Синхронизация регулярных, хаотических и стохастических колебаний: Учебное пособие

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Author: Анищенко В.С. | В.В. Астахов | Вадивасова Т.Е | Стрелкова Г.И.

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- . - / V ! " . . . . . . . . . . . 0| | U { * &"JK"+" U&UUNU&U&U&UNU&UU&UNU&U&U&UUNU&U&U&UNU&UU&UNU&U&U&UUNU&U&U&UNU&UU&UNU&U&U&UUNU&U&U&UNU&7 U 0| | U ~ * 14'BO5 !H!7& 1D I zJ.NH#+'B'B"+$7*G+ 1D_qH#" "%= 7O$&&7 &UDUNU&UUNU&U&U&UNU&UU&UNU&U&U&UUNU&U&U&UNU&UU&UNU&U&U&UUNU&U&U&UNU&UU&UNU&U&U&UUNU&U&U&UNU&7 U 0 | U | * 1%JK"+T+$&" &"+pz"Tw"7 JK*"+,-&Ty+!2_U U UBUNU&UU&UNU&U&U&UUNU&U&U&UNU&7 U 0 | U oX+&&_P"lS&F. 7, I w,&O1D "7JK *"+,!Hh7"+3g'B&U U UPU&U&U&UNU&U $ { | U 0 oX+&S&" &+$h+!H!7&!14 I y7"+1&; _qHw7"kJK"%2;:&_qH I , 2& $80 | U $ !H!7& 1D I aJK.NH!*;%$&$_qHy,-O2;"SN TtU U UUNU&UU&UNU&U&U&UUNU&U&U&UNU&U $A2 | U v "+1+.92;:9$))$&_[@,- "7'B"+ $))2:_qHs+%2"kJK&NOTMU UU&UNU&U&U&UUNU&U&U&UNU&U v ~ | U * "+pz w H 7 & 1D I w,NO1"+7JK*"+, Hy,-O2"+SN TtU U UdUNU&U v| | U 2Hq,-+"7 'B" $))2;:&_PTx;)2; 1l&"+pz"Tw H 7 & 1D I tU U U U U&U&U&UNU&U Lv $ / K

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,-& JLG&;'B"!"+$ZeG,-&"+*&Tf&,7 "+$&%$&e+&S&%$&&"+&Tg*;+$$&_$&&' = &+g+""+7;O$&7; >;$),w;O1D_P&"'(Oa&;Oa&SO2+$:g+!H!7&!14 I tU * JL &T 2K&"<'B_ 7; *'BZ & D "' *d$O2Q:&JK,-0"+7[&WG )&2;&t>^&T &S))2K2:O)== +$[!H!7 14 I 0"DO$&&'B&<& I !22 $&7;T 1.OYFlO1&&&". 7;O&" ">B. ,-&7&*"+_P"a. 7;&"+ J;2;C'B 2 $.DJK_ AFlO1_ \O2;&" "$&&'B&"E.!7&"+: " *dQJ"70WO2;tU X$&&S_ *7 "+1D'B"+7<" .&243e! $&:g 1+2;93#"" S&F. 7, I _qHw"7 "%HNJKe;]!2&+,-&+$h. 7;O2!= 6^z!HE;7;O'B"+$&7 &>'B_ "f&+$)O2&"'BZ ;w& D rJK ;'B, $"%= 'B_ &,7 "+$&%$&c$&*,C 'q& JL &N[ ,-"C 4 &,7"++$&+$:>M JK"E H&J $Z 2 t> 7O1D+7;* N6M#"SO2K%$& 1D)H!NO$) >JLO2"+ ,N)1D"+7JK!= *"+,!HRJK 35" T 6%"U !t $&"+7"+&N &'/. * $)O$"%26 7",-&'B"+ J.&"+' 'B&&+7;)= FG6 0 ~ # "+&7 g !H!7& 1D I a"+7JK* ", !HaO$&&,-O2"+SN T 1+2KO"+$Z 2 >k{ >k{4| >k{9v { 2 > 0 ~ $&&| U $ "%&% "(')'+* $%"(,.-0/ "%$"1324%$('65 #"&3-7298:/-02 ; 2 " 3-=<>/@?"' ACB /-7/9D%#"%"7E >F)-02 "&#" $"' G?7'+29HJI"?%$78KF LE N $ MO213E

\X7&"+"P"+"7O$&&7*dO#J"7fWO2;5'B43gX14 DO$:G7;O1D _qH]FG&7'()HL> * $l1D&+ $l$149JK f&7 'B7&&, JK;'B *"+,!Hf"+7"'B"+_qHg];7;'B"%$O= 7&f +$&"+'B_UtoXS&_P* 5.!7&"+ d* eJK"7wWO2;n7"kJK+$)OO2;6M$]FG&7'B" du 4 {&U {[6 du − α 1 − βu + ω u = 0, dt dt !2 4 {U ~ 6 y¨ − ε 1 − y y˙ + y = 0, !2 4 {U |86 x¨ − λ − x x˙ + x = 0. JK"+: J;'B* ", "e"+7"'B"+_P " ,-&α,@FGε,FGλ w ;"+ O7;$9> 'BH-"+7;$7O, _$&> "+. 7 7;1. 6MO2#6MT = # "e&u,1Sy,.-3]xJ""+'[$&,-O2"S&T ω s I 2

2

2 0

2

2

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!

S&+$&"!.!6 *;%$&$O.a" "7;O$&7; U B%2lM"7&'w. 7;" "+7"T!$&#,l&&'/.X&7"+'B" τ = ω t >$&MO2.!* ' . 7;& "" 4 {U ~ 6>f, O$&&7' y = √βu, y˙ = dy , ε = α U *R$)O, '& I !22;!$&&7" S&+$&"Oy*;O+$&O$) ω 7;;f"kJK I "NU \Xdτ7;" ω" 4 {&U A| 6 + 1Dfl. 7;"%= "+' 4 {&U ~ 62"J;.!6M#"+T]1D'B" &YT X x = √εy, λ = ε UOR. 7;O&" 6 *dGJK"7]WO2; DJ $&Zha$)O,h;O1_PN"+'B&".!7&"+" (@2"+ X 4 {&U 86 z¨ − (λ − z˙ )z˙ + z = 0. 0

0

0

0

2

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4 {&U 6q7 DJFGFl"+7" I 7&&O$&:gw&"+$w1D'B" . z˙ = √1 x >$& 3 O2.!* $e. 7;" " 4 {&U |A%6 U W7hO$&7 I O$&"2:&_qHs1;4*" !Hw. 7;O26^z"+;7;O'B"+$&7 (α, ε, λ) g!= +$"'()H 4 {&U {%6%>]4 {U ~ 6 >]4 {&U |86 >]4 {&U P6 $&,-O2"S& n"l&1S.-3]JLO6^$&ZVU 43]J"" 7"kJK"%2;:&&+ , 2Kg7 & HNJK!$f!Ha7 h"+7"%H&DJ"z;7'B"+$7;f* "7"%1z)2:U B%2n. 7;&"+"

I W 7 '()2_qHO243g $&"2:&_qHe1D;4*"+ H ;7'B"+$7;5;SO26 JK6^$Za,NO17!= 'B& *"+,"g$&, )2"SNO t>t7rS&O2:Op#!Hh1D;4*"+ !Hs&r+$) & $Zx7"%= 2K,- I & _P'BtUoMJK;,-a +$&"+'()H4 {U {[6M 4 {&U ~ 6^7 x "7"HNJK"];7;O'B"+$&7 *"+7"+1eO2:w743]JLO"+$&ZE7"kJK"%2;:&_PT I , 2 ,-&"+*&+y7;9JK!.&Dw "+h7;)1D'B"7 7 JL)2:"T pz"'. &"2*"+s;7'B"+$7;g"l'B"!"+$ZtU * ++$"'B " 4 {&U A| 6P 7"%= JK"2:&_PT I ,!2s7 935JL"%$&Zw 1 "+.+$&T*&FG&, .Dfw"+e79JK . . &"%2;*!= N"%$&Zz7&7 I &)2:& √λ [U *r@+$&Tz +$&"+'B"d7 HNJK $u+. "+7,7 $*"+, O S&F. 7, I ! ^ JK7& &N)= X&Fl U O'BO$&7'\. 7;" "e"+"7O$&&7 *drJ"7 WO2;E7E&"%p#"+'0O7'BO= * ",-&'Y&1%J"T%$&&V>L+O2:O1.w%2;"J. 6^ . 6 &7 'B7&&,!. 4 {U 0A6 x¨ − λ − x x˙ + x = b sin ωt, JK" λ ]. 7;O26^zTn;7;'B"%$&7aO$&&'B&5+""+7;O$&7; > b, ω f'B!2 $O.DJL n*;+$$)g"+pz"+e&1JK"T +$& tU W"7"+ pz"'Y. 7;"!Z 4 {&U 0L6qf2"J;.!6M#"+'R& JK" 2

x¨ + x = λ − x2 x˙ + b sin ωt x¨ + ω 2 x = ω 2 − 1 x + λ − x2 x˙ + b sin ωt.

