\<IJW-DHXMY-I-V UBa\<c<Y2cadDMxyUG-IZ[-V " c<xi6UacV Y2c\c<i-lCpBmUacB<c<V j-i-c<XN\<i6UGJXM\<BD`[-BPd6[-DN\<X%[XHj-I-i-UPdf<G-lzr1Ua\\i6UYJ\c<iK

./# , -/ ' -/ *!. ), ' -* ' -/ $ -*$+-% # /- ! ,# ' -* ' -*$% '*$ -*-!.1* ,

UaXHXHVca\<i-I-Vi-D ¡ I-VXHj-I-i-UPdf<G-cHc^p2Uac<XU^B¢pyc<iJc! c IZB<DMXM\<GJc<r1EI-G6UVIJW-DHXMY;crXHIJXM\<DMVD HXHXedDHi-U 2

x = −y − z, y = x+ αy, z = β + z(x− µ). JK O

Sd6[G6URW-DMG-I-r j6Ui-UVDM\i-c<Bα = β = 0.2

Iµ = 6.5

j-iJc<I-GJ\DHHiJI-i q DMV XMI-XM\DHV q ZK O XG6URW6UPdfG-lzVI q Xedc<B<IZ[-VI B8c<Y-i-DMXM\G-c<XM\I¤G q d6[(Y;cc<iEI-G6Ua\ "j-c<XN\<i-cI-Vj-i-cDHY-k-IJ mUacB<c<r \<i6UaDHYJ\c<i-I-I G-UjZd6c<XHY2c<XN\<fbj-DMi-DHVDMG-G-lCp

(x, y)I8i6UXMXHW-I

\PUDMV XMj-DHYZ\<i_Vc! GJc<XM\ISx(ω)

E6d-[ j-DMi-DHVDMG-G-c<rxKZDN q dfR\PUa\<lLj-i-DFEXM\PUBPdDHG-l

G6U i-I-XKJK O<KhnUYB<I;EGJc IJì-I-X q G-Y;U "B j-i-cDHY-k-IJI i-IJXKJK O" = C\<i6UDMYJ\<ci-IJ[bXMcB<DMi UDN\_B<i6U Ua\<Dedf<G-lzDÈB<I ¡ DMG-IJ[ B<cY-i q TG q d6[bY;cc<iEI-G6Ua\ " U^XHjJDHYJ\i Sx(ω)G6UmcG-DuXMjZdc!TG-c<HcTj-f<DeEDHXM\PUodU I-VDHDN\EcXM\PUa\c<W-G-c q Y2c<jJcadc<XMG-lzrB<lzx<i-cX G6UG-DMY;ca\<c<i-cr1W6UXN\<c\D

ω = ω0

i-I-XaKJK O" ReK j-I-i-UPdf<G-lzD Ua\a\<i6UYZ\<c<i-l \I-j6U i-I-XaKoJK O!" =B<caG-I-Y;U \uB i-DN q dfR\PUa\D Y;UXHY;UoEU

x<I-m q i-Y;Uk-IJr q EB<cDHG-IJ[_jJDHi-I-cHEUZK nX p<cHEG-lzr1jJDHi-I-cHEIJW-DHXMY-I-rbUP\\<i-UYJ\<ci jJi-D EDedf<G-lzrk-I-YZd CB IZ\<c<MDXN\PUG-c<BIJ\<X%[G-D q XM\c<r-W-I-BlzV XMDeE6dcB<lzV "6G-c \i6UDMYJ\<ci-I-I=G6U XMj-I-i6Uodf<G-c<VUa\a\<i6UaYJ\<c<iJD B<i-U U \<X%[=B<xPdIJI N\<c<McXMDeE6d6c<B<c<Mck-I-YZdUX8B<lzXHc<Y2c<rLXM\DHj-DMG-f<iJDH q d6[Ji-G-c<XN\<IK '\<cM\<c(cG-UoW-UDM\ ]XedI B j-i-cDHY-k-IJI

−15 −10 −5 0 5 10 15x−15

−10

−5

0

5

10

y

= R f/8+.( X-r(*fdg`8'*/ M/xdghNhNfd8.hY/Yf*dB'*dr`*/8X./YXh Wr*+-f+q +-'' (x, y) l6 (XIr*+.hNf q /8),'/YXh8( Sx(ω) .1/g`*+Ydg'( x(t)

4 5 6 7

TiT0

0

10

20

30

p(T i)

dgX-r*f+IRD+-`+-'(+BTNf+q +-' r*+-f+X+*+-'( d8Q/gTN/ hNfd8+.hN/Yf1(*+ X+. i )o+ y = 0 p(Ti)d8h8ShNfd8.hY/Yf*d XX-`+f*d f(*X8" " 0 6

Ua\a\<i6UYZ\<c<i6U(B<B<DHXN\<IXHDMY q q dIJG-I- y = 0Ii6UXMXHW-IJ\PUP\<f j-cXNdDFEc<ByUa\DNdf

G-c<XN\<f'Bi-DHVDMGTi

j-DMi-DHXMDHW-DMG-IJ[`\<i6UDMYJ\<ci-I-DHr N\<c<rTXHDHY q TDHrB G6Uj-i-UBadDMG-I-ITXMG-I q B<BDHi<p "\c1Vc ¡ GJc q x<DFEIJ\<fX%[ B1XedDeE q DHVzK UXMj-i-DFEDNdDMG-I-D_B<i-DMVDHG p(Ti) XMVzKCi-IJXK JK Z I-VDMDM\ [Ji-Y2c B<lzi6U ¡ DHG-G-lzr¤V]UaY;XMI-V q V j-i-I T = T0

"WJ\<c XMB<I EDN\<DNd6f<XM\B q DN\¤c j-i6UYJ\I-W-DMXHY-ILi-DH q d-[-i-G-cV B<i6U DHGJI-ILX jJc<WJ\<Ij-c<XN\<cR[-GJG-lzVj-DMi-I-cHEc<VzK d6DeEXN\<B<I-DMV M\c<r i-DM q d6[-iJG-c<XM\I§[JBad6[-DN\<X%[=W-DN\<Y-IJr8jJI-Y=B XHj-DMYJ\<iJDG6UTW-UXM\c\<D

ω0 = 2π/T0

";I-VDM IJrY;cG-DHW-G q "-G-cEc<XM\PUa\c<W-G-cV]Uod q I-iJI-G qdI-GJI-I

∆ω/ω0 1K6ACXHD N\<I1XMB<c<r-XN\<ByU c<x q XNdUBPdI-ByU \ c<XHc<xDHG-G-cXM\<IXHj-IJi6UPdf

G-c<McÛP\\<i-UYJ\<ci6U "Y2c\<c<iJlzr1G6UaHlzByU \ \PUY ¡ D`mUaHc Y2c<MDHi-DMGJ\<GJlzVzK

1;>1)2+J£+J>;+8+J)2¤:+6<>->-?g£+->-:;<£(+->-6 2+-,

UaXHXHVca\<i-I-V"iJDM q dfR\PUa\lªi6Ux<ca\<l, OR "j-cPdc ¡ I-B DHr=G6URW6Uodc_mci-VI-i-c<B<UG-I-XHcB<i-DHVDMG-G-lCpj-i-DFEXM\PUBPdDHG-IJr¤c§XMI-GZpZi-c<G-IZRUk-IJI(p2UcXU§G-U8c<XMG-c<B<D_YZdUXHXMI-W-D XHY2c<r\<DMc<i-I-IY;cPdDHxyUaG-I-rK \<Ii-DM q dfR\PUa\<lx<lCdIj-cad q W-DMG-lLE6d6[i-D ¡ IJV]U¢XMj-I i6UPd6f<G-c<Mc_pyUc<XUBMDHG-DMi6Ua\c<i-D G-I TDHG-Y2c XM\PUPp<c<ByU XHVzKDHY-k-IJ w "G-cVlj-i-cIZd-d XM\<iJI-i q DMV IZp(G6U§j-i-I-VDMi-D Ua\a\<i6UYZ\<c<i6U HXMXNdDMi6UJKzACBDeEDMVªB<G-DTG-DHDeUi-VcG-I-W-DMXHY2c<DTBcNEDMr-XM\B<I-D`B XMI-XM\DHV q ZK O

x = −y − z + C sin(ωext), y = x + αy, z = β + z(x− µ). JK Z

EDMXHfC"ωex

=UaVjZdIJ\ q EUI8Y-i q Hc<B<Ua[8W6UXM\c\PU1B<GJD GJDHHcB<ceEDHr-XN\<B<IZ[K U mnI-Y2XHI-i q DHV¤HG6URW-DHG-IZ[1j6Uai6UVDM\i-c<B α = β = 0.2

"µ = 6.5

K 9i-I M\<cVG6URW-DMG-I-DxyUaHc<B<c<rW6UXN\<c\lsXMj-I-i6UPd6f<G-c<McUa\\i6UYJ\c<i6U B c\<X q \XM\B<I-I8B<G-DTG-DHr XMIZdlx q EDN\

ω0 = 1.068 XMVzKi-I-XaK*JK O!" RNK*A lzx<DHi-DMV G-UoWJDHG-I-D UVjZd6IJ\ q El C = 0.05

I

1.055 1.065 1.075

ω

−35

−30

−25

−20

−15

−10

−5

0

S(ω),dB

123

ωex

ω0

ωex

ω0

#+.h Qd T:dghYd Nd8Q-/8TN/ &DdgXhN/8h ω0T X-(XhN+q + " " ωex = 1.06 .f(T:d 0 ωex =

1.064 .f1(TNd ( ωex = 1.065 .f(T:d

x q EDMV§UaG6UPdIJHI-i-c<B<Ua\<f9XMj-DHYJ\i¢Vc! G-cXM\<I Sx(ω)jJDHi-DMVDHG-G-cr

x(t)"aIJVDMGJ[J[

ωexB1IJGJ\<DMi-ByUPdD1.060 6 ωex 6 1.065

"*BY;ca\<c<i-cVωex

xadIJHYZUYω0

KDM q dfR\PUa\<li6UXMW-DM\PU XHj-DMYJ\<iJc<B E6d6[\i-DNpHG6URW-DHG-IJr

ωex

j-i-DFEXM\PUBPdDHG-l"G-U i-IJXKJK QJKhnUY B<I2EG-c IJiJI-X q G-Y;U "E6d-[ c\<GJc<XHIJ\DNdfG-c§xcadf!Tc<r i6UXMXM\i-c<r-Y-I W6UXN\<c\

YJi-I-ByUa[Ô BuXHjJDHYJ\i-Dzj-i-I-X q \<XM\B q \ndI-G-IZ[ XHIJHG6UPdUuB<ceEDHr-XN\<B<IZ[ ωex

I`d6I-G-IJ[xyUaHc<B<c<rTW6UXM\c\<l

ω0

K q VDHGJf! DMG-I-DHV§i6UXMXM\<iJc<r-Y-I W6UXN\<c\ N\<I dI-G-IJIXHxadI ¡ U \X%[ Y-i-I-B<Ua[ Z uI=G6UY2c<GJDHk=i-DUodIJ q DM\X%[ HmnmnDHYZ\UPpZByUa\PUxyUaHc<B<c<r W6UaXM\<ca\<lp2Uac\<I-WJDHXHY2c<Mc Ua\i6UYJ\c<i6UbB<GJD GJI-VsXMI-HG-UPdc<V

ωex = ω0

"Y-iJI-ByUa[ QNK UopJB<U \ q W6UXN\<c\lgc\<B<DMW6UDN\ G6URW-DMG-I-DbW-UXM\c\<l ωex = 1.065

"]G-Di6UB<GJc<DIJX pycHEG-c<rB<DedI-W-I-GJD

ω0 = 1.068K

A i-DN q dfR\PUa\<DRUPpZByUP\PU¢XMj-DHYZ\<i6UPd6f<G6Ua[dI-GJIJ[_xyUaHc<B<c<rW6UaXM\<ca\<l"Ua\a\<i6UaYJ\<c<i-UXHVDMXM\IZdUXMf I XN\PUPdUni6UB<GJc<r¢W6UXM\c\<D B<G-D G-DMHcTBcNEDMr-XM\B<IJ[ K EDHXHfnI-VDMDM\TVD XM\c j-cad6G6Ua[1UG6UPdcHIJ[X MmnmDMYJ\<cV RUPpZByUa\PU W6UXM\c\<lB YZdUXMXHI-W-DMXHY2c<rb\DHc<i-IJIXHIJGZpZi-c<G-IJUk-I-Ij-i-DeEDNdf<GJc<Hc§k-I-YZdUZKSUodf<G-DMr I-D_I-XMXNdDFEc<ByUaG-IJ[(j-c<Y;UaRUodI "WJ\c M\PU UG6Uodc<HIZ[LGJD8Xed q W6Ur-G-UJKCS`d-[XHj-I-i-UPdf<G-cHc pyUc<XHU "zY;UYLIE6d-[LjJi-D EDedf<G-c<Mc=k-I-YZdU " Vc ¡ G-c Mc<B<c<iJIJ\<f=c<x MmmnDMYJ\PUPp UPpZByUa\PU=W6UXM\c\<l I mÙPl9K EG6UY2c M\<c`\<i-DMx q DN\¢Y;ci-i-DHYZ\<G-c<McB<B<DFEDHG-IZ[ c<j-iJDeEDedDHG-IZ[ mUal p2Uc\I-W-DMXHY-IZpUB\c<Y2cadDMxyUG-I-r K

,2,,?¢ +-$ *>J&2+8+Z)2+8 2+-,AClzXMc<Y;Ua[ XN\<DHjJDHG-fi-DM q d6[Ji-G-c<XN\<IB<i6U DHGJIJ[^mUaHc<B<c<r¢\<i-UDHYJ\c<i-I-IUa\a\<i6UaYJ\<c<i-U MXHXedDHi6U " cx q XedUBadIJByU Ua[¤G6UodI-W-I-D [Ji-Y2c B<lzi6U ¡ DMG-G-c<Mc j-I-Y;U B XHj-DMYJ\<iJD

Vc! TG-c<XM\I i-I-XaKJK O "j-cHB<cad6[JDM\_j-iJDeEXN\PUB<IJ\fY2cadDMxyUG-IZ[x(t)

Iy(t)

B BI;ED!"Y2c\<ci-lzrbI-XMj-cadfHc<ByUPd6X%[G-UVIjJi-I1UaG6UPdIJHD`XMI-GZpZi-c<GJIJRUkJI-IbHDMG-DHi-Ua\<c<i-UÂ UGEDMi 9cad6[ XHVzK-DHY-kJI-s 2

x(t) = A(t) cosΦ(t), y(t) = A(t) sinΦ(t). JK Q

UVDMG6U j-DHiJDHVDHGJG-lCp JK Q ¢cG6URW6UaDM\ XNd6DeE q TDHDaK ©nU8EB q VDMi-G-c<rj-cB<DHi<pZG-cXM\I j-i-cDHY-k-IZ[=Ua\\i6UYJ\c<i6U_G6U^jZd6c<XHY2c<XN\<f_j-DHiJDHVDHGJG-lCp

(x, y)"iJI-XKZK O" = Vl

B<B<cHEIJV cj-i-DeEDNdDMG-IJ[_VHGJc<B<DHGJG-c<r_UVjZd6IJ\ q El A(t)I mUal

Φ(t)K VjZdIZ\ q EU

Y2cadDMxyUG-I-r B Vc<VDHGZ\ B<i-DMVDHG-Itc<j-i-DFEDNd6[JDM\X%[=YZUaYE6dI-G-U^B<DHYZ\<c<i6U "XMc<DeEIJGJ[

TDHMc`\<cW-Y q G6U9mUacB<c<r\<i6UaDHYJ\c<i-I-I X G q d6DHV Y2c<ci<EI-G6Ua\oK MG-c<B<DMG-G-c<r mUaDΦ(t)

c\<BDHW6UDN\ q Hcad "ZY2c\<c<iJlzrc<x<i6Ua q DN\¢B<DHYJ\c<i A(t)X c<XHf<LY2c<c<iEI-G6UP\

y = 0K

STi q HI-VI Xedc<ByUaVI "RUVDMG6U1j-DMi-DHVDMG-G-lCp JK Q 'cG6URW6UaDM\ j-DMi-DNp<cHE(G6UjZd6c<XHY2cXM\I mUac<BlCpY;cc<iEI-G6Ua\

(x, y)Ybj-cPd6[-i-GJc<r1XMI-XM\DHVD c\XHW-DN\PUJKA M\c<V¤XNd q W6UaD

Vl Vc ¡ DHV"B<B<DMXM\I cj-i-DeEDNdDMG-IJ[ VHGJc<B<DHGJG-c<r UVjZdIZ\ q El A(t)I mUal

Φ(t)XNd6DeE q TI-V¤c<x<i-Uac<V2

A(t) =√

x2(t) + y2(t),

Φ(t) = arctgy(t)

x(t)± πk, k = 0, 1, 2, . . .

JK

'nc<j-i-DFEDNd6DHG-IJ[VMG-c<B<DMG-G-c<r_mUalΦ(t)

XNdDFE q DN\^cj-i-DeEDNdDMG-I-DnVHG-cB<DHG-GJc<r_IXHiJDeEG-DMrbW6UXN\<c\ Y2cadDMxyUG-I-r/2

ω(t) =dΦ(t)

dt, ωmean = 〈dΦ(t)

dt〉 = 〈ω(t)〉. ZK t

+-<>->-,-#%+J)2+J4(<,£,*)2+-34 +J , £+-<+9i-c<IJHB<DeE6[ RUVDMG q j-DMi-DHVDMG-G-lCp JK Q B q i6UBG-DHG-IZ[Zp JK Z "j-cad q W-IJV q i6UBG-D G-IJ[XMI-XM\DHVl MXHXedDHi6U B BI;ED2

A =1

2αA− 1

2αA cos 2Φ− z cosΦ+ C sin(ωext),

Φ = 1 +1

2α sin 2Φ+

z

AsinΦ,

JK w

z = β + z(A cosΦ− µ).

UmnI-Y2XHIJi-c<ByUB§G6URW-DMG-IJ[¤j6Uai6UVDM\i-c<Bα = β = 0.2

"µ = 6.5

IC = 0.05

"j-i-cI-GJ\<DMHi-IJi q DMV8XMI-XM\DHV q JK w W-I-XNd6DHG-G-c "RIJVDMGJ[J[W6UXN\<c\ q B<G-D G-DMHc B<cNEDHr XM\B<IJ[

ωex

K nXMj-cadfa q [¢jJcad q W-DHGJG-lzDzEUG-G-lzD E6d-[ Φ(t)"B<lzW-IJXNdI-V§XHc<ca\<B<DN\<XM\B q

TI-DnG6URW-DMG-IJ[XHi-DFEG-DHrW6UXN\<c\lωmean

JK t!NK6DM q dfR\PUa\lj-i-DeEXM\PUBad6DHG-l"G6U

1.060 1.065 1.070 1.075 1.080 1.085

ωex

0.990

0.995

1.000

1.005

1.010

Θ

12

dgT8(*X-(q /YXh /8hY'*/ ,+-'(+DdgXhN/Yh Θ = ωmean/ωex .f(TNd ( Θ = ω0/ωex

.f1(T:d /Yh *d8XhY/YhTY/YQIR*+ *Xh8TY( ωex

0fd8XX(*h8dg''AR` WX-(*XhY+q "

i-I-XaKJK "-Y2c\<c<iJlzr1IZd-d6uXM\i-I-i q DM\UB<I-XMI-Vc<XM\I1c\G-c!TDHG-IJ[W6UXN\<c\ ωmean/ωex YJi-I-ByUa[LO Iωmean/ω0

Y-i-I-ByUP[ Z ca\ B<DedI-W-IJG-l W6UXN\<c\l XHI-MG6UPdU8B<cNEDHr XM\B<IJ[

ωex

KhnUY"B<I;EG-c¤IJ Hi-UmnI-Y2c<B"uIJVDHDM\VDMXM\<cRUopJB<Ua\XHi-DFEG-DMrsW6UaXM\<ca\<l

ωmeanB<G-D G-IJV XHIJHG6UPd6c<VB Y;cG-DHW-GJc<r c<xadUXN\<I W6UXN\<c\1.065 6 ωex 6 1.078

K 9i-IM\c<V i6UXMXHW-IZ\PUG-G-lzDuHG6URW-DHGJIJ[ XMi-DeEGJDHr W6UaXM\<ca\<l

ωmean

I xyUaHc<B<c<r W6UXN\<c\lUa\\<i-UYJ\<ci6U

ω0

XHc<B<j-UoEU \ B¢j-iJDeEDedUPpc! IJx<Y-IB<lzW-I-XedDHG-IJrK9XHj-cPdf q [EUaG-G-lzDTi-UXHW-DN\<c<BVMG-c<B<DMG-G-c<rmUal Φ(t)

"j-c<XN\<i-cI-VHi6UmnI-YJIRUaB<I-XHIJVc<XM\ILVMG-c<B<DMG-G-c<ri6UaHG-c<XN\<ImÙP

∆Φ(t) = Φ(t) − ωext HDMG-DHi-Ua\<c<i-U

MXHXedDHi6UÌ B<GJD GJDHHc`XHI-MG6UPdUBnRUaB<I-XHIJVc<XM\I^c\B<i-DMVDHG-I KZ©nU9i-I-XaK;JK t'j-iJDeE XM\PUaBadDHGJl M\<IRUBI-XHI-VcXM\<IE6d6[_\<iJDNp_G-UoWJDHG-I-rW6UXN\<c\l"B<G-DTG-DHMc^XMI-HG6UodUJKhnUYB<I;EG-c IJ Hi6UmnI-Y;U "oB c<xadUXN\<I`XHI-GZpZi-cG-IJRUak-I-I YJi-I-ByUa[TQ I-VDMDM\'VDMXM\c

HmnmnDHYZ\RUPpZByUa\PU mUaHlHDHGJDHi6Ua\c<i6U_DMXHXedDHi6UB<GJD GJI-VLXMI-HG-UPdc<VzK niJI M\<cVi6UaHG-c<XN\<f^mUa

∆Φ ' 0IG-D9RUaB<I-XHIZ\ ca\^B<iJDHVDHGJIK

6+6<>- <,*£,)2+J3 +J,a £+J<+ACB<DFEDHVBbXMI-XM\DHV q DHXHXedDHi-U JK O j6Uai6UVDM\i Ω " q j-i6UaBad6[- I-r xyUPc<B<cr W6U XM\c\<cr1XHjJDHYJ\i6U^Vc! TG-c<XM\I

ω0(Ω)Ii6UaXHXHVca\<i-I-VG-DUBa\<c<G-cVG q XMI-XM\DHV q 2

x = −z − Ωy + C sin(ωext), y = Ωx + αy, z = β + z(x − µ). JK

UmnI-Y2XHIJi q DMV j6Uai6UVDM\i-l α = β = 0.2"µ = 4.0

Iωex = 1

K 9i-c<IJHB<DeEDHVRUaVDHG q j-DMi-DHVDMG-G-lCp ZK Q IB<lzj-cPdG-I-V|GJDHc<xpycHEIJVlzD i6UXMW-DM\l \<IJj6U(j-iJDeE

8k

0 2000 4000 6000 8000 10000t

−10

0

10

20

30

40

50

60

70

80

90

100

∆Φ(t)

123

w '/8TY+-''*d fd8Q-'/YXh ud8Q ∆Φ(t) = Φ(t) − ωextTQMdJTY(X-(*q /YX-hY( /Yh TNf+q +-'( TWX-(XhN+q +

" R*` hYf+ BQ-'Dd *+I'( *d8XhN/8h T8'*+ >'+ M/AX-('DdJ`Dd ωex = 1.06 .f1(TNd 0 ωex = 1.064 .f1(TNd ( ωex = 1.065 .f1(T:d

XM\PUaBadDHGJG-lCp G6Ubi-I-XaKJK I JK t_E6d6[ TI-i-c<Y2c<r c<xPdUXM\I G6URW-DMG-I-r(UVjZdIJ\ q ElB<G-D G-DMHc XHI-MG6UPdU

CIxyUacB<c<rW6UXN\<c\lUa\\i6UYJ\c<i-l

ΩKDM q dfR\PUa\<li6UXMW-D

\<cB j-iJI-B<DeEDHG-lG-U x<IJm q i-Y;UkJI-c<G-G-cr_EI6UaHi6UVVD i-I-XKZK w eK xadUXN\<f"j-c<VDMW-DHGJG6Ua[(G6U EI-UHi6UaVVD1k-I-mni-crO"]c\<BDHW6UDN\ i-D ¡ I-V q XMI-G

pZi-c<G-GJc<Hczp2UcXUJKoA M\c<rc<xadUXN\<Ii-DUodIJ q DM\X%[ HmnmnDHYJ\ UPpZByUa\PU x<Uac<Bc<r9W6UXN\<c\<l(Ua\a\<i6UYZ\<c<i6U

ω0

B<G-D G-I-V8XHIJHG6UPd6c<V=G6U'c<XHGJc<B<G-c<Vb\c<G-D "ω0 ' ωex

K EG-c<B<iJD VDHGJG-cuB M\c<rTc<xPdUXM\ITI-VDMDM\ VDHXN\<cuUPpZByUa\ mÙP X q XNdc<BI-DHV ∆Φ = Φ(t)−ωext =0K$9DHiJDNpycHEUaV"IJ^cxadUXN\<IO B1cxadUXN\<f Z c\B<DHW6UaDM\bjJDHi-DepycHE(c\i-D ¡ I-V]U_XHI-G pZi-c<G-GJc<Hc pyUc<XU§Y G-DHXMI-GZpZi-c<GJG-c<V q K \<I-V j-DMi-DNp<cEUV XMc<c\B<DM\XM\<B q \ \<c<W-YJIj-DMi-DHMI-xyU^Y-iJI-B<lCpG6U¢Hi6UamIJY;Dì-I-XaKZK "-\c^DMXM\<f G6Ui q TDHG-I-Dì-D ¡ I-V]U¢RUPpZByUa\PUW6UXN\<c\l"ImUaHl9KA c<xPdUXM\I Qî-DHUPdIJ q \<X%[=iJD ¡ I-Vl"j-DHiJI-cHEI-W-DMXHY-IZp Y;cPdDHxyUaG-I-r jJi-DeEDedf

G-lzD kJI-YZdl"j-DMi-I-cHEUT = 5 2π

ωex

Ibk-I-YZd6l q EB<c<DHGJG-lCpbjJDHi-I-cHEcB NKALc<xPdUXM\I 1′B<G-cB<f_B<cGJI-Y;UDM\i-D ¡ I-VLXHI-GZpZi-cG-G-c<Mcp2Uac<XU " q EcBadDM\B<c<iy[- I-r B<XHDMV q XedcB<IJ[JVc<xPdUXM\I§OK

XNdI c<Mi6UG-IJW-IJ\<fX%[V]UPd6lzVI HG6URW-DHGJIJ[-VI(UVjZdIZ\ q El BcNEDMr-XM\B<IJ[ C 6