4 {U $86

B%2A*;%$&$)x&"+pz"+ &O1%JK"+T+$& ASO2 1,!x, &S&%$&&"+&TA*;%$&$"y$O= &'B &]"+"7O$&&7 (ω ∼ 1) >!;7;'B"%$&7n&1S.-3]JK"+ λ O2; "+$Zw'(92&T O= 243g $&"2:&&Tf&"2* &Tt>N5&"%p#"+"^1%JK"+T+$&" %2S&&"4 'B!2 $O.DJL b '()= 2KOe&"%2;*;A6%>-$&57;ONOg*;+$:z. 7;& " -4 {&U $86B7"J+$)O2"+$f+&S&&Tg'()2&" &1'/.-#" "P7'B& *"+,-&G& I !22;!$&&7Xq*;%$&$&T ω #"$&&'B_PT5+"%= "+7;O$&7 O2"+$Z?,&O1DO7'B&* ",-&T 2 2KS&& "%2 "TT %$&"'BTtU * @+$&'2;. *;"l7"+pz""G. 7;& " 4 {U $86q'B43g5+,!)$&:e]&-JK" X 1 x(t) = Re a(t) exp(iωt) = a exp(iωt) + a∗ exp(−iωt) 2

4 {&U vA6

uJK&)2 $"%2:& _P'.&%2;&&"+'

x = ρ(t) x˙ = cos(ωt + ϕ(t)) −ωρ(t) sin(ωt + ϕ(t)) ρ(t) = |a(t)| ϕ(t) = Arg a(t) & 1

4 {&U 6

a˙ exp(iωt) + a˙ ∗ exp(−iωt) = 0.

. 6

.!* "+$&'RJKO2!$&"%2;:&&+n.%2& s14 p#"+'Q"+7. 6 s$&7 . 6 7&!1D&J;=

X

i 1h a˙ exp(iωt) + a˙ ∗ exp(−iωt) + iωa exp(iωt) − iωa∗ exp(−iωt) = 2 i 1h = iωa exp(iωt) − iωa∗ exp(−iωt) , 2

x˙ =

i 1h iω a˙ exp(iωt) − iω a˙ ∗ exp(−iωt) − ω 2 a exp(iωt) − ω 2 a∗ exp(−iωt) = 2 i ω2 h = iω a˙ exp(iωt) − a exp(iωt + a∗ exp(−iωt) . 2

x¨ =

W JK+$) O2.!* '

x¨, x, ˙ x

l. 7;&"+" 4 &{ U $86(e&_P7;O1

sin(ωt)

* "7"%1@,- &"!$&_>

i iω h 1 2 ω2 − 1 h a exp(iωt) + a∗ exp(−iωt) + λ− a exp(2iωt)+ 2 2 4 i 2 ∗ 2 ∗ + 2|a| + (a ) exp(−2iωt) × a exp(iωt) − a exp(−iωt) + b + exp(iωt) − exp(−iωt) , 2i

iω a˙ exp(iωt) =

i hλ 1 ω2 − 1 h a + a∗ exp(−2iωt) + a2 exp(2iωt) + 2|a|2 + − 2iω 2 8 i b + (a∗ )2 exp(−2iωt) × a − a∗ exp(−2iωt) − 1 − exp(−2iωt) . 2ω

a˙ =

#)2"+">q7,7 _PNO+,-&S&,R.7"kJK R1D "+7J 7;O.!6 2"+. 6R*;+$5.!7&"+ t>O2;. *'r. ,-&7&*"+&"d. 7;O&"T "q=J;2;2π/ω 5,-&'B!2;",-T 'B!2; $.DJK_ 4 {&U 286 λ 1 b ω −1 a + a − |a| a − . a˙ = −i W7"J+$)O2 w, 'B!2",-+ . 6 &2ω "2*!.a2g)2;87 _qHa,-2ω &&7JK;)$))H 4 {&U { 6 a(t) = ρ(t) exp(iϕ(t)), O2.!* 'Y. ,-&7&*"+_P"l.!7&"+ J;2;w'B!2; $.DJK_ nFlO1D_ 2

2

&

1 λ ρ − ρ3 − β cos ϕ, 2 8 β ϕ˙ = −∆ + sin ϕ, ρ

ρ˙ =

4 {&U {&{[6 4 {&U { ~ 6

JK" ∆ = ω − 1 HN7;,!$&"7 1."+$w7;%$&7T, .w'B"+35J.a*;%$&$&Ts&"%p#"+&O1+= JK"+T+$& ?2ω+&S&%$&&"+&T *;O+$&O$&&T +""+7;O$&7; >/ β = b x $&"+&+$: 2ω &"%p#"+g&1%J"T%$&& VU jG&'B 'P>t* $n7 "+p#"+"#J;2;r H&JK&+n. 7;"!E+""+7;O$&7;*dhJK"7 WO2;a'B"+"+$e& J 2

x(t) = Re [a(t) exp(iωt)] = Re [ρ(t) exp(iωt + iϕ(t))] = ρ(t) cos [ωt + ϕ(t)] .