0.35"*\c1Vlªj-cPd q W6UDHV XNdDFE q q Y;UiJ\I-G q 2 IJVDHDM\bVDHXM\cp2Uai6UYJ\DHi-G6UP[ c<x

dUXN\<fXHI-G;pJiJc<G-IJUk-I-I=p2Uc<XHUB_B<I2ED YZd6uByU " Y;UY8I1E6d6[=XNd q W-Ua[8j-DHi-IJcEI-W-D XHYJIZpUB\<cY;cPdDHxyUaG-I-rK \dI-W-I-DB8\c<V "WJ\<c8NEDMXHf I(B<G q \<iJI " I(B<G-Dc<xadUaXM\<IXHIJGZpZi-c<G-IJUk-I-I Y2cad6DHxyUGJIJ[§[JBad6[J \<X%[8p2Uca\<I-W-DMXHY-IJVI " G-cXM\i q YJ\ q i6UbIZp§i6UP dI-WJG6UJK

0.92 0.93

0.02

0.03

0.04

0.01

0.05

0.000.91

22

3

Ω

C

4

1

1

( i f.1d(/8''Dd R*(*d Mfd8q q3dBX-(XhN+q " '*dor`/YX./8XhY(xr*d8fd8q +hNf/8T (C,Ω)

-!. ! " '+-* *-,#$% ) '' -*$%#' $

9i-I"BUI-VcHEDHrJXM\<BI-I"EB q p"XHcB<DHi TDHG-GJcLI;EDHGJ\I-W-G-lCp"pyUc\<IJW-DHXMY-IZpXMI-XM\DHVVc ¡ GJcG-Uxad EUP\<f_[-BadDMG-I-D¢j-cad6G-c<r=XHI-G;pJiJc<G-IJUk-I-I=p2Uc<XHU 2G6URW-I-G-Ua[8XG-DHY2c\<ci-c<HcHG6URW-DHG-IZ[§j-Ui6UVDN\<i6U_XHB[ZI " Y;cPdDHxyUaG-IJ[=j6UiJk-I6UPdfG-lCp XMI-XM\DHVXN\PUG-cB[Z\<X%[j-cadG-cXM\<fI2EDHGJ\I-W-G-lzVI K UaXHXHVca\<i-I-VXHI-XN\<DHV q cHEG-ca\<I-j-GJlCp1BaRUI-VcHEDMr-XM\B q TIZpbcXHk-IZdJd6[J\c<i-c<B 2

x1 = F (x1,λ1) + γg(x1,x2),

x2 = F (x2,λ2) + γg(x1,x2), JK ]

EDλ1

Iλ2

B<DHYZ\<c<i-GJlzDj6Ui6UVDN\<i-lc<XMk-IZd-d-[J\<ci-c<B-K#XNdIλ1 = λ2

"\<c j-Ui k-I6Uodf<G-lzDXMI-XM\DHVl j-cPdG-c<XN\<f<ªI;EDHGJ\I-W-G-l9K~ q G-Y-k-IZ[ g(. . .) cj-i-DeEDNd6[JDM\p2U i6UYZ\<DHi=XMB[JHI "jJi-I-W-DMV

g(x1,x1) = g(x2,x2) = 0KA XNd q W6UaDj-cadG-crbI;EDMGJ\<I-W

G-c<XN\O@FHS5IEGFL=W? IE.@S nK~UacB<lzD`\i6UDHYZ\<c<i-IJI "dD ¡ U TI-DTBU"XMc<c\<BDM\

XM\B q \^j-cadG-cXM\<fsXMI-GZpZi-c<G-GJlzV Y2cadDHx<UG-IJ[JVzKXNdIj-i-DFEDNd6f<G-c<D VG-c ¡ DMXM\<BcU[-Bad6[JDM\X%[sjJi-IJ\R[-MI-ByU I-V B¤mUaHc<B<c<V

j-i-cXM\<i-UG-XM\B<DuXMI-XM\DHVl JK ] "\c`DMXM\<fn[-BPd6[-DN\<X%[Ûa\\i6UYJ\c<i-c<V "y\<ci-DHUPdIJ q DM\X%[,)C E ODS XH?DS IAB?@FHST?BA :<=&BABA6K UaXHXHVca\<i-I-V HmnmnDHYZ\¢j-cadG-cr XHI-G;pJiJc<G-IJUk-I-IB XHIJXM\<DMVDuEB q p XMI-VVDM\i-I-W

G-cnXHBa[JRUaG-G-lCp I;EDMGJ\<IJW-G-lCp c<XHkJIZd-d6[Z\<c<i-cBT HXHXedDHi-UJKy¨'i6UBG-DHG-IZ[¢XHI-XN\<DHVl U j-I TDHV B¢B<I2ED2

x1 = −z1 − y1 + γ(x2 − x1),

y1 = x1 + αy1,

z1 = β + z1(x1 − µ), JK

x2 = −z2 − y2 + γ(x1 − x2),

y2 = x2 + αy2,

z2 = β + z2(x2 − µ). UaXHXHVca\<i-I-V i-D ¡ IJV XHj-IJi6UPdf<GJc<Hc§pyUc<XHU B Y;U ¡ Ec<r(IZp(BRUIJVcEDHr-XN\<B q

I;pjJcEXHI-XN\<DHV "j-cad6c ¡ I-Bα = β = 0.2

Iµ = 6.5

K 9i-cXNdDFEI-V MB<cad k-I-j-i-DFEDNd6f<G-c<McVG-c ¡ DMXM\ByUmUaHc<B<lCp1\<i-DMYJ\<c<iJI-r XMI-XM\DHVl JK X q B<DNdIJW-DHG-IJDHVj6Ui-UVDM\i6U§XMB[JHIK M\c<rkJDNdf<gx q EDMV XN\<i-c<IZ\<f j-iJc<DHY-kJI- UP\\<i-UYJ\<ci6U§G6UjZdcXHY2c<XM\f j-DMi-DHVDMG-G-lCp

(x1, x2)IZd6I

(y1, y2)K2D ¡ I-V q j-cadGJc<r XMI-GZpZi-c<GJIJRUkJI-I

j-i-I N\<c<Vsx q EDM\=c\B<DHW-Ua\<f8XHI-VVDN\<i-IJW-G-c<Dj-cHEj-i-cXM\<i-UG-XM\B<c U "d6D ¡ U TDHDG6UEI6UaHc<G6UodIx1 = x2

IZdIy1 = y2

NK©nU§i-IJXK ZK §jJi-DeEXN\PUBadDMG-l XMc<c\B<DM\XM\<B q I-D8i-DN q dfR\PUa\<li6UXMW-DM\c<B§E6d6[

G-UoWJDHG-IJ[ Y;c!HmnmnI-k-I-DMGJ\PU XHB[ZIγ = 0.2

KW9i-I-X q G-Y;U B<I;EG-c "WZ\<c mUaHc<ByUa[

−15 −10 −5 0 5 10 15

X1

−15

−10

−5

0

5

10

15

X2

U

−15 −10 −5 0 5 10 15

Y1

−15

−10

−5

0

5

10

15

Y2

U

= R f/8+.(( d8Q/gTN/ hYf*d8+-.hN/8f(( dghNhNf*dg.hN/8f*d X-(*XhY+q " 'Ddr`*/8X./YXh r+f+-q3+I'' (x1, x2)

l6 (y1, y2)

\<i-UDHYJ\c<i-IJ[ "XM\PUaiJ\ q Ua[=IJTG6URW6UPdf<GJc<r\c<W-Y-I "G-DTjJi-I-G6UoE-dD ¡ U DHr U "XHj q XM\R[=GJDHY2c\<ci-c<D¢B<i-DMV [ j-DHiJDNpycHEGJc<Hcj-i-c<kJDHXHXHU Ec<XM\I-eUDN\1I-GJByUi-I6UaGJ\<G-cHc_j-cHE j-i-cXM\<i-UG-XM\ByU

UI \PUaVcXM\PUDN\<X%[K]DUPd6IJ q DM\<X%[(i-D ¡ I-V"jJcadG-c<r pyUc\I-W-DHXMY2c<r

XHIJGZpZi-c<G-IJUk-I-I "6Y;c EUE6d6[^d6ux<lCpt > t0

t0 B<i-DMV [j-DMi-DNp<cEG-c<Hc¢j-iJc<k-DHXMXU

x1(t) ≡ x2(t)"y1(t) ≡ y2(t)

Iz1(t) ≡ z2(t)

K Y2XHj-DMi-I-VDMGJ\<lj-c<Y;UaUPdI "6WZ\<c¢i-D ¡ I-V j-cadGJc<rXMI-GZpZi-c<GJIJRUkJI-I B ca\dI-W-I-D

c\ mUaHc W6UaXM\<ca\<G-c<r ^Vc ¡ DM\ i-DUodIJcB<lzByUa\f<X%[G-Db\<cPdf<Y2c E6d6[XHjJI-i6UPdfG-c<Hcp2Uac<XU "G-cI B1Xed q W6UD x<cadDMD^XNd6c ¡ G-lCp=p2Uac\<I-WJDHXHYJIZp UBa\<c<Y2cadDMxyUG-IJrK ©nUjJi-I VDHi ";iJDM q dfR\PUa\l\I-j6Ui-I-XK;JK 9j-cad q W-DMG-l¤I¢E6d-[^BUI-VcHEDMr-XM\B q I;p XMI-XM\DHVc<i-DMG-k6UJK

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XM\I-k-lB EB q p;[-VG-cVXMI-VVDM\i-I-W-G-cVj-ca\<DHGJk-I6UPdD U(x)X x<cad6f! I-V \i-DHG-I

DHVj-cHE EDHr-XN\<B<I-DMV"EB q p XMIZd 2XM\<cpyUXM\I-W-DHXMY2c<r F (t) q V I j-DHiJI-cHEI-W-DMXHY2c<r

A cos(2πf0t) XMI-HG-UPd NK Agc\<X q \<XM\B<I-D XMI-HG6UodU(I q V]U(XMI-XM\DHV]UIJVDHDM\ EByU

q XN\<c<rJW-I-B<lCp XHc<XN\<cR[-G-IZ[ i6UaB<G-c<B<DMXHIJ[ x1Ix2"*XHc<ca\<B<DN\<XM\B q TIZp VI-G-IJV q V]UaV

j-c\DHG-kJI6UPdU8Bb[-V]UPp " I G-D q XM\c<r-W-IJB<c<D!"XHcc\<B<DN\<XN\<B q TDHDbV]UaY;XMI-V q V q j-c\DHG k-I6Uodf<G-c<r HG-DMi-HIJI

x0

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< T >j-cHEW-I-GZ[-DM\X%[ÛY2c<G q i-i-DHGJI q

XU "JB<lzi6U ¡ U TDHV q X%[_B HY2XHjJc<G-DHGJk-I6UPdfG-c<rRUBI-XHI-VcXM\<I_c\ B<DNdIJW-I-G-lj-c\DHG k-I6UodG-c<Mc xyUi-fDHi6U

∆UI I-GJ\<DMG-XHIJB<G-c<XN\∝ exp(∆U/D)

K9DHVB<l TDj-ca\<DHGJk-I6UPdfG-lzrxyUai-f<DHi I G-I ¡ D q iJc<B<DHGJf q V]U " \<DHV"Ecadf D MY;XMj-cG-DMG-k-I6UPd6f<G-cTEcadf D W6UaXM\<IJk6Ux q EDM\¢m'd q YJ\ q IJi-c<ByUa\f BTj-c\<DMG-k-I6Uodf<G-c<r[JVDK

S UG-G6UP[_XHI-XN\<DHV]U G-DnI-VDHDN\^XMc<x<XM\B<DHGJG-c<r_EDM\<DMi-VI-G-IJi-c<ByUGJG-c<rbW6UaXM\<ca\<l "cEG6U Y2c`X q DHXN\

IXMB[JUG-G6UP[ X'G-DMrPFHC ;>?0 =tIE STEK= ODCPFHC1 >X CI?BA fs = 1

2<T>

"ZI-Mi6U Ua[ iJcadf XMc<x<XM\B<DHGJG-c<r W6UaXM\<ca\<l9K \PU B<DedI-W-IJG6U HY2XMj-c<G-DMG-k-I6UPd6f<G-cW q B<XN\<B<I \<Dedf<G6U YIJVDMG-DHG-IJ q i-c<BGJ[ q V]U "\c DMXM\fuY2c\<iJcadI-i q DM\X%[ q Vc<VzKRA YZduWJI-VBi6UXHXMVc\<iJDHG-I-DuXNdUxlzr j-DMi-I-cHEI-WJDHXHYJI-rXHI-MG6UPd*K dUx<lzr zcG6URW6UaDM\ "2WJ\<cTUV jZdIZ\ q EUbj-DMi-I-cHEI-W-DMXHY2c<r XHI;dlªV]UPdUG6UXM\cadf<Y2c"WJ\cbXHI-MG6UPd XHUV"j-cXHDHxD^G-DVc ¡ DM\j-DMi-DHxi-c<XHIZ\<fTW-UXM\I-k q IJ cHEG-c<r¢j-ca\<DHGJk-I6UPdfG-c<r [JVlB Ei q q ";U9j-i-I B<cHEIJ\=dI Tf§Y j-DMi-I-cHEI-WJDHXHY2c<rVcE q d-[-k-I-I(j-c\DHG-k-I-UPdUJK I-MG6UPd B<G-c<XMIJ\8j-D i-I-cHEIJW-DHXMY q Y2c<Vj-c<GJDHGJ\ q BXNd q W6UrJG-lzr1j-iJc<k-DHXMXnj-DHi-DMXHY2c<Y2c<B W6UXM\I-k-l9K XNd6IXHiJDeEGJ[Z[ W6UXN\<c\PUbjJDHi-DMXHY2c<Y2c<B=XHc<Bj6UoEUDN\bX W-UXM\c\<c<r jJDHi-I-cHEIJW-DHXMY;cHc=XHI-MG6U dU "

fs = f0

" \<c_j-iJc<k-DHXMX¢j-DHi-DMXHY2c<Y2c<BBXMi-DeEG-DHVLx q EDM\XNd6DeEc<B<Ua\<fmUaDj-DMi-I cHEI-W-DMXHY2c<r XMIZdl9K©nUxad EUDM\X%[ X q TDHXM\B<DHGJG-lzr i-c<XM\c\YZdI-Y;UxI-XM\PUxIZdf<G-crXHIJXM\<DMVl G6U jJDHi-I-cHEIJW-DHXMY;cD B<caV q DMG-I-D!"\<c_DHXN\<fG6UxPd EUDN\<X%[8i-DMHc<G6UGJXHG-c<D[-BPdDHG-IJDK6~ q G;EUaVDHGJ\PUPd6f<G-c<D9c\d6I-W-I-D N\<c<Mc HmnmnDHYJ\PU ca\ YZdUaXHXHIJW-DHXMY;cHci-DNcG6UGJXU B \<c<V "PWJ\cuBuG-U DMV XNd q W6UD XHI-XN\<DMV]U'XUV]U j-cuXMDHx<D G-DI-VDHDN\9XHc<xXM\<BDHG-G-cr;<CIE CQF "AB?BA@FHSW=W?B?DS =tIE STE" "Ubc<xPdUoEUDN\bdI Tf8Y2c<GZ\<i-cad6I-i q DMVc<r q VcVXHiJDeEG-DMr W6UXN\<c\c<r jJDHi-DMXHY2c<Y2c<B-K 9cM\c<V q M\c[-BPdDHG-IJD xlCdcG-UaByUaG-c E S$D= IEKA C1 A i-DNc<G-UG-XHcVzK "J\PUaY-I-V c<x<i6UaHc<V "2[-Bad6[JDM\X%[_c<x<c<x! TDHG-IJDHV YZdUXMXHI W-DMXHY2c<Hc=[-BPdDHG-IZ[ i-DMHc<G6UGJXU=G6UbXNd q W6Uar XMI-XM\DHV "]c<xadUoEU IZp§p2Ui6UaYJ\<DMi-G-lzVB<i-DMVDHG-GJlzV V]UaXT\PUax<c<V " q j-i-UBad6[JDHVlzV q Vc<VzK DMc<i-DM\I-W-DMXHY-I-D1I HY2XHjJDHi-I VDHGZ\PUPdf<G-lzDI-XMXNdDFEc<ByUGJIJ[j-c<Y;UaRUodI "-WJ\c¢j-i-IV]UPdlCp_UaVjZdIJ\ q EUPp_I_G-IJHY-IZpW6UXN\<c\PUop XHI-MG6UPdUbca\<G-c!TDHG-IJDXHI-MG6UPd q V " "W6U D`B<XHDMHc I-XHj-cPdf q DHVc<DnE6d6[Y2cadI-W-DMXM\B<DHG-GJc<r=c<k-DHGJY-I $"c<jJI-XHlzByUaDM\<X%[1mnc<i V q dc<r SNR ∝

(A∆U

D

)2exp(−∆U/D)

KhnUoWJDHXM\B<DHGJG6Ua[ RUB<IJXHI-Vc<XN\<fSNR

c\I-GJ\DHG-XMI-B<G-cXM\<I q V]UX p<DHV]Ua\I-W-DHXMY-I j-c<Y;UaUG6UG6Ui-I-XaKO [ K Z K q B<DNdIJW-DHG-IJDHVq i-c<B<GZ[ q V]U D c\<G-c DHGJI-D XHIJHG6UPd q Vi6UXM\DM\ "Ec<XM\I-eUDM\_V]UaY;XMI-V]UPdfG-c<HcG-UoWJDHG-IJ[§j-i-I8c<jJ\I-V]UPdfG-c<V q i-c<BG-D q V]U "XHcc\<B<DN\<XMB q IJViJDMcG6UG-XMG-c<V qq Xedc<B<IJ "IUa\<DMV q x<lzByUDN\oK DHrJW6UX'IJHB<DHXN\<G-c "JWJ\c G-Uxad EUaDM\<X%[_GJD \<cPdf<Y2cBTx<IJXM\PUx<I;df<G-lCp XHI-XN\<D

V]UPp "GJc9I B9x<cadDMD XNd6c ¡ G-lCpTXHI-XN\<DMV]UPp "yG6Uj-i-IJVDHi¢B'XHIJXM\<DMV]UPp¢XEI-G6UVI-WJDHXHYJI-Vp2Uac<XHc<V "aI Bnx<cPdDHDzj-iJc<XM\lCp XHI-XN\<DHV]Uop K XHcx<DHG-GJcIJGJ\<DMi-DHXMG-l¤Ij-DHiJXHj-DMYJ\<I-B G-l"j-iJIZdc ¡ DHGJIJ[ ¤B^xI-cadc<MI-IKALiy[;EDTi6Uxc\ O ] x<lCd6c^j-c<Y;UaUG-c"WJ\c Vc ¡ DM\ x<l \<fI-XMj-cadfac<ByUaGE6d6[ cx [JXHG-DMG-IJ[1mDMG-c<VDMG6U i6UXMj-cG-UByUG-IZ[bx<I-cPdcHIJW-DHXMY-I-VI c<x DMYJ\PUVIW-i-DMHB<lzW6UrJG-cuXedUx<lCpXMI-HG6Uodc<B"Rj-i-UYJ\<IJW-DHXMY-I XHYJi-l \<lCpB q VcB<c<V c<Y-i q ¡ DMG-I-IK 9i-c<XM\DHr Ua[VcHEDNd6f G-DHr-iJc<G6U " Y;ca\<c<i-lzr [JBad6[JDM\<X%[jJi-c<B<cHEG-IJY;cV nI-G-mnc<i-V]Uak-I-I1B ¡ I-B<lCpc<x DMYJ\PUPp "j-i-DeEXM\PUBad-[-DM\ XHc<x<crx<IJXM\PU x<IZd6f<G q XMI-XM\DHV q XTEB q V [ q XM\c<r-W-IJB<lzVI XHc<XN\<cR[-GJIJ[-VI "*XMc<c\<BDM\<XN\<B q I-VIG-DMB<cx q ¡ EDMG-G-c<V q I¢B<cHx q ¡ EDHG-G-cV q Hc<iy[ TDHV q G-DMr-i-c<G q K U9XHW-DN\ HmnmnDHY

k

0.00 0.10 0.20 0.30 0.40 0.50D

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

SN

R (

dB

)

D

SNR

dJTY(*XI(*q /YXh rf/g(*Q-TY/8`8' +>+IR('( /8hB('hN+-'X-(T8'*/YX-hY( i q3d D " ∆U = 1/4

\PU ¤W q B<XN\<B<IJ\DNdfG-c<XM\f\PUY2c<rbXHIJXM\<DMVlsYV]UPdlzVU q V dDMG-G-lzVXHI-MG6UPdUaVVc ¡ DM\ x<l \f j-cB<l DMG6U B EDMX%[J\<YJI1IXHc\G-I1i6UP

. " .1- &!(' !.1-

Sd6[ j-i6UYJ\I-W-DMXHY-IZp j-i-IZd6c ¡ DHG-I-rÎ-GJ\<DMi-DHXMG-lXHI-MG6UPdl "ZG-DHX q TI-D'I-GJmci-V]U k-I-II-VDM TI-D`Xedc ¡ GJlzrXMj-DHYZ\<i6UPd6f<G-lzrbXMc<XM\PUBJK6A¤XHB[ZI_X M\I-VVlLI-XHXedD Ec<B<UPdI E6d6[ Xed q W6UDHB UVjZdIZ\ q EG-c<r(I mÙPc<B<cr VcHE q d6[Jk-I-IK] lCdc q XM\PU G-c<BPdDHG-c WJ\<c "-E-d6[\PUaY-IZpXHI-MG6UPdcB" ¤\PUaY ¡ DG6Uxad EUDM\X%[ "\oK DaKX q TDHXN\<B q DM\j-i-IJG-k-I-j-I-UPdf<G6UP[B<cHVc ¡ G-c<XN\<fI-XMj-cadfac<ByUaG-IJ[ E6d6[^c<x<i6Uax<c\<YJI XNdc ¡ G-lCpXHIJHG6UPd6c<B-K A i6Ux<ca\<D O ] Vl j-cYZUPRUPdI " WJ\c ªB<cHVc ¡ DHG E6d-[XMI-HG6Uodc<B XY2c<G-DMW-G-c<r TI-i-I-GJc<r XHjJDHYJ\i6UPdf<GJc<rdI-G-IJIKC cadDHD_\<cHc"x<lCdc8\<DHci-DM\I-W-DHXMY-Ij-i-DFEXHY;UaUG-c"yWZ\<c'RU XHW-DN\ IJi-I-G6U'XMj-DHYJ\i6UPdfG-c<rdI-G-I-IG-U'B<lCpycHEDzx<IJXM\PU x<IZd6f<G-c<rXHI-XN\<DHVlVc ¡ DM\^x<l \<f q VDMG-f!TDHG6U u9G-lzVI1Xedc<ByUVI "XMI-HG6Uod=Vc ¡ DM\x<l \f G-D \cadf<Y2c q XMIZdDMG " G-cbI DHHc1YZURW-DMXM\B<c=Vc ¡ DM\ x<l \f q d q W DMG-c RU1XMW-DM\ K

!. #$%- ' #$+ #- , ' -*

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\<lωmn = nω0 +mΩ, n,m = ±1,±2, . . .

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*.

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s v

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OBA <=& E * S FHC *CI?B? )C*CIE ST; A = X + S FDABE M A IODSYXDJ@: LC )C O@FBA FHC$# CI?BABA :<=@;>=& @S5IEK=@?DSIX C ?BA *ST;<C XDJ ?DS ;>AB?B= "A C LS IA E C XHABR<SC C B= FL=& E CPFDA EKA ODS S>;>?DS *CPFB? NFHC UC ?B? F0B;>= mL FDA@ST;>ABE O@FBA *CPF LST? FHC EK?DS + S O@FDA *CI?DCI?BA STOBA =W?B? V= X + S FDABE*S; X FHC LST? IEGF s BABAEGFHC* *CPFD?DS ;>A ,,CQFHCI? BAB= XHJ ?DS A IE C ,C XHCPF =s

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dIi-DUPd6f<G6Ua[XHIJXM\<DMV]U RUREUG6UJK6¦uc<i-c!Tc IJHB<DHXN\<G-lzr jJi-I-VDHi Y2cad6DHxyUa\DNdfG-lzrLC− Y2c<GJ\ q iK6©nUcXHG-c<BD HG6UG-IZ[^X p<DHVlY2c<GJ\ q i6UÌ Nd6DHYJ\i-I-W-DMXHY-IZp^RUY2c<GJc<BTG-Dj-i-DFEXM\PUaBad6[-DN\^\i q EU¢Uj-I-XHUa\<f c<XMG-c<B<c<jJcadUeU I-DnXMc<c\<GJc! DMG-IJ[IjJcad q W-IJ\fq i6UB<GJDHG-IJ[YZdUaXHXHIJW-DHXMY;cHc^cXHk-IZdJd6[J\c<i6U 2

x + ω20x = 0, ω2

0 = 1/LC. O Z K O

D DMG-I-DHV q i6UBG-DHG-IZ[ O Z K O[-Bad6[JDM\X%[ eUi-VcG-I-W-DMXHY-I-r Y;cPdDHxyUP\<DNd6f<G-lzrªB<cB<i-DMVDHG-I jJi-c<k-DMXHX!" W6UXM\c\PU Y2c\<ci-c<Hccj-i-DeEDNd6[JDM\<X%[=j-Ui6UVDN\<i6UVI=Y2c<GJ\ q i6U LICK$9i-I8UoEUGJG-lCp§G6URW6UPdf<GJlCp q XNdcB<IJ[Zp x(t0)

Ix(t0)

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K EG6UY2c9c<WJDHG-fnW6UXN\<cnj-i-I;pycHEIJ\X%[^XN\PUPdY-I-B<Ua\<f<X%[ Xzx<cad6DHD Xedc ¡ G-cr XMIJ\ q Uk-I

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a(t)"6IJVDMi-DHG-G-Ua[1B

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DHr 9XMI-XM\DHVl "Uj-i-IbEI-XHYJi-DM\IJRUkJI-I X UMc<V∆t2a(i∆t) = ai

"i = 1, . . . , N

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j-i-DFEj-cad6c ¡ DHG-I-D c\<c<V "WZ\<c_G6Uxad6 EUDHV]UP[a(t)

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c<x<i-Ua\<G-lCp_UoEURW "6i-D DMG-I-DTY2c\c<i-lCpG-DnVc ¡ DN\^x<l \f cHEGJcG6URW-GJlzVzK

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c\i6U ¡ UDN\MB<cad k-I-c<GJG-lzr j-i-ck-DHXMX i-DHUPdf<GJc<r S " q j-i6UBPd6[-DMVc<r=EDN\<DMi-VI-G-I i-c<B<UG-G-lzVI EI-G6UVI-WJDHXHYJI-VI RUY2c<G-UVIKWXM\<DMXM\B<DHG-GJc" WJ\c \PUY2c<D1jJi-DeEjJcadc¡ DMG-I-D1X q ¡ UDM\ YZdUXMXi-UXHXMV]Ua\<i-IJByUDHVlCpXHIJHG6UPd6c<B-K A W6UXN\<G-cXM\<I "CDHXedI B<iJD VDHGJG-c<r i2[2EDMXM\f`XedDeEXN\<B<IJD'Ux<XMcad \<G-cXNd q W6Uar-G-c<Mc q Vc<B<c<Mc j-i-c<kJDHXHXHU "<\<cHcB<c<i-IJ\f c i-DMY;cG-XM\i q Y-k-I-I G-DnI-VDMDM\ XHVlzXedU " Z t K9c<j-l \PUDMVX%[ j-cGJ[J\f c<XMG-c<B<G-lzD(j-i-c<xPdDHVl "nX Y;ca\<c<i-lzVI XHB[ZRUG-ci-D D