f "%$&:g" "7;O$&7y+&&"7!p5"%$,-O2"+SN n;]*;%$&$"z& "+p# "e&1%J"T%$&& >d,-& JL ϕ(t) = const U *q1'B93g_ $), 35"hJKO.-H!*;O+$&O$&_P"?,-O2;"SN !2 ω $), ;)1D_PN"+'B_P"eS& " V>,-& JKy'B!2!$.DJLh?FlO1Dw'B"kJ;2" y'B"!6^$Z & &7"+'B"VU !H!7 14 I XJK%$&7&T ,!X*+$$&_ +""+7;O$&7;u,z*+$$&" &"+pz"Tz!= 2_ =V&1'B93g;M#7#*":^%2KOS&&'s&1JK"T +$&tAU *x@+$&'r%2;. *;" &1'/. z" " 'B!2; $.DJK_R)2;"%$&Z'(92_P'P> G+.-z"+$&" _P'A O2;"%$&Ze&1D'/.-z""uF )1D_U W7 I "+l H 7 & 1D I w'B93#"+$gS&_d$&:f&nfFlO1&&&'C7 SO2 35"+tU W7"JO243g'P>N* $& ++$"'B"u. ,-&7*" _qHg. 7;O&" TfJ;2;g'B 2 $.DJK_R FlO1D_ 4 {&U {&[{ 6%+> 4 {&U { ~ 6B'B!2 $O.DJL#&"%p#"+#&O1%JK"+T+$& β O2;"%$&Z '(92&Te"%= 2* &TtU ^'B!2 $O.DJLM,-O2;"SN T ρ(t) Jz&"%p#'w&1JK"T%$&& "'y%. z"%$&&"+!= # "u 1'B"!2:5z7;" 6 X'B 2 $.DJKTa$&, )2"SNOT O$&&'B& "+"7O$&&7 U "%$&:A'B!2 $O.DJL ρ(t) 7;,!$&*"+,c&$&"+$+$.&"%$<7;9JK .+. 7"kJK"%2;:&&+ I ,!2Ks$&&'B &s+""+7;O$&7; >B,-$&7_PT 'B93gs& 7"JK"2 $: 1G. 7;& " 4 {&U {48| 6 1 λ ρ˙ = ρ − ρ . $) I &;7 _P' 1D;4*"+ ' 'B!2; $2.DJK_ (8ρ˙ = 0) $&&"%$&%$&. 6^$Y&%$&4 7;&&"+ 4 &+&S&_P" $&&*, !2e "JK&!3e_P"^$&&*, 6(. 7;"! 4 {U {4A| 6%U *q_q= HNJ5;P+$) I &;7 &" 1;4*" ">)7"+pz" " ρ(t) >4& _PNO6^#"+" "7"HNJK&Tz7O= I "+ 1(",-$&7&+u;4*;92:&&+ &%$&4! ρ(0) >D'B43e M7"J"%2 $:>9 $"7 != 7 .OaJK FlFG"+7" I ;)2:& &"l. 7;O&" " 4 {&U {48| 6%U oX7"kJK"%2 'R&%$&4!h7;O&&&"+ wn!Ha.&%$&&T* &&%$&:U * +&+$9 y7O= &"!, 4 !2z^" DJ& 3g&TG$&&*,-" 67 1D&JKO ρ˙ = 0 U & JLM7;&Ol*;+$: JKFGFG"7 " I )2:&y. 7;& " 4 {&U {48| 6X7;O&;w!.92;6z> O2.!*"'[)2;"S7;!= *"+,-&"G.!7&"+"XJ;2;n" DJ& 3g_qHa$&&* ",

3

4 {&U { 6

1 λ ρ − ρ3 = 0, 2 8

1l,-$&7&g;)H&DJ'A$7h"+JK& 3g_P"G$&&* ,

P1 , P2 , P3

ρ1 = 0,

√ ρ2,3 = ± 4λ,

G,-&&7JK;)$)'B 4 {&U {[0A6 4 {&U %{ $86

$&&"%$&%$&&"+U W"7&Or" DJ& 3g;Oy$&*, P : [ρ = 0] %.-#"%$&."+$s 7r2;6uS&_qHs1;4*"%= Hr;7'B"+$7; λ > gJK&"zJK7!.!+!H P : hρ = ±√4λi =P$O2:&,-n7 λ ≥ 0 U W7 λ = 0 &"$7 $&&* , %2&6^$Z sDJ .&U=* O$&& &'B&'[& I !22;!$&&7 " *d JK"7yWG)2;hfFlO1&&&Tw!2&+, +$& 7"+pz" " $&"*;O"+$ "O= JK&!3eTf$&*,-">-7;+O2435"+&Tfl;4*;)(x,2" x)˙,-&&7JKO$9U ρ"+pz="0 ρ = ±√4λ $&"*;O6^$A7;9JK .+.<7 "JK"2:&&+ I ,!2K >P,-$&7_PTQ&1D , "+$C7 "+7"%H&DJ" *"+7"+1 S&F. 7, I &"G1D;4*"+" λ = 0 a7;%$&"+$e7& &7 I ;)2:& √λ U (+$&T* &&+$:a"JK 3e _qHy$&*", P , P , P %$&"+'B_ 4 {U {4A| 6PH-O7;, $"7!= \ 1."+$e&S&%$&&"+&"G1D9* "" 1

1

2,3

2,3

1

2,3

1

∂ µ= ∂ρ

2

1 λ ρ − ρ3 2 8

3

4 {&U {9vA6

, ρ=ρi

,-$&7&"X7 "JK%$)O2; "+$]S&&Te 7& 1&JK . 6cJK;O'B*"+,-&T "7 "'B" &T $ 7;O&&T5*;%$&#. 7;"! 4 {&U {48| 6>O_P*%2;" . 6<X"+JK& 3g&T#$&*,-" P d,-O= &7JKO$)'B ρ (i = 1, 2, 3) U l2;s"+JK& 3g&Ty$&*, P : [ρ = 0] O2;. *;"+' U 2"kJK&N)$&"%2;:&>L7 λ < 0 +&S&+$&" &"z1;4*" " µ $&7 I O$"%2:O= µ = λ/2 &" 5$&*, P .+$&T*N!U!W7 λ > 0 &S&%$&&"+&" 1D9* "" µ O243g $&"2:O= &"G $&*, P "%.&%$&&T* N U l2;a" DJ& 3g_qH $&*", P : hρ = ±√4λi S&+$&" _P" 1D9* " wJK;O, &_P" a7;O&_ i

1

i

1

1

1

1

1

1

2"kJK&hNO$"%2:> 7i

√ P2,3 : ρ2,3 = ± 4λ

2,3

U

µ2,3 = λ > 0

2,3

λ 3 2 λ 3 − ρ = − λ = −λ. 2 8 i 2 2

7435JL6^$ZQ.+$&T*_P" "+JK& 3g_P"s$&&* ,

j "7"kJK+$&" J;2 +$&"+'B_ 4 {&U {D|86]!1+2435"+&"s&_dpz"y$&&"%$&%$&."+$ S&F. 7, I e&!2 >Nf. 7;" " 4 {&U {4|86 7"kJK+$)OO2;"%$5S&&T]&7'(92:& . 6l 7"JK%$)O2"+;O ; 7 U]{&U {&>&, O1_PN)= ' "+$9>(* $ 7 λ < 0 %. z"%$&."+$ JK;s.&%$&&T* NOY"JK 3eO $&&*, P : 1

&

4.0 3.0

ρ2

2.0 1.0

ρ

0.0

ρ1

−1.0

ρ3

−2.0 −3.0 −4.0 −2.0

−1.0

0.0

1.0

2.0

λ

9

3.0

4.0

dU WG7 "7"HNJK"x*"7"%1rS&F. 7, I &"s1D9* "" λ = 0 $&*, P $&"+7N"+$ .+$&T*&+$: hf""l&,7"++$& &+$x7 935JL"%$&Zw;7;].&%$&&T* &_qHh' = 'B"+$7* _qHYJK7 . aJ7 . +. "JK&!3e_qHC$&*", P , P > ,-&7&J;O$_ ,-$&7_qH 'B"!6^$Zr#7&%$&&'R. 7;OO2;6M#"+a;7;O'B"+$&77&7 I &)2:& ,-&7 6 1 ;9J,7 $*&%$&Y X

[ρ1 = 0]

1

2

3

√ ρ2,3 = ± 4λ.

\X . 760&Fl"+7'/"g.$%.!; 7"74 ,{&7 U {4 |8$&6Xn* ", ,-'B&Tr!2S&" ,-FG+. &7'\, & J "g^ 7 JK"JK7%&$) &ON2;z = "+$sX&+& S&F & Tx4U &7'(92:O= * $&&'B&'h" "7;O$&7"d* GJ"7#WOI 2 MFlO1&&&Tl!2&+,-&+$ (x, x)˙ 7 x

N7 ; I . 7;92;6M#"+e;O7;'B"+$7; λ S.DJ.-$f;SO2;6 JLO$&:Za%2"kJ. 6^# "z"+7"%= +$7&T, AFlO1&&_qH &7 $&7 "+$&U(W7 'B""%$&ZADJ;h.&%$&&T* NO $&*, 7;&&"+ V>L7;OO2;93#";)ye9*)2λ"z<,-&07&J;O$9U . &"%2 *" "' ;7;'B"%$O= 7; λ 7s "7"HNJK"5*"+7"+15 O2:+&+$9 "57;O&&&"+ w$&"+7- "+$a.&%$&&T *&&%$&:U *<"]&, 7"%$&&%$&y7 935JL"%$&Za.&%$&&T *&_PTy 7"JK"2:&_PT I ,!2 >;7;√9JK .G, O$&O= 7&+e7;O+$&"%$ 7&7 I &;)2:,-&760 1z;9JK,7 $&* &+$ ρ¯ = 4λ 4 J" ρ¯ = 7;9J .&7 "JK"2:&&+ I ,!2K# 2 +$) I &;7 &"X1D9* ""GO'B!2 $O.4J_ $&,-O= 2"+SN?T 6%U