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a(t)KohnUYB<B<DMXM\I G-c<B<lzD]Y2c<c<iEIJG6Ua\<l IXHY2cadfY;c IZp'Ecad ¡ G-c

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c\B<DM\lG-U N\<I EB<U`B<c<jJi-c<XUnj-c`X q \<I EDNdUI^DHXN\<f XMcHEDHi ¡ UGJI-Dui6UPNEDedU\DHc<i-IJIEI-G-UVI-W-DMXHY-IZp^XHIJXM\<DMV "Zi-UXHXMV]Ua\<i-IJByU TDHMcj-i-c<xad6DHV q i-DMY;cG-XM\i q Y-k-I-I^S j-ccHEG-c<VDMi-G-lzV Bi-DHVDMG-G-lzV iy[;EUVzKA O ]\[ FKxlCdUc<j q xPdI-Y2c<ByUG-Ui6Uxc\PU1©K$nUY;UiEU "*BY2c\c<i-c<r j-c<Y;UaRUaG-c"

WJ\c mUac<Blzr j-c<iJ\i-DM\EI-G6UVIJW-DHXMY;crXMI-XM\DHVlVc ¡ DM\ x<l \<f Bc<XHXN\PUG-c<Bad6DHG j-cXHY;UPd-[-i-G-cV q B<i-DHVDMG-G-c<V q i2[2E q ai

"6DMXNdIB¢YZURW-DMXM\B<DTG-DFEc<XN\PU TIZpY;cc<iEI-G6Ua\B<DMYJ\<c<i-UXMc<XM\co[JG-IJ[ I-XHjJcadf q DM\X%[ \<ca\ ¡ DzXUVlzr i2[2E ai

"<BH[Z\<lzr X GJDHY2c\<ci-lzVRUaj6UaNElzByUG-I-DMVzKAsO ] OTFKx<lCdU Ec<Y;UaRUaG6U^\<DMc<i-DHV]U " q \<B<DMi ¡ EUa UP[ " WZ\<c j-ccHEG-c<VDMi-G-c<r§i-DUodIJRUak-I-I

a(t)S "c<xadUoEU DHr Ua\\i6UYJ\c<i-c<V

A"*j-i-I-G-UoE6dD

¡ U TI-V dUoEY2c<V q d VDMi-G-c<V q VG-c<Mc<c<x<i-UaI- "ZVDN\<cHEc<V RUREDHi ¡ Y-I Vc ¡ GJc j-cd q W-IJ\f n VDMi-G q iJDHY2c<G-XN\<i q Y-k-IJ AR

I-X p<cHEG-c<McÙP\\<i-UYJ\<ci6U'Y;UYVG-c ¡ DHXN\<B<cB<DMYJ\<c<iJc<B

x(t)BRn jJi-I n ≥ 2d+ 1

\<DMc<i-DMV]U UaY;DMG-XU 2

x(t) = Λn(a(t)) = (a(t), a(t+τ), . . . , a(t+(n−1)τ)) = (x1, x2, . . . , xn).O Z K Z

c< dUaXHG-c^\DHc<iJDHVD!"c\c<x<i6U ¡ DHG-I-DΛn : A → AR

[-Bad6[JDM\X%[1 dUREY-I-V¤Ic<x<i6Ua\I VlzVG-U

AR

j-c<WJ\I1j-i-Id x<c<rRUoEDHi ¡ Y2Dτ DHXedI

N → ∞ eK9c<j-l \PUDMVX%[ i6UPc<x<i-Ua\<f<X%[8BXHcHEDMi ¡ UG-I-I8\<DHci-DHVl UY2DHG-XHUJK G6Uc<x<c<XMG-c

B<lzByUaDM\ BB<DeEDMG-I-D B`Y;URW-DHXN\<B<DuG-c<B<lCp Y2c<c<iEIJG6Ua\XMc<XM\co[JG-IJ[^XHI-XN\<DHVl¤HG6URW-DHG-IJr

8kcHEG-c<VDMi-G-c<Mc B<iJDHVDHGJG-c<Hc8iy[;EU

a(i∆t)"]BM[J\<lCp WJDHi-DN1G-DMY2c\<c<iJlzrI-GJ\DHi-B<UPd

B<i-DMVDHG-Iτ2

x1(t) = a(t), x2(t) = a(t+ τ), x3(t) = a(t + 2τ), ...O Z K Q

9I-XNd6cY;cc<iEI-G6Ua\ XHc<XN\<cR[-G-IZ[_Y;cG-DHW-GJcI Ecad ¡ G-c`j-i-DHBl Ua\f i-UVDHi-GJc<XM\f¢Ua\\<i-UYJ\<ci6U

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τ hUaY-I-VXHj-c<XMc<x<c<Vc<j-iJDeEDedIJ\<fW-I-Xedc

n"Y2c\<c<iJc<DRUoEUaDM\i6UaHVDHi-GJc<XM\f1Vc

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i-DN\<I-W-DMXHY-I;p iJDM q dfR\PUa\c<B1jJcad q W-IJ\fG-DedfM[ " \i-DHx q DM\X%[ j-i-IJBadDHWJDHG-I-DTEUaG-G-lCpHY2XMj-DHi-IJVDHGJ\PUPd6f<G-lCpI-XMXNdDFEc<ByUGJI-rK9c<jJl \PUDHVX%[ÊUP\<f c\B<DM\lG6U N\<IB<c<jJi-cXHl9KO<K \<cW-Y-Ini-DMG-IJ[`j-cadc ¡ DMG-I-r9\<DMc<i-I-IRUoEDMi ¡ Y;U

τVc ¡ DM\ x<l \<f j-i-cIJB<cPdf

G-c<r K EG6UY2cXHc<B<DMi DMG-G-c_[-XMG-c"WJ\cDMXNdIτXNd6I Y2c<VLV]UPdc" \c

i [=I

i + 1 [

Y2c<c<iEI-G-Ua\<l\<c<WJDHYbmUaHc<B<c<r\<i6UDMYJ\<ci-I-I=j-i6UYZ\<I-W-DMXHY-I=G-DMc\dI-WJI-Vl Ei q c\Ei q eUJKCDMY;cG-XM\i q I-i q DMVlzr"Ua\a\<i6UYZ\<c<i¤B M\<cV Xed q W6UDi6UXHjJcadUeUDN\<X%[B<xPdI I_ dUaB<G-c<r EI-UHc<G-UPdIj-i-cXM\<i-UG-XM\ByU¢Badc ¡ DHGJIJ[ dI-G-IJII2EDHGJ\I-W-G-cXM\<I NK M\c<HcbEc<j q XHY;Ua\f G-DedfM[(B XMIZd q cj-i-DeEDNdDMG-IJ[2]Y2c<c<iEIJG6Ua\<l XHc<XN\<cR[-G-IZ[(DMXM\fG-DNRUB<IJXHI-VlzDj-DMi-DHVDMG-G-lzD " cHEG-cHG6URW-G-cbc<jJi-DeEDed6[- I-DXMc<XM\cR[-G-I-D^XHI-XN\<DHVl9K Ei q Mc<r XM\c<i-c<G-l "DHXedI B<i-DMV [§UoEDHi ¡ Y-I c<WJDHG-fbBDNdI-Y2c"Y2c<c<iEI-G-Ua\<l c<Y;U lzB<U \X%[ G-DMY2c<i-i-DedI-i-c<B<UG-G-lzVI "I=i-DHY2c<GJXM\<i q IJi-c<ByUGJG-lzr Ua\a\<i6UYZ\<c<i GJDc\i6U ¡ UDM\ I-XN\<I-GJG-c<r EI-G6UVIJY-IK ©nU c<XMG-c<B<D MY;XMj-DHiJI-VDHGJ\c<B q XM\PUGJc<BadDMG-c" WJ\<cc<k-DMG-Y;U'c<jJ\I-V]UPdf<GJc<Hc9B<iJDHVDHGJI RUoEDMi ¡ Y-ITVc ¡ DM\9x<l \f9j-cad q W-DMG6U'IJCi6UXHWJDM\<cBUB\c<Y2c<i-i-Ded6[-kJI-c<G-G-cr m q GJY-k-I-I ψ(τ0) =< a(t)a(t+ τ0) >

";Y2c\c<i6Ua[ E6d-[^Xedc ¡ G-lCp¢GJDHj-DMi-I-cHEI-W-DMXHY-I;p j-iJc<k-DHXMXHc<Bx q EDN\`XMj6UoEUa DMr BcBi-DHVDMG-I τ0

K G-UoWJDHG-I-D

τ0"<XHc<ca\<B<DN\<XM\B q TDHD'B<iJDHVDHGJI Ec<XM\I ¡ DHGJIJ[ j-DHi-Bc<HcnG q d6[¢m q G-Y-k-IJI ψ(τ0)

"aI-X j-cad6f q DM\<X%[B MY2XHj-DMi-I-VDHGZ\PUPdf<G-lCpIJXHXNd6DeEc<B<UG-IJ[;p E6d6[^c<k-DMG-Y-I RUREDHi ¡ Y-I τ B O Z K Z Cj-i-IB<B<DeEDHG-I-IG-c<B<lCpY;cc<iEI-G6Ua\ XHcXM\<cR[JG-IJ[VcEDNdf<GJc<rXMI-XM\DHVl9KZ K DHj-DMi-fnG q ¡ GJcY;UYJI-V \c`XMj-c<XHcx<c<V8G6UrJ\I n "yWJ\<c<xlc<j-iJDeEDedIJ\<fì6UaHVDHi G-c<XN\<f8i-DHY2c<GJXM\<i q IJi q DMVc<r XHIJXM\<DMVl9KW9\DHc<iJI-I XNdDFE q DN\ " WJ\c1E6d6[ c<jJi-DeEDedD G-IJ[

nG-DMc<xpycHEI-Vc HG6Ua\<fi-UaVDMi-G-c<XN\<fUa\\i6UYJ\c<i6Ui-DHUPdf<G-crEI-G6UVIJW-DHXMY;cr

XHIJXM\<DMVl "zY2c\<c<i-Ua[Vc ¡ DM\ xl \<f c<j-iJDeEDedDHG6U XedDeE q I-V c<x<i6UaHc<VzK AClzx<i6UBj-i-cIJB<cPdf<G-c<D G-DHxcadf!Tc<D HG6URW-DHG-IJD i-UaVDMi-G-c<XN\<I jJi-c<XM\i6UG-XN\<ByU_Badc ¡ DMG-IJ[n0"Pi-DMY;cG-XM\i q I-i q DMV Ua\\i6UYJ\c<i I c<j-i-DFEDNd6I-V§DMHc9i6UPVDHiJG-c<XM\f dF

XCj-c<Vc f<XHjJDHk-I6Uodf<G-lCpÛPdMc<i-IJ\Vc<B-K Ua\DHV j-cB\<c<iJI-V8EUGJG q Lj-iJc<k-DeE q i q " q BDNdI-WJI-ByUa[n0

Ec¢\<DNpjJc<i "6j-c<Y;U¢B<DNd6I-W-I-G6U i6UaHVDHi-G-cXM\<IdF

G-Dnj-DMi-DHXN\PUG-DN\_j-i-DM\DHi-jJDHByUa\fIJHVDHG-DMG-I-rK c<c\<BDM\<XN\<B q DHDbHG6URW-DHGJI-D dF

E6d6[ p2Uc\I-W-DMXHY-IZp Ua\a\<i6UYZ\<c<i-cBx q EDN\^G-DMk-DNd6lzVzK-S UPd6DHD9c<j-iJDeEDed6[-DN\<X%[bxPdI ¡ Ur DHDnXMB<DHi<p q k-DNd6c<DnW-I-XNd6c"6I_G6U pycHEIZ\<X%[

d i6UaHVDHi-GJc<XM\f j-i-c<XN\<i6UaG-XM\ByU " B Y2c\<ci-c<D q Y;dUoElzByUaDM\<X%[ G6U |Ua\

\<i-UYJ\<ciK j-c<Vc f<mnc<i-V q dl UG

n ≥ 2d+ 1 O Z K

c<j-iJDeEDed6[-DN\<X%[ i-UaVDMi-G-c<XN\<f j-iJc<XM\i6UG-XN\<ByU8Badc ¡ DHGJIJ[ "G-DMc<xpycHEI-V]UP[ E6d-[ i-D Y2c<G-XN\<i q Y-k-IJI¤Ua\\i6UYJ\c<i6U U1G-UoWJIJ\ "I i6UaVDMi-G-c<XN\<f§VcEDNdf<GJc<r EI-G6UVIJW-D XHY2c<rXHI-XN\<DMVl "I_\PUaY-I-V c<xi6UacV(c\B<DM\ G6UB\c<i-c<rB<c<j-iJc<Xnj-cad q W-DMGK'XHY;UaRUaG-G-c<Mc XNdDFE q DN\ " WJ\c¢E6d-[jJcad q W-DHGJIJ[EIJG6UVI-W-DMXHY2c<Mc c<j-I-XHUG-IJ[G6Uc<XMG-c<B<DcHEG-c<VDMi-G-c<Mc B<i-DMVDHG-G-cHc iy[;EU§G q ¡ G-c§iJD IZ\<f XNd6DeE q TI-DbRUREURW-I2c<j-iJDeEDedIJ\<f i6UaVDMi-G-c<XN\<fLj-i-c<XN\<i6UGJXM\<B<U Badc ¡ DMG-IJ[ "9iJDHY2c<G-XN\<i q I-i-cByUa\<fUa\\<i-UYJ\<ciB G-c<Bc<VVcEDNdf<GJc<VsmUaHc<B<c<Vj-iJc<XM\i6UG-XN\<B<DI " G6UaY;cG-DHk "RUj-IJXUa\f[-BG-lzr B<I;E¤VcHEDNdfG-c<rXMI-XM\DHVl9KzUXHXMVc\<iJI-V \<DMj-DHi-f N\<I N\PUj-lgx<cad6DHDj-cEi-cx<G-c-K