# 9 2

x˙ = mx−x3

z˙ = (m+iω)z −z|z|2 )

l2;a$&&+* $&S&_

& DO$:e7 I "+ .+$)O2" s+$) I &;O7&TsO'B!2 $O.-= JK_ O$&&,-O2;"SN Tt>7&!$&"+77 ."' JKFGFG"7"+ I ;92:&&"]. 7;& "" 4 {&U {4|86U )1%JK"2'R&S&"*+$w. 7;& " n; a. 'B43g'Y5= ~ U ρ 4 {&U { 6 1 1 −2 dρ = −λ + . ρ dt ρ 4 W"7"+ pz"'R@+$f. 7;O&" " ]&-JK" 1 d 1 4 {&U %{ 286 1 = −λ − . ρ 4λ JK"2Kg1D'B"!. "7 "'B" dt_qH ρ 4 {&U ~f 6 1 1 r= − , ρ 4λ O2.!* ' 4 {&U ~ [{ 6 dr = −λr. "%p#"+" . 7;O&" h 'B""+$g& J dt 4 {&U ~f~ 6 r(t) = C exp(−λt). &%$) $. C & 7"JK"2'C 1;4*;92:&_qHg.&2&&TVUW7 t = 0 O2;. *' r(0) = 2 r =C 3

3

2

2

2

2

0

JK"

C = r0 =

1 1 − , 2 ρ0 4λ

l1D;4*"+"^'B!2 $O.DJK_O$&&,-O2"+SN Tg;4*;92:&_PTf'B&'B"!$G&7"'B"+tU W"7"+ pz"'7"%p#"+"GJ2;n HNJKTh"+7"'B"+&T ρ0

ρ

1 1 1 1 − exp(−λt). = − 2 ρ2 4λ ρ0 4λ

4 {&U ~ |86

jG&'B 'P> !* $&?%$) I &7&"y1;4*" "s'B!2; $.DJK_ $&,-O2"+SNTR7& √ U .! * "+$&'@+$&g"7"+ pz"'R7"%p#" " ]&-JK" ρ¯ = 4λ 1 1 1 1 − = 2 − 2 exp(−λt), ρ2 ρ¯2 ρ ρ¯ 02 2 ρ¯ ρ¯ − 1 = 2 − 1 exp(−λt), ρ2 ρ0 ρ¯ ρ(t) = rh . i ρ¯2 − 1 exp(−λt) + 1 ρ2 0

4 {&U ~ 6

&

#"f_P7;O35"+"f&7"kJK"%2; "+$h7

I "+].&%$)&O2;" ?'B!2!$.DJK_ $&,-O= 2"+SNTzuO$&&'B&'y+""+7;O$&7" *dXJK"+75WO2;tU9W7 &17;+$)O#&7 "'B" f%$&"'B"G.&%$);)2N"%$&Zs%$) I ;7"l1;4*" " ρ(t) → ρ¯ = √4λ U jGz7%. ,-"f{U ~ 7 &!226^+$77&&w 7 I "+X.&%$)&)2"!w%$) I ;7T 'B!2; $.DJK_[O$&&,-O2"+SN Tw7 w7;)1+2* _qHw;4*;)2:& _qH 1D;4*"+ H ρ U *q"+7"'BZ,Y. , 7&*"+_P' .!7&"+ ' J2;R'B 2 $.DJK_ FlO1_ 4 {&U {&[{ 6> 4!{2U !{ $~ .D6%JLU&W 752K&S&$'x&"+$+"+$pz.&"%"$a'x7;9JK7 'B.&+ .n*7"+"k,-JK&"%'2;:&&O&1%+JK "+T+$, &2KefOO$&2K&O'B"'P&>*!f$&l"+' "%== I ρ(t) √ >/ 7 I "a+!H!7&!14 I Y'B435"+$xS&_d$:r&A 7;O$&7; ρ(t) FlO1D&&'C7=SO2;ρ¯ 35=" 4λ 0

4 {&U ~ 0A6

dϕ β = −∆ + √ sin(ϕ). dt 4λ

#;)EJK;O'B*"+, OC%$&"'(x'B""%$xDJ .xJK ;'B*"+, . 6

"7 "'B" . 6 ϕ t>O&$&"+$+$&" >-7;O1D'B"+7&%$&:lFlO1D&&X 7&+$7;%$&NG7;;u"J I "OU O= +$4"PJK;O'B*"+,-&Te++$"'B_<'B43eG7 "JK%$)O2;!$&: 2 S&G;XJ"T%$&& $"%2:)= &Tsy$ −∞ J +∞ >2;S&;,7 .-3e &+$s7;9JK .D ρ¯ = √4λ >.!* $&_PN) "+7JK* &+$: FG. , I sin ϕ UKW&&"kJK" "z++$"'B_\14 $e$g;7;'B"%$&7 >O,-$&7_P"qH-O7;, $"7 1+. 6^$ 7;%$&7T, .#'B"%3]J;.z*;O+$&O$&&T5& "+p# "G&O1+= ∆, β, λ JK"+T+$& *;%$&$&T\O$&&'B&C"+"7O$&&7 4 ∆6%> O'B!2 $O.4J;.Q"+pz"+ &1JK"T +$& Z4 β 6qw&"JK"+" $&&'B]"+"7;)$&&7; 4 λ6%U 0.90

0.80

ρ(t)

0.70

0.60

0.50

0.40 0.0

20.0

40.0

t

60.0

&

ρ λ = 0.1 0

80.0

)

O 'BO$&7'RJK;O'B, .yFlO1_ 1D&+'B&%$&x$a7;++$&7 &T, ∆ x'B 2!= $.DJK_c&"+pz"+g1%JK"+T+$& β 7nFl , +7&&&' 1D9* "w;7;O'B"+$&7 λ > $&"*;O6^# ',NO17'B*"+,'R,-O2;"SN 'Y$&&'B&+g+""+7;O$&7; U jGw7 +. ,-"y{&U |w+$&7 &"_ 1D&+'B&+$?FlO1D_ ϕ $y7"'B"+ t 7 λ = ?7;O1%2*_qH 1D;4*"+ H ;7'B"+$7;y7;++$&7 &T,tU X1e7+. , 0.1, β = 0.01 & J>d* $A 7<'()2_qH1;4*" !H 7;O+$7&T, QFlO1D "s'B" "+$&Z & &7 "%= 'B" t>d&%$)"+$Z<&+$4T "%2* &T 4 2 0E $ 7 ∆ = ∓0.036U . &"%2;*" "'<7;++$7&T,wf*;+$$&"O>L,-& JK 7"+&_dp5O"+$ ",-$&7&"l,7!= $& *"+, "n1;4*" " |∆ | >7& HNJK $E, 9* "+$&|∆| " &"n 1D'B"+" "nh&&"kJK" FlO1D_ ϕ(t) U B"r&"%2;*;C;4*;O"+$A'B" !$&:&Z & &7"+'B"VU * 1D&+'B&+$ $]1D, ]7;'B"%$&7; 4 "2w*+$$)f"+pz"+g&O1%JK"+T+$& wS&O2:pz" 4 'B"+:O= p#" 6l+&S&+$&" &T?*;O+∆$&O$&_ "+"7O$&&7 >$ ∆ > 0 4 ∆ < 06#6X&"%2 *;nF )1D_ 2S&r. 'B":p#"+$Zt>(2S&x;7;O+$)"%$?&r&7"+'B"VU W7C'()2TA;4JK,7!= ϕ(t) $& *&%$& |∆ − ∆ | & 7"'B"+&T7 "D)2!14 I ϕ(t) 'B43e &_ JK"2 $: 7O= JKO23e $"%2:_P"z!$&"7 N)2_>Kf$"*"+" , O$&&7_qHwFlO1D]+$)"%$&Zs7, $& *"+, &%$&4 &Tt>*"7 "J. 6^z!HNZ ],-&7O$&,'BE $"7N92K'Bt>&a&7"+'/ ,-$&7_qH 7 HNJK $Y 1D'B"+" "EFlO1_ 2π U 7&%$&&' ;9J,7 $*&%$& |∆ − ∆ | $"7&)2_ &+$9 +$N FlO1D_[. 'B":p#6^$Zt> +7"JK! r,-&7 &+$:n!1D'B" " FlO1D_ . &"%2 *N"%$&ZVU

c

c

c

25.00 20.00

3

2

1

15.00

4 10.00

5

5.00

ϕ

0.00 −5.00

6

−10.00

7

−15.00

10

−20.00 −25.00 0.00

9

8

500.00

1000.00

t

9 : (ϕ 0 @ t ∆ = ∓0.035 ∆ = ∓0.032 ∆ = ∓0.03

1500.00

2000.00

λ = 0.1, β = 0.01 ∆ = ∓0.04 ∆ = ∓0.07

&

/O,'0&S&7;O1&'P>/%. z"%$&."+$ $"7&)21;4*" T7;++$&7 &T, |∆| < ∆ > JK"xFlO14?"r'B"+ "%$&Z (ϕ(t) = const) >q""r7&!1D&JK;)t>d+, 7&+$:Y 1'B""= CFlO1_ 7;;s .426zU $ 1;4*;"+$9>(* $E+""+7;O$&7Y+&&"7!p5"%$?"+7JK!= *"+,"l,-O2"+SN a#*;+$$&"l&"%p#"+f&1JK"T%$&&!t>;;S)26 JL"%$&ZnO2"+" !H!7& 1D I VU !t7"kJK"%2'Bg $&"+7N)2Kl H 7 & 1D I gF )14G'B" "+$&Zg& &7"+'B"V>/,-O2"S& +$) $ZA, NO1D "7 DJ*"+,'BtU 7 "JK ! ,-&7 &+$: 1'B""+ rFlO1D_ hϕ(t)i & 7"JK"2;"%$a$&7 . 6 "+1D&+'/. 6 *;%$&$O . n*;O+$&)= ˙ $.S&" TtU X1G7%.! ,!a{&U |#& JK >* $f 7a'()2&Ta;9J,7 $*&%$&y+7"J ,-&7 &+$: 1D'B"+"!#FlO1D_ *":^ 1,!)t>)* $&X+&$&"+$+$&O.&"+$l&* ":^'()2&T *;)= +$$&"gS&"+TtU 7 &+$&' ;9J,7 $*&%$& 7 & HNJK!$h. &"%2;*" "e*;O+$&O$&_ S&"+TtU W7cFlO1&&&' &c!H!7_P' JK& 3#" ' $&"*;O6^$+&+$4 7;&&"+ w!2y "JK&!3e_P"l$&&* ,nJK'B*"+,-&Ts +$&"+'BH _ 4 {&U ~ 0A6U "+3g!= 'B_c+!H!7& 14 I nJKO23e_ $&&"%$&%$&&&&O$&:e.+$&T*_P'R&%$&4!'Y7O= &"!t U l2;h++$"'BH _ 4 {U ~ 0L6PT JK"+'&%$&4 y7;O&&&"+ V>&7"kJK"%2 'R!H &SO2+$:g%. z"%$&&&& hg7;'B"%$&7;'P>; %2;"J."'R.&%$&&T* &&%$&:aS&F. 7!= , I w" DJ& 3g_qH $&*",w7n 1D'B"+" y;7;O'B"+$&7 &g++$"'B_U j"JK 3e _P" $&*,s +$&"+'B_ 4 {&U ~ 0A6PS.DJK"'R7 "JK%$)O2;!$&: JK'B"7' FlO1D&&'c 7&+$7;%$&&"e;w&, 7 .-3e+$&E7;9JK!.&D ρ¯ = √4λ > &+$7&"+&T? !2,-&+$yG,-&&7JK;)$)'B ρ sin(ϕ), ρ cos(ϕ) 4 +'PU;7OU {&U 86%U \(%2;&&"+'C%.-#"%$&&N a+&+$4Ta7;O&&&"+ gO2"+$Z 7;"%$&&5 .N= 26[7;O&&Tn*;+$n. 7;O&" Z 4 {&U ~ 0A6

c

X1

4 {&U ~ $86q;9HNJK'

β −∆ + √ sin(ϕ) = 0. 4λ

4 {&U ~ $86

JK&"G" DJ& 3g_P"l$&*,nG,-&&7JK;)$)'B √ ∆ 4λ ϕ¯1 = arcsin , β √ ∆ 4λ ϕ¯2 = π − arcsin . β √ ∆ 4λ β |∆| ≤ √ ≤1 β 4λ

4 {&U ~ vA6

4 {&U ~f 6

Xo %.-#"%$&. 6M$9>B"%2; !2; U 2"JKNO$&"2:& > 7 FG,-7 &N &' λ +7; I _&SO2KO+$&f+.-z"+$&&NO "JK&!3e_qH]$&*"+,g; !2,-&+$y;7'B"+$7& (β, ∆) &7 "JK"2;"%$g2 t>1D9JLO . 7;&"+"+' √ 4 {&U ~ 286 β = 4λ |∆| .

1.0

1.0

ρsinϕ

ρsinϕ

0.5

0.5

0.0

0.0

−0.5

−0.5

−1.0 −1.0

−0.5

0.0

0.5

−1.0 −1.0

1.0

ρcosϕ

−0.5

0.0

0.5

1.0

ρcosϕ

1.0

ρsinϕ 0.5

0.0

−0.5

−1.0 −1.0

−0.5

0.0

∆ = 0 ∆ =0.03

0.5

1.0

ρcosϕ

L (•) (×)

λ = 0.1, β = 0.01 ∆ = 0.032

Xo SO2K%$&:C H 7 & 1D I t> +7; I _b,-$&7&T &7"kJK"%2; "+$A.!7&"+" 4 {&U ~ 286> &%$&7"_\;57 U{&U 0!U 2"J."' .&+$&T* &&+$:Q "JK&!3e_qHc$&&* ",tU +'B$&7 ' &&"J"" +$&"+'B_ 4 {&U ~ 0A6] &,7"++$& &+$ " DJ& 3g_qHR$&*", ϕ¯ (i = 1, 2) 2;"T = &' 7 SO2 35"+tUtWG7 "JK%$)&'RJK;'B *"+, . 6 "7"+'B" . 6 & JK" ϕ(t) = > J" ϕ(t) H-O7;, $"7 1+.&"%$ '()2&"O$&,!2" " $f+&+$9 n7;&&"= ϕ¯ + ϕ(t) ˜ ˜ ! ϕ¯ UW"7"+ pz"'Y. 7;" " 4 {&U ~ 0A6q%2"kJ. 6^z'R&S&7;)1D&' i

i

i

d β (ϕ¯i + ϕ) ˜ = −∆ + √ sin (ϕ¯i + ϕ) ˜ , dt 4λ β dϕ˜ = −∆ + √ (sin ϕ¯i cos ϕ˜ + sin ϕ˜ cos ϕ¯i ) . dt 4λ cos ϕ˜ sin ϕ˜ ϕ˜

O,!2K4JK_PNO E7--J +" T!2&7 A. * $_PNOR2K"+'B_P"s"h_q= p#"f"+7&&+h&7N-JK, w'()2+$&E > O2.!* ' %2;"J. 6^z""g2;"DO7 1DN"

&

0.50

β 0.40

0.30

0.20

0.10

0.00 −1.0

−0.5

0.0

0.5

∆

1.0

:

λ = 0.1

. 7;& "" dϕ˜ β = −∆ + √ (sin ϕ¯i + ϕ˜ cos ϕ¯i ) . dt 4λ

X1l&7"kJK"%2"+ w" DJ& 3g&Ta$&&*, h%2;"J."+$9>K* $& β −∆ + √ sin ϕ¯i = 0. 4λ

*<7"+1+.92;:9$))$&"#)2;. *;"'C. 7;&"+ dϕ˜ = dt

ϕ(t) ˜ ∼ exp

7"%p#" "z,-$&7&+g'B"+"+$g& J

4 {&U | 6

β √ cos ϕ¯i ϕ, ˜ 4λ

4 {&U | {[6

β √ cos(ϕ¯i ) × t . 4λ

/O,'[&S&7;)1D&'P> .+$&T*&+$:?&%$&4 TY7;O&&&"+ 1D&+ $x$r1D;O,! U (2 cos ϕ¯ > 0 >$& '(92&"zO$&,!2" " ϕ(t) ;7;%$)"+$a&7"+'B"t>L cos ϕ¯ ˜ +$&4 " 7;& &&"+ ϕ¯ O2; "+$Z5"+.+$&T*_P'PU B2 cos ϕ¯ < 0 >)$'()2&" $,!2&"+"#1DO$.NH-O"+$a&7"'B"+t>Ky&%$&4 "57;& &&"+ nO2; "+$Zy.&%$&&T!= *_P'PU X1y_P7;O35"+ J;2;,-&7&J;O$ "JK&!3e_qHC$&*", 4 {&U ~ vA6%> 4 {&U ~f 6 %2;"J."+$9>/* $ ϕ¯ > 0 "JK&!3e;)E$&*, ϕ¯ "%.&%$&&T* NOVU *, ϕ¯ )2;"%$&Zn.&+$&cos T* &&Tt>K&+, )2:&, . cos ϕ¯ < 0 U

i

i

i

i

1

1

2

2

Gj 57U {U > > S]&, O14O;].+$&T*&Oha"%.&%$&&T* NOh"JK 3e _P"G$&*!= , 5FlO1&&&'? 7&+$7;%$&&"l +$&"+'B_c7 7;O1+2;*_qHe1D;4*"+ !H;7;O'B"+$&7 UG,z& JKu 1d&_P7O35" T4 {U ~ vL64 {&U ~f 6%>97 ,-&&7JK;)$)G "+.&%$&&T = ∆ 0 *&Ts$&&*, x7&; ϕ¯ = 0 > ,-&&7JK ;O$) .&+$&T∆* =&&Tr $&&*, E7;O&; ϕ¯ = π √ 4 7 UP{&U > 8 6%U W7 .!"%2* "C7;++$7&T,V> ,-& JK ∆ 4λ +$7"'B!$&ZA,C"kJK!= I "> "+.+$&T*N)A.+$&T*&O"+JK& 3g_P"a$&&* β,"+7"'B"%56M$&ZR &,7!. 3g&%$&n;&%$&7"+* .eJK7 . uJK7 . %+. X"%.&%$&&T* NO, e7$]*;&&Ta+$7"%2= ,t>L.&+$&T* NO wa*;+&&&TtUVW7 ϕ = π &s2NO6^$&ZV>y14O$"'c *"%14)= 2 6^$9>-,-& JL ∆√4λ 7"_dp5"%$#"J I . 4 7OU;{&U > 6U *"%1D&" "^"+JK& 3]= _qHh$&&*"+,r7β& H&JK $n7s&_qH&JK"5 15&S)2K+$s H 7 & 1D I tU X1&S&7;O35)= 6^#Ox$&&*, n&&"+7 p5O"+$s&7O5O$"%2:&"#JK& 35"+"gw,7 .-3e &+$t>+7"J & 7"JK"2;"%$e*+$$.nS&"+TtU ,-&7 &+$:e,-$&7& hϕ(t)i jG?7%.! , "C{&U $?˙&%$&7&"+; 14& 'B&%$&: Ω = hϕ(t)i $ 7;++$&7 &T, ˙ *;%$&$" ∆ ++$"'B_ 4 {&U ~ 0A6u7 β = 0.01 λ = 0.1 U $"7&)2?1;4*" T ∆ > JK" Ω = 0 >&O$&&"%$&+$.&"%$ SO2K%$& !H!7& 1D I V>/14h7 "JK"2K'B ,-$&&7 &T ;S)26 JL6M$&ZyS"!al*+$$&&T Ω U 1

2

0.04 0.03 0.02 0.01

Ω

0.00 −0.01 −0.02 −0.03 −0.04 −0.04

G β = 0.01 λ = 0.1

−0.03

−0.02

−0.01

Ω = hϕ(t)i ˙

0.00

(

∆

0.01

0.02

G.

0.03

0.04

∆

3 24%$7/ "%0 " E 2 "02@?&$"1>* $0*#8 &H E F B /-0/D&#" "0E ,

7$ M F F)-72 "%# "%$

B

#

5O1D&&"X& D "XO2;" !H!7 14 I hu&'Bz:&6c. 7;& " 4 {&U ~ 0A6 JK&!.&+$'BG$&O2:, G7 f*":2KS&'x"+pz"'x&1JK"T%$&& t>N,-& JL&1'/. z"%= "+' 'B!2 $O.DJK_ ,-O2;"SN T<$&&'B?+""+7;O$&7; 'B93gE7" "S&7"+*:U W4O2"+"w!2+*"+1&&"+"y+!H!7& 14 I R!14 nS&F. 7, I "+TR7O= 35JK"!]!2f*"%1D&"!g.+$&T*&Tgf"+.+$&T*&Te"+JK& 3g_qH5$&*"+, rJK&'B"+7&' F )1D&&' 7 &+$7;%$&&"OUqjGs!2&+,-&+$A. 7;)2;6M#!HC;O7;)= 'B"+$7& (β, ∆) 'B""%$&Z m 1D_P, n!H!7 14 I t> 7; I _`,-$&7&E149JLO_ 7N'B_P'B]2!'B β = √4λ |∆| >N;,-$&7_qHf]7& HNJK $ .!, O1D;)g"kJK!= +$&" ;O?SF.!7 ,! I xJ2;rJ.NH "JK&!3e_qHx$&*",V]U \u*"+$y'B!2 $O.DJK&+ . 7;& " V>&$&G"%$&:l7;+'B$7" "^++$"'B_Y. ,-&7&* "_qH]. 7;O&" T5J2;f' = !2!$.DJK_ GFlO1D_ 4 {&U {&[{ 68> 4 {U { ~ 6L7DJ $u,lS&O2;""B%2;93g_P'wS&F. 7, I & _P' )2"!'Yf!2;&,-&%$&w. 7O2; 6^# Hn;7;'B"%$&7 (β, ∆) U l2;f%$&"+'B_ . ,-&7&* "_qHg. 7;& "T 4 {&U {&[{ 6N> 4 {&U { ~ 6/7"%3e'(O' !H!7&!= 14 I E$)O, 35"]$&"*;6M$y+&+$9 E7;& &&"+ ϕ(t) = const ρ(t) = const > .-35"5;FlO1&&&Ts!2;&,-&%$& 5,-&7&J;O$)'B ρ sin ϕ ρ cos ϕ UtjG 7%. != ,-"{&U vf;f!2&+,-&+$r. 7O2; 6^# Hy;O7;'B"+$7& (β, ∆) 7yFG,-7 &N &' 0.0225 0.0200

a

1

f

l3’

0.0175

2

b l3

B

0.0150

β

l2

0.0125

5

0.0100

d

3 l1’