/. /& -#$+ !.- -#$ ) *$%$ *$ -

9i-DeEj-cadc ¡ I-V "<WJ\cB9iJDM q dfR\PUa\D MY2XHj-DMi-I-VDHGZ\PUTj-cad q W-DMG-l(HG6URW-DHGJIJ[ G-DHY2c\<ci-c<rmnIJHI-W-DMXHY2c<r B<DedI-W-IJG-l

a"\<c8DHXM\f G6Uxc<i

ai = a(i∆t)"i = 1, . . . , N

K q EDMV(XMW-IJ\PUa\f"-WJ\c Bi-DHVDMG-G-c<riy[;E ai

jJc<i-c ¡ EUDM\X%[_G-DHY2c\c<i-c<r^S X'G-DHjJi-D i-lzB<GJlzV IZdI¢EI-XHY-iJDM\<GJlzV Bi-DHVDMG-DHV "2j-i-DFEXM\PUaBad6[J[^XHc<xc<rEI-XHY-iJDM\<IZc<ByUaG-G q X UMc<V

∆tcHEG-cVDHi-G q j-i-c<DMY-k-I- mUacB<c<r=\<i6UDMYJ\<ci-I-IK \PU\i6UDHYZ\<c<i-IZ[

j-i-IJG6UoE6dD ¡ IZ\"Ua\a\<i6UYZ\<c<i q XHIJXM\<DMVl "Ti6UPVDHiJG-c<XM\fY2c\<ci-c<HcLi6UB<G-U d K c dUXMG-c UY2DHG-X q "zRUoEUa\<f(B<DHYJ\c<iXMc<XM\co[JG-IJ[ Vc ¡ G-c j-c(VDM\<cHE q O Z K Z NK Ua VDHiJG-c<XM\fjJi-c<XM\i6UG-XN\<ByU_Badc ¡ DMG-IJ[

nc<j-i-DFEDNd6[JDM\X%[§G-U c<XHG-cByUG-I-I mnc<iJV q d6l

UG O Z K "\cuDHXN\<f \<i-DMx q DN\nj-i-DeEByUi-IJ\DNdfG-c<HcuB<lzW-I-XedDHG-IZ[ B<DedI-W-I-GJl d Ko©Uj-i6UaYJ\<I-Y2DG6URW-DMG-I-Dn"Rc<jJi-DeEDed6[-DMVc<D]mnc<i-V q dc<r O Z K "HW6UaXM\<c c<Y;UaHlzByUDM\X%[Ù

B<l TDHG-G-lzV " I Vc ¡ GJc1c<Mi6UG-I-WJIJ\<f<X%[ j-i-c<XN\<i6UGJXM\<Bc<VsVDMG-f!TDHr i6UPVDHiJG-c<XM\In < d

eKACG6URW6UodD¤G-DHcxRp<cHEI-Vcc<j-i-DFEDNdIZ\<fi6UaVDMi-G-c<XN\<f Ua\\i6UYJ\c<i6U

dKnSd6[ DHD

c<k-DMG-Y-Ic<xlzW-G-c'B<lzWJI-XNd6[J \'\PUaYG6UalzB<UDHV q Y2c<i-iJDNd6[Jk-I-c<G-G q ¤i-UaVDMi-G-c<XN\<f"I-XMj-cadf q [mnc<i-V q d q

Dc = limε→∞

limN→∞

lgC(ε,N)

lg ε,

O Z K t ED

C(ε,N) = N−2∑

i6=j v(ε− | xi − xj |) Y2c<iJi-DNd6[Jk-I-c<GJG-lzr I-GJ\DHMi6UPd "

ε

i6UaHVDHin[-W-DMr-Y-Ii6UaHx<I-DHGJIJ[mÙPc<B<cHczj-i-cXM\<i-UG-XM\ByU "NuW-I-Xedcz\<c<W-DMY "oI-XHjJcadf

q DHVlCpnE6d6[`c<k-DMG-Y-ITi-UaVDMi-G-c<XN\<I " v 9m q G-Y-kJIJ[T¦uDHB<I-XHUr;EU " xi = x(i∆t)KRSd6[

c<j-iJDeEDedDHG-IZ[Dc

XM\<iJco[Z\ÙB<I-XMI-Vc<XN\<flgC(ε,N)

c\lg εI I q \TG-UnGJDHrdI-G-DMr

G-lzr q W-UXM\c<Y "*G6UYZd6c<G§Y2c\c<i-c<McI8c<j-iJDeEDed6[-DN\bI-XMY;cVc<D HG6URW-DHGJI-D i-UaVDMi-G-cXM\IKZhniJc<VDz\<c<Mc"ZIJG-c< EUÙG6UPdIZI-i q \TRUaB<I-XHIJVc<XM\f Dc(n)

I q B<DNd6I-W-I-ByUa \ nEc¢\<Depj-c<i "6j-cYZUDc

GJD'Ec<XM\I-HG-DN\ G6UaXHl DMG-IJ[ K'B<DMXM\G-c"CWZ\<c j-i-IB<lzW-I-XedDHG-IJI

Dc

X q TDHXN\<B q \(c<Mi6UG-I-WJDHG-IJ[ G6U§B<DedI W-I-G q ε KBXNdI ε j-iJI-xadI ¡ UDM\X%[(Y i6UaHVDHi6UaV Ua\a\<i6UaYJ\<c<i-U εmax

"\<c dI-G-DMr-G6Ua[RUaB<I-XHIJVc<XM\f

lgC(ε,N)c\

lg εj-i-c<j6UREUDM\ "WJ\<c cx[-XMGJ[-DN\<X%[ Bad6IJ[-G-IJDHVHi6U

G-I-k Ua\\i6UYJ\c<i6U " ED W-I-XedcbXMc<XHDFEDHr q Y;U ¡ Ec<r8\<c<W-YJI c<x<lzW-GJc1VDHGJf! D " W-DMVB XHDHiJDeEI-GJD FKA j-i-DFEDNdD "DHXedI

ε = εmax"\<c

lgC(ε,N) = 1K Ei q Hc<r XN\<c<iJc

G-l ";j-i-I q VDMG-f!TDHG-I-I G6URW-DMG-IJ[ ε X q DMXM\<B q DM\ G-DMY2c\<c<iJc<D εmin

<\PUY2c<D!";WJ\<c9E6d6[ε < εmin

XM\<i q YZ\ q i6U Ua\a\<i6UYZ\<c<i6U8c<XM\PUaDM\<X%[G-DHi-Uai-D DMG-G-c<rK hnUY XNdDFEXM\B<I-D!"B<G-cB<f¢G6Ui q UaDM\<X%[dI-G-DMr-G6Ua[_UB<I-XMI-Vc<XM\f lgC(ε,N)

c\lg εK

EG-c<r IJ j-i-c<xad6DHV jJi-Iî6UaXHW-DN\<D'i6UPVDHiJG-c<XM\I^[-BPd6[-DN\<X%[ Blzx<c<i B<DNdIJW-I-GNI

∆tK nDedDHXMc<c<x<i-UaG-DMD'IZp j-cHEx<IJi6Ua\<fTG-Duj-c`c\HEDedf<G-c<XN\<I "-U q W-IJ\lzByUa[i6UPVDHi

B<i-DMVDHG-GJc<Hc§c<Y-G6UN∆t

K 9i-I N\<c<V XNdDFE q DN\ j-i-I-GJI-V]Ua\<f§B<c8B<G-I-V]UGJI-D_X q D XM\B<c<ByUaG-I-Dnm q G2EUVDHGZ\PUPdf<G-lCpcHi6UGJI-W-DHGJI-rG-U B<DNdIJW-I-G q Dc

"Zc<j-i-DFEDNd6[JDHVlCpmnc<i-V q dc<r

Dmax =2 lgN

lg(1/r), r =

ε

εmax,

O Z K w Y2c\<ci6Ua[=cHG6URW6UDM\ " WJ\<cUPdMc<i-IJ\VLi-UXHW-DN\PUi6UaVDMi-G-c<XN\<I8G-DVc ¡ DM\ EUa\<fHG6U W-DMG-I-Dx<cadf!TD!"W-DMV

Dmax

jJi-I(UoEUG-GJc<VsWJI-XNdD_\<cW-DHYNK9G-lzVI XNdcByUVI "

DHXedIr = 0.1

IN = 1000

"\<cDmax ≤ 6

DHXedIN = 100000

"\cDmax ≤ 10

KAi-DN q dfR\PUa\<DB<cHG-I-Y;U \§cj-i-DeEDNdDMG-G-lzD_XNdc ¡ G-c<XN\<I B XNd q W6UaD!" Y2c< EUb\i-DHx q DM\ X%[ ca\dI-W-IJ\fÈDM\DHi-VIJG-I-i-c<B<UG-G-lzrj-iJc<k-DHXMX'BXHI-XN\<DHVDuX Ec<XN\PUa\<cW-G-c x<cPdf! IJVW-I-Xedc<V XN\<DMj-DHG-DMrbXHBc<x<cHEl ca\ IJXM\<IJG-G-c^Xed q W6Ur-GJc<Hc j-i-c<kJDHXHXHUJK G6UGJI-D i6UaHVDHi-GJc<XM\I(j-caB<cad6IJ\=G6UVi-DMY;cG-XM\i q I-i-c<ByUP\<f UP\\<i-UYJ\<ci(VDN\<c

Ec<V O Z K Z "-Y;UYx<lCdc¢c<j-I-XHUG-c B<l DaK DN\<cHERUREDHi ¡ Y-I UaY;DMG-XU [-Bad-[-DM\X%[G6UaI-x<cadDMD IJHB<DHXN\<G-lzV " G-cGJD§DFEI-G

XM\B<DHGJG-lzV XMj-c<XHcx<c<V"RUoEUG-IJ[ BDHYJ\c<i6UbXHcXM\<cR[JG-IJ[K dfR\<DMi-G6Ua\I-B<c<r DHV q XNd q ¡ IJ\\PUY_G6UaHlzByUDHVlzrVDM\cHEbjJc<XNdDFEc<ByUP\<DNd6f<G-c<McEIJmmnDMi-DHG-kJI-i-c<ByUaG-IJ[ "JI-VD TI-rcj-i-DeEDNdDMG-G-lzDbj-iJDHI-V q DMXM\ByU j-i-I(i-DTDHG-IJI(RUREURW-Ii-DMY2c<G-XM\i q Y-k-I-I K EDM[_EUG-GJc<Hc VDM\<cHEU XedDeE q Ua[ K q XN\<fI-VDMDM\X%[ B<i-DMVDHG-G-crbiy[;E a(i∆t) =ai

"i = 1, . . . , N

K UoEUG-IJDTB<DMYJ\<c<i-U XHc<XN\<cR[-G-IZ[=B mUac<Bc<V j-i-cXM\<i-UG-XM\B<D¢j-i-cIJHB<cHEIJ\<X%[1XNdDFE q TI-V¤c<x<i6UPc<V2

x(t) = (a(t), da(t)/dt, . . . , dn−1a(t)/dtn−1) = (x1, x2, . . . , xn).O Z K

9c<XHY2cad6f<Y q IJHB<DHXN\<G-l G-UoWJDHG-IJ[ ai

\cadf<Y2c'B EI-XMY-i-DM\G-lzDzVc<VDMGJ\<l B<iJDHVDHGJI

i∆t"2Y2c<c<iEI-G-Ua\<l

xj

B<DMYJ\<c<i-Uxc<jJi-DeEDed6[- \<X%[^j q \<DHV WJI-XNdDMG-G-c<McEI-mnmnDHiJDHG

k-I-iJc<ByUG-IZ[¢I-X pycHEG-cHcTBi-DHVDMG-G-c<Mcìy[;EUnjJcjJi-I-xadI ¡ DHG-GJlzV V]Ua\<DMV]Ua\<IJW-DHXMY-I-V

mnc<i-V q dUVzK W-DMB<I;EGJc"WJ\<c¢\<c<WJG-c<XM\f B<lzW-IJXNdDMG-IJ[bjJi-c<IJHB<cHEG-lCpx q EDN\^c<jJi-D EDed6[J\f<X%[¤V]UPdcXM\<f B<DedI-W-I-GJl UeU=EIJXHY-i-DN\<IJUk-I-I

∆tKC©9DFEc<XM\PUP\<Y2c<V VD

\<cHEU=[-Bad-[-DM\X%[jJc<B<l DMG-G6Ua[(W q B<XN\<B<IJ\DNdfG-c<XM\f Y q V q "]WJ\<c§c<Hi-UG-I-W-IJByUDM\DHMc j-i-I-VDMG-I-Vc<XN\<f E6d6[j-i-c<XN\<i6UaG-XM\B(Bad6c ¡ DHG-IJ[x<cadf c<r i-UaVDMi-G-c<XN\<I j-cY-i6Uar-G-DHrVDHi-D "6x<DMj-iJc<B<DeEDHG-IJ[j-i-DFEByUiJIJ\<Dedf<G-c<rj-i-c<kJDeE q i-l"mnIZdfR\i6Uk-I-I NK

-*$) - , #$%"!