0.0075

c

4

A

l1

C

0.0050 0.0025 0.0000 −0.050

e −0.030

−0.010

L

L

L λ = 0.1 β, ∆

∆

0.010

0.030

0.050

G

9

&+$7&" _ &SO2K%$&E!H!7 14 I ?s2 7;O1%2* _qH SF.!7 ,!9= I Tn !H!7&_qH JK 35" TtU "%3e 'R!H!7 14 I x+.-#"++$&O.&"+$w7y1;4*" !Hy;7;'B"%$&7 1#&S)2K)= +$ A f&SO2+$ B >&,-$&7_P"^,7 .-3]"%$ SO2K%$&: C ,N)1D"+7JK* ", !H5JK&!= 35"+TtU *CSO2K%$& A ;lF )1D&&Tg!2;&,-&%$& 'B"+"+$&Ze$&7g"JK 3e _qHg$&*!= ,Y X;.+$&T*&_PTs.-1D"2 P 4 'B" @+$)]$&&* ,!g$&&"+*;"%$7"%3e'/.a+!H!7& 14)= I ? 6%>+"J;2NO$&*, P "+.+$&T*N)n$&*, P > ,-$&7;Oa#14O& 'B&+$ $w1D;4*"+T 'B435"%$yS&_d$:w2Sh"%.&+$&T* &_P'0.-1%2&' 4 7 ""22"7 & ' 6%> 2Sf "+.&%$&&T *β&_P∆'RFG&, .&'PUjl57 +. ,-"]{&U > 57"kJK+$)OO2"wFlO1D&_PTn &7!= $&7 "+$9> HN7;,!$&"7 _PTAJ;2;A&SO2KO+$&A H 7 & 1D I A U * &SO2+$ B 7"+3g'/. !H!7& 1D I ]$), 3#"q$&"*;O"+$.&%$&&T *NO] "JK&!3e;)z$&*, P >OJK;O, +$7 . , $. 7;FlO1&&&+]7&%$&7; +$&&g+$)92Kg7 I ;)2:& g &T 4 7OU {&U > S 6U !JK"+:zO$&+.-$+$&O.!6M$f"+.+$&T*&_P"X$&*, P U oXSO2+$: C $&&"%$&%$&."+$ P

λ = 0.1

N S

R

N

R

1.000

S

1.000

ρsinϕ

ρsinϕ

0.500

0.500

PN

0.000

x

* PR

PN

0.000

PS

−0.500

−0.500

−1.000 −1.0

−0.5

0.0

0.5

ρcosϕ

−1.000 −1.0

1.0

−0.5

0.0

0.5

ρcosϕ

0.5

ρcosϕ

1.000

1.0

1.000

ρsinϕ

ρsinϕ

C

0.500

0.500

* PR

0.000

C * PN

0.000

−0.500

−0.500

−1.000 −1.0

−0.5

0.0

0.5

−1.000 −1.0

1.0

ρcosϕ

−0.5

0.0

1.0

: G

β ∆ A, B C λ =β 0.1 = A 0.01, ∆C= 0.001 β = 0.02, ∆ = 0.01 B β = 0.01, ∆ = 0.03 C β = 0.02, ∆ = 0.037

3

,N)1D"+7JK* ", ' 7"+3g'('P> , O$&&7_P' r%$&"'B"a. ,-&7&*"+_qH . 7;"%= T 4 {&U {&[{ 6%> 4 {&U { ~ 6#$&&"+*;"%$ .&+$&T* &_PT<7"J"%2:& _PT I ,!2 C 4 7 UM{&U > 6U jGgFlO1&&&'A&7 $7"+$"5%$&"'B_['B'Bg.+$&T*& 7"kJK"%2;:&&+ I ,!2 C 'B"+"+$&Zh"+.+$&T*_PTsFl, .& P U O'BO$&7'ES&F. 7, I #HN7;, $"7 _qHf7"+3g'B&X++$"'B_A7zJK& 3#" f!2&+&,-&+$h;O7;'B"+$7& (β, ∆) U jGu7%. ,-"l{U 2u &+$7&"GSF.!7 ,! I &O JK;O7;'B'(X^14& 'B&%$&]$ ;7'B"+$7;u7;++$7&T, ∆ 7zFG,- 7&N &' β = 0.01 U9jG !2&+,-&+$5;7;)= 'B"+$7& 4 β, ∆6(@+$&]+&$&"+$&%$&."+$e"7"+'B"+z"6[)JO2:z2 cd U W7 '()2&T 7;++$7&T,-" ∆ 4 r $"7N92"w'B"%3]J;.?$&&* ,!O'B|s >B$&'B"+*" _qHA'B&)2K)= 'B #;n7 UB{&U v C{U 286X"+JK& 3g;Os$&*, O2; "+$Z "%.&%$&&T* &_P' .-1+2×' !2 7"+"%2 2"7'PU 5O1&&_PTC&7 $7"+$E +$&P"+'B_ J;2; 7+'(O$&7 N"'BT !$.N I &, O1D ;w7 U {&U > U . &"2*"+"'0!2?. 'B"+:p#"+"+' ∆ 7 "+7"%H&DJ"h*"+7"+1n$&&* ,Y|s r7""22"+7Y7"+&7;O#"+$Zr"%.&+$&T* &_PTFGO= , .UBW7xJK)2:&"+T p#"+' !1D'B" " ∆ > ,!26u*O??_qHNJY 1 &SO2+$ A 7 "+7"+"*"+?2 T l l > "+.&%$&&T *&_PT FG&, . P , ,!HES&F. 7, I T " 7"+$"7 "N"%$9UNjG7 .-p#"+"d7 "+3g'(^+!H!7&!14 I 5&7"kJK"%2; "+$ &&"J"" "+JK& 3g_qHy$&*"+, P P UtjGfF )1D&&'R&7 $7"+$"5"+.+$&T*&_P"f'B&+O= &S&7O1D y"J2K P 14'B_P, 6M$&Zx;e.&%$&&T *&_PTr.-1D"%2 P U . &"2*"+"' ∆ "kJ;2 h.+$&T*_PTr.-1"%2xS)2 3]6M$&Zhy;e+7; I " l 2N6M$&Zsg"kJ;2O= .-1D"2U #)2"+"7& HNJK $ES&F. 7, I ?743]JK"+ ?.+$&T*&r7"kJK"%2;:&&+ I , 2KE7 A+*"+1&&"+243e &TA&S&T $&*,A"kJ;2O= . 1"%2U W7"J"%2:& _PT R