9c<XNd6DzB<c<XHXN\PUG-c<BPdDHG-IZ[ mUaHc<B<c<Mc9j-c<iJ\i-DM\PUUa\a\<i6UYZ\<c<i6U'S d x<lzV8IJ B<l DMj-DHiJDHW-I-XedDHGJG-lCp_VDM\cHEc<BVc ¡ DM\`x<l \<fi-DTDHG6UnRUoEUoW-Ui-DMY2c<G-XM\i q Y-k-I-I_c<jJD i6Ua\c<i6U HBcadukJI-IK6©nUI-xcadDHDuj-i-c<XN\<c<r_XHjJc<XHc<xTE6d-[ M\c<Hc XHcaNEUa\f

n]VDMi-G-c<D

EI-XMY-i-DN\<G-c<D`c\<cx<i6U ¡ DMG-I-D

x1,i+1 = F1(x1,i, x2,i, . . . , xn,i),

. . . . . . . . . . . . . . .O Z K ]

xn,i+1 = Fn(x1,i, x2,i, . . . , xn,i),

EDxj,i

Y2c<c<iEI-G-Ua\<lB<DHYZ\<c<i6U XMc<XM\cR[-G-IJ[ "i6UXMXHVc\i-DHGJG-c<Hc_B VcVDHGJ\lsB<iJD VDHGJI

i∆t"Fj

^G-DNdIJG-DHr-GJlzD m q GJY-k-I-IKAi-UVY;UPp UPdHci-IJ\<V]U( dc<xyUPd6f<G-c<riJDHY2c<G-XN\<i q Y-k-IJIE-d6[j-cPd q W-DHG-IZ["Y2c<G

Y-i-DN\<G-cHc8B<I;EU MB<cad k-I-c<G-GJc<Hc c<jJDHi6Ua\c<i6U m q G-Y-k-I-I Fj

"j = 1, . . . , n

jJi-DeE XM\PUaBad6[- \ B B<I;ED i6UPMdc ¡ DMG-IJ[j-c G-DMY2c\<c<iJc<V q xyUaHI-X q " c<Mi6UG-I-WJI-ByUa[JXHf j-i-IM\c<V Y2c<G-DHWJG-lzV W-I-XNd6c<V WZdDMG-c<BTi-UaMdc ¡ DMG-IJ[ K-A(jJi-c<XM\DHr TDHV XNd q W6UaDuRUoEUG-I-DFj

Vc ¡ DN\ c<X q DMXM\Bad6[J\f<X%[ j-cPdI-G-c<V]UaVIGJDHY2c\<ci-c<r1XN\<DMj-DHG-I ν 2

Fj(xi) =

ν∑

l1,l2,...,ln=0

Cj,l1,l2,...,ln

n∏

k=1

xlkk,i,

n∑

k=1

lk 6 ν, O Z K

EDCj,l1,l2,...,ln

nG-DMIJBDHXM\G-lzDzY2cHmnmnI-k-IJDHGJ\l "Y2c\<ci-lzDC\<iJDHx q DM\<X%[¢G-UrJ\<I KoSd6[Uj-jJi-c<Y2XHI-V]Uak-I-I Vc< q \bj-i-IJVDHGJ[Z\<f<X%[ jJcadI-G-cVl D ¡ UG2Ei6U_d6I-x<cVc ¡ DM\I-X j-cad6fc<B<Ua\<f<X%[ x<cadDMD^XNd6c ¡ G6Ua[§VDM\cHEI-Y;UJKSd6[8RUoEUG-IJ[

Fj

Vlªc<Hi6UaG-I-W-I-VX%[mnc<i-V q dc<r O Z K NK I-XN\<DHV]U q i6UB<GJDHG-I-r O Z K ] Ec<j q XHY;UDN\Uj-I-XMfÊ6d6[1d x<c<McG-c<VDHi-U i KSd6[G6UPp<c ¡ EDMG-IJ[=Y2cMmmnI-kJI-DHGJ\c<B O Z K G-DHcxRp<cHEI-Vci-DTIJ\<f_XHIJXM\<DMV q N dIJG-DHr G-lCpUPd6HDHxi6UI-W-DMXHY-I;p q i6UaB<G-DHGJI-r

xj,i+1 =

ν∑

l1,l2,...,ln=0

Cj,l1,l2,...,ln

n∏

k=1

xlkk,i, i = 1, . . . , N

O Z K O [

X^GJDHIJHB<DHXN\<G-lzVICj,l1,...,ln

"*BY;ca\<c<i-crN=W-I-Xedc\c<W-DMY XHY;UPd6[Ji-G-c<McbB<i-DMVDHG

G-c<Mciy[;EU "I-XHjJcadf q DHVlCpE6d6[ Uaj-j-i-c<Y2XMI-V]Uk-I-I=j-i-UB<lCp1W6UXM\DHr " ν XM\DHj-DMG-fj-cad6I-G-c<V]UJK9i-I RUREUG-G-lCp

nIνW-I-XNd6c8Y2cMmnmIJk-I-DHGZ\<c<B

KjJcadI-G-cVc<B O Z K nB c<x

DMVXNd q W6UDTVc ¡ DM\ xl \<f c<jJi-DeEDedDHGJcj-c mnc<i-V q dD K = (n+ ν)!/(n!ν!)KhnUY

j-i6UaB<IZdc"N K

"j-cN\<c<V q E-d6[8Y;cG-Y-i-DN\<IJUk-I-I MB<cad k-I-c<G-GJc<Hcc<j-DHi-Ua\<c<i-UXHIJXM\<DMV]U q i6UBG-DHG-IJr O Z K O [ i-D UDN\<X%[bVDN\<cHEc<V(G6UI-VDMG-f!TIZpY-ByUoEi6Ua\<cB-K 9cd q W6U Ua[JX%[V]Ua\DHV]Ua\I-W-DHXMY;Ua[ VcEDNdfu[JBad6[JDM\<X%[ Hi-c<VcaNEY2c<r "PG-c'jJi-I q Xedc<B<IJIq EURW-G-c<Mc Blzx<c<i6Uc<x! TDHMc BI;EUG-DNd6I-G-DHrJG-lCpm q G-YJk-I-rDHDni-DTDHG-IJD`B<cXHj-i-cIJ B<cHEIJ\ XHIJHG6UPd1XnB<lzXHcY;crXN\<DHjJDHG-f<\<c<WJG-c<XM\IK G-UPdc<MI-W-G-lzV(c<x<i6UaHc<V Vc ¡ G-ciJDHY2c<G-XN\<i q I-i-cByUa\<f G-D'\<cPdf<Y2cTEI-XHYJi-DM\G-lzD

c\c<x<i6U ¡ DHG-IJ[ "G-cI V]Ua\DHV]Ua\I-W-DMXHY-I-D VcHEDNd6I8BBI;EDXHIJXM\<DMVl ST¨gO Hc^j-ciy[;EY;U 2

dx1

dt= F1(x1, x2, . . . , xn),

. . . . . . . . . . . . . . .O Z K O<O

dxn

dt= Fn(x1, x2, . . . , xn).

VlzXNdbm q GJY-k-I-rB jJi6UB<lCpW6UaXM\R[Zp_\c\ ¡ D!"-WZ\<cÎ_i6UaG-DHDaK UY_Y;UYG-U¢jJDHi-B<cVM\PUaj-DUPdMc<i-IJ\V]U x<lCdU c<X q DMXM\BadDHG-Ui-DMY2c<G-XM\i q Y-k-IJ[=mUaHc<B<c<r\<i6UaDHYJ\c<i-I-I "M\c§HG6URW-IJ\ "WJ\<c§B<XHD

xi

IJB<DMXM\G-l XedDeEc<ByUa\DNdf<GJc" Vc ¡ G-c=c<j-i-DFEDNd6IJ\<f jJi-cIJHB<cHEG-lzD=c\ G-IZpK 9cN\<c<V q O Z K O<O XHG-c<B<U DHXM\f(GJIWZ\<c I-G-c<D " Y;UYXHI-XN\<DMV]UdI-GJDHr-G-lCp=UPdHDMx<i6UIJW-DHXMY-IZp q i6UBG-DHG-IJr8ca\<G-c<XMIJ\<Dedf<G-c_G-DMIJBDHXM\G-lCp8Y2cHmnmnI k-I-DMGJ\<cB1K

XNdILBY;URW-DHXN\<B<D§XHj-cXHc<xyU RUREUG-IJ[LB<DMYJ\<c<i-U(XHcXM\<cR[JG-IJ[B mUacB<c<V jJi-cXM\i6UG-XN\<B<DnI-XMj-cadfa q DN\<X%[VDM\cHEbjJc<XNdDFEc<ByUP\<DNd6f<G-c<McÈI-mnmnDHi-DMG-k-I-i-cByUG-IJ[ "J\<cV]Ua\DHV]Ua\I-W-DHXMY q VcHEDNd6f Vc ¡ G-c¢B<c<XHXN\PUG-c<BIJ\<f B xcadDHDnj-iJc<XM\c<V B<I;ED

dx1

dt= x2,

dx2

dt= x3, . . . ,

dxn

dt= f(x1, x2, . . . , xn)

O Z K O Z B=XHIZd q \<c<Mc"WJ\c8BaRUI-VcXHB[Zf=VD ¡ E q Y2c<ci<EI-G6Ua\PUVI(cHEG-cHG6URW-G-c=RUoEUaDM\<X%[i6UBDHG-XN\<B<c<V O Z K eK

$ -*#$ ) 1 - , $ !

UaXHXHVca\<i-I-V¤Y2c<GJY-i-DM\G-lzrbj-iJI-VDHij-i-I-VDMG-DHG-IZ[bc<j-IJXUG-GJc<Hc UPdMc<i-IJ\V]UJK M\c<r kJDNdf< I-XMj-cadfa q DMV"IJBDHXM\G q EIJG6UVI-W-DMXHY q XHI-XN\<DHV q DHXHXedDHi-U "cj-I XHlzB<U q i-D ¡ I-V(G-DHjJDHi-I-cHEIJW-DHXMY-IZp1Y;cPdDHxyUaG-I-r2

x = −(y + z), y = x + ay, z = b+ z(x− c). O Z K OQ

9i-I G-UoWJDHG-IJ[;pj6Ui-UVDM\i-c<Ba = 0.15

"b = 0.2

"c = 10.0

XHIJXM\<DMV]U O Z K ORQ p2Uai6UYJ\DHi-IJ q DN\<X%[8i-D ¡ I-Vc<V XM\i6UG-GJc<HcÛa\\i6UYJ\c<i6UJK9XHj-cPdf q DHV BYZURW-DMXM\B<D9cHEG-c<VDMi-G-c<Mc B<i-DMVDHG-G-cHc iy[;EU ai

UB<I-XMI-Vc<XM\f B<cB<i-DMVDHG-IcEG-c<rIZ1Y2c<c<iEIJG6Ua\

y(i∆t)"]j-cad q WJDHG-G q W-I-XNd6DHG-G-lzV I-GZ\<DHMi-I-i-c

ByUGJI-DHV q i6UBG-DHG-IJr O Z K ORQNK] q EDMV XHW-IZ\PUa\<f"CWJ\<c§B<I;EXMI-XM\DHVl O Z K ORQI(DHDi6UaHVDHi-GJc<XM\f G6UVG-DMIJBDHXM\G-l9K©nUxad EUaDHV]Ua[

a(t) = y(t)"RUoEUG-G6Ua[ G6U Y;c

G-DMW-G-c<V IJGJ\<DMi-ByUPdDB<i-DMVDHG-I0 ≤ t ≤ 100

"-jJi-DeEXN\PUBadDMG6U^G6Ui-IJXKO Z K O" =KSd6[ UoEUG-IZ[_B<DHYJ\c<i6UXHcXM\<cR[JG-IJ[i-DHY2c<GJXM\<i q IJi-c<ByUGJG-c<r1XMI-XM\DHVlBc<XHj-cPdf

q DHVX%[ \DHc<i-DMVc<r UaY;DMG-XU O Z K Z eKUXHXMW-IJ\lzByUa[ j-cbG-Uxad EUaDHVc<ra(t)

UBa\<cY2c<i-i-Ded6[-kJI-c<G-G q |m q G-Y-kJI- "G6UPp<cEI-V"B<i-DMV [ XMj6UoEUGJIJ[ DMD¢EcG q d6[ τ0 ≈ 1.6II-XHjJcadf q DHV M\ q BDNdI-WJI-G q BY;URW-DHXN\<B<D B<i-DMVDHG-IRUoEDHi ¡ YJIsB O Z K Z NK'©nUi-I-XaKO Z K O" R j-i-DFEXM\PUaBadDHG-ULj-i-cDHY-k-IZ[siJDHY2c<G-XN\<i q I-i-cByUG-G-cHc"Ua\a\<i6UaYJ\<c<i-UG6UjZdcXHY2c<XM\f¢EB q pjJDHi-DMVDHG-G-lCp#2 x1(t) = y(t)

Ix2(t) = y(t+ τ)

KSd6[(c<j-i-DFEDNdDMG-IJ[i6UPVDHiJG-c<XM\I¤VcHEDNdfG-c<r¤XHIJXM\<DMVlG q ¡ G-c§i6UXHXMW-IJ\PUa\f

i6UaHVDHi-GJc<XM\fÙP\\<i-UYJ\<ci6U'I i6UaHVDHi-GJc<XM\f9j-i-c<XN\<i6UGJXM\<B<UnBPdc ¡ DHG-IZ[KS`d-[Tc<kJDHG Y-I i6UaHVDHi-G-cXM\<I Ua\\i6UYJ\c<i6U§BlzW-I-XNd6I-V DHMc§Y2c<iJi-DNd6[Jk-I-c<GJG q i6UaHVDHi-GJc<XM\f

Dc

O Z K t "6I-XMj-cadfa q [1XHj-DMk-I6UPd6f<G-lzD UodHc<iJIJ\<Vl9K©nU i-IJXK*O Z K Z j-i-I-B<DFEDHGJl"i-D q dfR\PUa\lLi-UXHW-DN\PU RUB<IJXHI-Vc<XN\<I Dc

c\lg ε

"2 EDεi6UaVDMi^[-WJDHr-Y-Ii-Uax<IJDHG-IJ[

mUacB<c<Hc^j-i-c<XN\<i6UGJXM\<B<UJKhnUYbBI;EG-c IZ Hi-UmnI-Y2c<B" BG-DÙB<I-XMI-Vc<XM\I c\i6UP VDHiJG-c<XM\I¢j-i-c<XN\<i6UGJXM\<B<U'Badc ¡ DHGJIJ[

n"aI-VDMDM\X%[ j-cadcW-Y;U G6U q iJc<B<G-D Dc ≈ 1.9

"Y2c\<ci-lzr1Ij-i-I-GJI-V]UDHV(RU G6URW-DMG-I-DÌ-XHY2c<Vc<ri6UaHVDHi-GJc<XM\IK

UaY-I-Vbc<xi6UacV "PiJDHY2c<G-XN\<i q I-i q DHVlzrÛa\\i6UYJ\c<iI-VDHDN\9i6UaHVDHi-GJc<XM\f D ≈ 2I8Vc ¡ DM\x<l \<f Badc ¡ DMG B \i-DNpZVDHiJG-c<D^mUaHc<B<c<D¢j-iJc<XM\i6UG-XN\<B<c-K \c1cHG6URW6U

−20.0

−10.0

0.0

10.0

20.0

y

0.0 25.0 50.0 75.0 100.0t

−20.0 −10.0 0.0 10.0 20.0−20.0

−10.0

0.0

10.0

20.0

x

x2

1= R ! f+q +-''Dd cQMdgT8(*X-(q /YXh .1/8/YfNR*('Ddgh y(t) X-(XhN+q +XXI`*+fd " 6 M0Nf+./8'*X-hNf i (Sf/gT:dJ'' d8hNhYf*d8.hN/YfATcr*f/Y+.(('*d,r`*/YX-.1/8Xh x1, x2

0 R*+ x1(t) = y(t) 0 x2(t) = y(t+ τ ) "! f+q QMdMRD+faB.1(p/gr*f+ RD+-` 1`/YX .1d8.LTNf+-q R*/YXhY(a,+-'( Lr+fTY/ M/ ' i ` dJTNhN/8./Yff+-` (/8''*/ i '.(( *hY/BX/8/YhYTY+hNXhYT i +hBrf(8`(Q-(*hY+-`8'*/ Nd8Q-/8TN/ M/ r*+f1(*/R5do./8`*+ Ndg'(

−2.7 −1.8 −0.9 0.00.0

1.5

3.0

4.5

n=2

n=10

D

lg

c

ε +-Q i `ghYdghmf*d8X*+h8d./Yff+-` (/8''*/ Af*d8Q-q3+-f'*/8XhY( rf(ATNd8f8(*f/8T:dJ'(( fd8Qq +f1'*/8XhY(r*f/YXhYf*dg'XhYTNdTY`/Jao+I'( n " /g` i *+-'/#QI'Dd *+-'(*+ Dc ≈ 1.9 X-/Y/Yh8TN+hNX-hYT i u),++ <r*/g`*/ .+ "cdg''*/Y+Q-'*d +-'(+r/YQ-TY/8` +h,Tor*f('(r*+/ Mf*dJ'((*h NX Sjq +f' q r*f/YXhYf*dg'XhYTY/YqR` BT8`*/Ja,+-'( dghNhNfd8.ShN/8f*dc(&0X-/Y/Yh8TN+hNX-hYTN+-''*/10/ Mf*dJ'((*h YX X-(XhN+q / S oZ S M/Br/Yf :RD.1dorf( q3/R*+-`(f/8TNdg'((DM\ "WZ\<c8Vl Vc ¡ DMVsI-XMY;Ua\<f8VcHEDNdfG q |S B=B<I;EDXMI-XM\DHVl ST¨ \i-DM\f<DHMcj-c<iy[2EY;U

n = 3eK 9XHY2c<V q sXMI-XM\DHV q RUaj-I DMV¤B BI;ED O Z K O<O "6I-XMj-cadf q [j-c

dI-GJc<VI6UPdfG q Uaj-j-i-c<Y2XMI-V]Uk-I- O Z K I=c<Hi-UG-I-W-IJB! I-XMfHG6URW-DHGJI-DHVν = 2

KDN q dfR\PUa\l|i6UXMW-DM\c<BbY2cHmnmnI-k-IJDHGJ\c<B" c<jJi-DeEDed6[- IZp B<I;E j-i6UB<lCp W6UXN\<DMrq i6UB<GJDHG-I-r O Z K O<O]E6d6[ n = 3

Iν = 2

"6j-iJI-B<DeEDHG-lB\PUxPdI-k-DaK

G-UoWJDHG-I-DY2cHmnmnI-k-IJDHGJ\PU Uaj-j-i-c<Y2XMI-V]Uk-I-Il1 l2 l3

O D q i6UaB<G-DHGJI-D Z D q i-UB<G-DMG-I-D Q D q i6UB<G-DMG-I-D[ [ [ [ K [ [ K [ [ K O ] [ [ O [ K [ O<K [ JK ZO[ [ Z [ K [ [ K [ [ K [[ O [ O<K [ [ K [ [ K t [ [ O O [ K [ [ K [ O<K [\[ Q[ Z [ [ K [ [ K [ [ K ORt [O [ [ [ K [ [ K [ %O [ K [ O [ O [ K [ [ K [ [ K O tO O [ [ K [ [ K [ O<K [ Z wZ [ [ [ K [ [ K [ [ K O t

C+J3+(!6Khnc!HmnmnI-k-I-DMGJ\<l"Uaj-j-i-c<XMI-V]Uk-IJI1G-DedI-G-DMr-G-c<XN\<DHr B¢Xed q W6UDj-cad6I-G-c<VI6Uodf<G-c<Mc jJi-DeEXN\PUBadDMG-IJ[1m q G-YJk-I-r Fj

O Z K NK nUi6UaVDM\<iJli-DMY2c<G-XM\i q Y-k-I-I n = 3

"ν = 2

K2S`d-[iJDHY2c<G-XN\<i q Y-k-IJI=Ua\\i6UYJ\c<i6UI-XHj-cPdf q DM\X%[bVDM\cHE=j-c<XedDeEc<ByUa\DNdf<GJc<HcEI-mnmnDHi-DMG-k-I-i-cByUG-IJ[ O Z K !NKACDNdIJW-I-G-l

ljc<x<cHG6URW6U \^XN\<DHjJDHG-IbjJDHi-DMVDHG-G-lCpXMc<XM\co[JG-IJ[

xiB jJi6UB<lCp

W6UXN\R[ZpXMI-XM\DHVl O Z K OO

0.0 25.0 50.0 75.0 100.0−20.0

−10.0

0.0

10.0

20.0

t

x 1

dgT8(*X-(q3/8Xh x1(t)0r/8` i +-''*d (*XI`*+-'' qL('*hY+ Mf(f/gT:dg'(*+q f+./g'*XhNf i (*f/gT:dJ''*/ X-(XhN+q

Ai-DN q dfR\PUa\DjJi-c<k-DFE q i-lLiJDHY2c<G-XN\<i q Y-k-IJIS j-c cHEGJc<VDHi-GJc<V q Bi-DHVDMG-G-cV q iy[;E q Vl|jJcad q W-IZdI \<i-DepZVDHi-G q S BI;EU O Z K O<O TXY2cHmnmnI-k-I-DMGJ\PUVI "j-i-IJB<DeEDMG-G-lzVI B1\PUxad6I-k-DKB DHjJDHi-f j-iJc<B<DeEDHVXHi6UaB<G-DHGJI-Di-DM q dfR\PUa\<c<B "Y;ca\<ci-lzD Vc ¡ G-c^jJcad q W-IJ\f" I-GZ\<DHMi-I-i q [8YZUaY1\<DMXM\<cB q ªVcEDNdf O Z K ORQ "-\PUaY IbVcEDedf<G q S O Z K O<O eKCDM q d6fo\PUP\<l I-GZ\<DHMi-I-i-cByUG-IJ[VcHEDedf<G-cr S O Z K O<O j-i-DFEXM\PUaBadDHGJlG-U¢iJI-XK*O Z K Q B B<I;ED9RUaB<I-XHIJVc<XM\I

x1(t)K

i6UBG-DHG-IJDbEUG-G-lCpiJI-XK'O Z K Q XEUG-G-lzVIi-I-XaK'O Z K O" = j-c<Y;UalzB<UDM\ Y;URW-D XM\B<DHGJG-c<DX pycHEXN\<B<c_i-DHUPdf<GJc<Hc I1VcEDNdf<GJc<Hc Y2cad6DHxyUa\DNdfG-lCp j-iJc<k-DHXMXHc<BJK E G6UY2c ByU ¡ G-lzV "-Y2c<G-DMW-G-c"6[-BPd6[- \<X%[Y;cPdI-W-DMXM\<BDHG-G-lzDXMc<c\B<DM\XM\<BIJ[KACcHVc ¡ G-c¢dIbXj-c<Vc! Tf< i-DMY2c<G-XM\i q I-i-c<B<UG-G-c<r XMI-XM\DHVlc<X q TDHXM\B<IJ\fjJi-c<HGJc MB<cd k-I-I XMI-XM\DHVlªB<c_B<iJDHVDHGJI§RUj-iJDeEDedUVI IJGJ\<DMi-ByUPdU "G-UY2c\c<i-c<VLG6UVIJ B<DMXM\<G-UG-Uxad EUaDHV]Ua[ N\<c<rk-Dedf< j-iJc<B<DeEDHVgXNdDFE q TI-r MY2XHj-DMi-I-VDHGZ\oKACcHf<VDHV B`Y;UoWJDHXM\B<D9G6URW6UPd6f<G-c<Mc`G-UoWJDHG-IJ[ Y;cc<iEI-G6Ua\ q jJc<XNdDFEG-DMr_\c<W-Y-I^G6U

−20.0

−10.0

0.0

10.0

20.0

t t +60.00 0

y,x1

! /8Qq /gao'*/8Xh rf+IR*X.1d8QMdJ'( r/8TY+IRD+-'( LX-(XhN+q " r/YX-`+ /Y./g'DdJ'( 'Dd g` SRD+I'( QMd M+-'+f(f i +q qR5dJ''*/ LX-(*X-hN+q / LX-('DdJ`*/8q" f(XY" 0th8" +Y"&R` TNf+-q3+I' t0 ≤ t ≤ t0 + T 0<RD+ T yWTNf+-q rf+IR*X.1d8QMdJ'( X i XhNf*dJ(TNdu),+ .X-r+f(q +-'*h8d8hN/8f*dBX-hN+-r*+I' hY/ '/YXh8(t" i '.hY(fyf+Q i `ghYdghc('*hN+ f(*f/8TNdg'( X-(*X-hN+q " 0DXIr`*/ >'Dd `('( S3f+ ,+-'(+>f+./8'XhNf i (f/gT:dg''*/ q /RD+-`Y'*/ X-(XhN+q

xad EUDHVc<r i-I-XaK O Z K O!" = ¢B§Vc<VDHGJ\ B<i-DMVDHG-It0 = 100

K S UPdDMD1j-iJc<I-GJ\DHHiJI i q DHVYZUaY I-X pycHEG q "]\PUaY I VcHEDedf<G q XMI-XM\DHVl X^G6URW6UPdf<GJlzVI q XNd6c<B<IJ[JVIj-i-I

t = t0IXHi6UBG-I-V i-DN q dfR\PUa\lE-d6[ t > t0

K©nU¢i-I-XaKO Z K j-i-IJB<DeEDMG-l"XHcc\B<DN\<XM\B q TI-D_Hi6UmnI-YJI RUBI-XHI-VcXM\<DMr y(t) E6d6[ \DHXN\<c<B<cr XHI-XN\<DHVl O Z K OQ 'Ix1(t)

E6d-[1i-DMY2c<G-XM\i q I-i-c<B<UG-G-c<r1S KhnUY1XedDeE q DM\_IJì-I-XaKO Z K "6j-i-c<MG-c MB<cd k-I-I XMI-XM\DHVlB<cB<i-DHVDMG-I^cX q DMXM\<BPd6[-DN\<X%[X GJDHY2c\<ci-c<r c!TI-x<Y2c<r "2Y2c\c<i6Ua[XHc¢B<iJDHVDHGJDHV G6Ui6UXN\PUDM\oKh9c<G-YJi-DM\G-c<D9B<i-DMV [_j-i-c<MG-cU Vc ¡ G-c q Y;UaRUP\<f"-UoEUB\<cW-G-c<XN\<fTjJi-DeEXMY;UaRUGJIJ[K3' i-DM q dfR\PUa\<c<BTi-I-XK6O Z K XedDeE q DM\ ";WJ\c`DMXNdI c<Hi6UaG-I W-IJ\f<X%[c! IJx<Y2c<r ±5%

"J\c¢B<i-DHV [_jJi-DeEXMY;UaRUGJIJ[1BGJc<i-VI-i-cByUG-G-lCpDeEI-GJI-k6UPpx q EDN\¢XHc<XN\PUBad6[Z\<f¢j-i-I-VDMi-G-c T = 12.0

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