R

0 1

1

N

R

S

S

N

1

0.80

PN 0.60

ρ

PS

0.40

PR 0.20

0.00 −0.02

4

3

−0.01

0.00

∆

0.01

0.02

PN , PS PR & ∆ #

β = 0.01

λ = 0.1

I , 2[&1, "+$ 35"++$, xc7O1.<35"s14CS&F. 7, I &Tc$&*,-&T 'B""+$ ,-&"+*_PT7;4JK .&4 7 U{&U > 86%U *qSO2; 1DgS&F. 7, I & &Te$&*,g&$&"+$+$.N= 6^z"M, )2"SNO f,-&"* &TO'B!2 $O.4J_'B"6M$z*":lS&O2;:p#&Tf"+7JU1* %= HNJK &TE++$"'B"zJ;2;y"7"+'B" &T x(t) @+$aS.DJK"+$a+&$&"+$&%$&&NO$&:n35"%$&,-&'/. &1, &&" 6\'B"J;2;"_qHS&"+TtU +'B$&7 "_P"GS& FG. 7, I a;z7!= I "^&SO2K%$&"+T A C & 7"JK"2;6M$5, ,'B"H-!1D' H 7 & 1D I a*"+7"+1X1D)H!NO$9U oXSO2K%$&: A 1D_P, ] H 7 & 1D I h$)O, 35"G;O1_PN6^$g&SO2+$:&6\14)H!NO$)!U jG57+. ,-"f{&U { &%$&7&"+;]S&F. 7, I & ;OJK;+7;'B'(zJ;2;a"+JK& 3]= _qHs$&*"+,E++$"'B_ 4 .!7&"+ 4 {U {&%{ 6^ 4 {&U { ~ 6#6^7r 1'B""+x;7;O'B"+$&7 EFl , +7&&&'Q1D;4*"+ ∆ = 0.01 UjGa!2&+,-&+$?;7;O'B"+$&7 & (β, ∆) β 4 7 U!{&U Lv 6@+$&O$&&"%$&+$.&"%$#"+7"'B"%#"+6<)JKO2;:^2 T &U 7* "kJK&"%SO22K:O+$&_P T C ,!!2 = 35"P2 l ; FlO1&&&Tf!2&+&,-&+$'B"+"+$&Ze.&+$&T* &_Pef I ; 7 O 2 4 5 3 " P _ T & . & $ 7 + " s % " & . % & $ & T * & P _ C T l F , & . U 7 & + $ & 0 ' ' = C !2!$.DJK_ &"+pz"+s&1JK"T%$&&! β "+.+$&T*&_PTCFG&, . PP "7 "'B"+#"+$Z XFlO1&&&T !2&+&,-&+$tU&W7z"7"+"+*" 52; l 's, ,!HzS&F. 7, )= I Ts "#7& HNJK $9UjGe7 I "z&SO2K%$&r !H!7& 1D I r7 "JK"2:&_PT I ,!2 +*"+1D"+$9>743]JK"+$Z 243e;)E&SN)s$&&*, n"J2O=m.-1"%2U *q_dp#"52 l C .&%$&&T *&_PTy.-1D"2 n"kJ;2 7; H&DJ; $&Zy;fFlO1&&&Tn!2,-&+$tU . &"%= 2* ""+' β 14a$&&P* , Tt>O$&'B"* P"T?'BO2&' “ × “ > "+.&%$&&T *&_PT FG&, . 7"7;O5O"+$&ZQ?"%.&%$&&T* &_PTc.-1D"2QRSO2!3]"+$Z y"kJ;2&' P UdW7 P "+7"+"*"+2 l 7& H&DJ $]"kJ;2O= . 1%2&N)eS& FG. 7, I t> 7 "+1.42:4$)O$" ,-$&7&T]"+.+$&T*_PTg.-1D"2 P 5+"J;2 P 2N6M$&Z]5+*"+1D6^$9U-jGuFlO1O= 1

R

R

1

1

N

S

R

S

2

R

S

0.80

ρ

PN 0.60

PS 0.40

PR

0.20

0.00 0.000

0.005

0.010

0.015

β

0.020

L 0

β

∆ = 0.01 λ = 0.1

9

3

1.00

PN

0.80

0.60

ρ

2 0.40

1

4

3

0.20

0.00 −0.05

−0.03

−0.01

∆

0.01

0.03

0.05

: ∆

P Nβ = 0.02

λ = 0.1

&&Tg!2&+,-&+$ &+$)"%$&Z JK &+&SNOf$&&* ,!" 5.+$&T*&_PT .-1D"2 >-,-$&&7 _PT &SO2+$ B ;e7OU/{&U vg7JK)2;3]"%$a$&&"+*;O$:n7 "+3g'/.w&_P .-3]PJ"Ts!= H!7&!14 I Y"+"7O$&&7 *dO?J"7AWG)2;tU(oXSO2+$: B !1D_P, r!H!7 14 I ;O1_PN"%$&Zn&SO2KO+$&:6 JLO2;" VUw"H-!1D'Y7 .-p#"+ 4 2 &1,&"%= ? 6d!H!7 14 I h1JK"+:g &Tt>;*"+'Y;]7;O I "G&SO2+$"T A C U jGx7 +. ,-"?{&U {&{a&%$&7"; SF.!7 ,! I &O J;7'B'(E"+JK& 3g&T $&*, P 7 1D'B"+" R;7;O'B"+$&7 ∆ FG,-7 &N &' 1D;4*"+ β = U jGx!2&+, +$&R7;'B"%$&7& (β, ∆) 4 7OU^{&U Av 65@+$?&O$&&"%$&+$.&"%$Y "7"= 0.02 'B"+z" 6 )JKO2;:a2 ab ]U * !$&"7 N)2"f1D;4*"+T?7;O+$7&T, t> &+7; *"!= &'A$&&*, 'BE{ ~ +&+$4" 7;& &&"+ P )2;"%$&Zh.+$&T*_P'<.-1%2&'PU !t $&&*,-&T ~ JKA "7"+"* " 2 .&%$&&T* &_PTc.-1D"2[7"+&7;O#"+$Z .&%$&&T *&_PT FG&, .UWG7 "+7"+"*"+ l 2 l 7& H&DJ $s+. "+7,7!$&*"= , O?S&F. 7, I ^ JK7&N[ X&Fl KU *, P $&"+7- "+$r.&%$&&T *&&%$&:U X1 "%.&+$&T* &&aFG&, .Dee"e&,7"++$& &+$x7 935JL"%$&Zy.+$&T*_PTx 7"JK"2:O= _PT I ,!2>)7;9JK .d,-$&7&^.!"%2* N"%$&Z]7&7 I &;)2:X,-&76A 1d9J;= ,7!$&* &+$tU &$&"+$+$&O.!6M#TfFlO1D&_PT#&7!$&7"%$G&, O14Oz;u7 U {&U > k&U *q &7"+'B" &Te7")2 1D I x(t) +""+7;O$&7; *dO#J"7gWG)2;]7f&"%p#"+'?&1JK"T!= +$&n@+$f+&$&"+$+$&O.&"+$'/,-&'/.e&1,&" 60S" Tt?U * $&7;Oa"%14%= '(OE'BJLw,&O1D "7JK *"+,!H JK 35" T?;7+$)"%$s!2KO&hO$y .42;VUW7 &S&7O$&' JK 35" 5;7'B"+$7 . 7 )HNJK"G#&SO2+$: B 7& H&JK $gO2= &"fJL)2" "eO$&&,-O2"+SNO$"%2:& &T 'BJK_ x +$&"+'(h&"7 p#"+$h,-O2"S& N

N

N

3

3

N

0.04 0.03 0.02 0.01

Ω

0.00 −0.01 −0.02 −0.03 −0.04 −0.04

−0.03

−0.02

−0.01

0.00

∆

0.01

0.02

( Ω = hϕ(t)i 1.11) − (1.12 ˙ λ = 0.1 β = 0.01

0.03

0.04

∆

;f*;+$$&"z&"+pz"+&1JK"T +$& t>K* $&e&1D'B43g]$&O2:, g7ySO2:pz!Hw' = !2!$.DJL)Hw&"%p#"+Th+!2_U jGX7%.! , "G{&U { ~ +$&7 &";14O& 'B&+$: Ω = hϕ(t)i $G7;++$7&T,f*;9= ˙ +$$&" ∆ J;2;r +$&"+'B_ . ,-&7&*"+_qHr. 7;" T 4 {&U {&[{ 6 +4 {&U { ~ 6 7 λ = 0.1 NU !&7 1D&!$))2:&_PTy. *;O+$&,s&$&"+$+$.&"%$wSO2K%$&y+!H!7&!14 I t> β = 0.01 ,-& JL Ω = 0 #,-O2;"SN x(t) 7& H&J $l;X*;O+$&O$&"P&"%p#"+G&1JK"T +$& U 7#O2,e;z7FG,-"u14 $#$#;O7;'B"+$7; U W 'B"7"X"+z. &"2!= ω *"+ f&SO2KO+$&:G+!H!7&!14 I e7;%p#7N "+$&ZVPU * _PS&7; β_P" 1;4*" β = 0.01 λ = 0.1 &$&"+$+$. 6^$a&SO2+$h14)H!N)$) 4 SO2K%$&: A ;f7U{&U Av 6>;7y&_q= HNJK"l 1l,-$&7&Th1D ,!O6^$gS&" n*;+$$&T Ω U $7M F - B 24%$7/ %" "0E 2 "02@?&$"1 "%&2 8:/ "0/9H "7/ 5 #"&3-028:/-02 2" 3-N<>/@?"'

5 /

q"+7"'BZQ,<O2 _P' . 7;& " ' 4 {U 0A6e+""+7;O$&7;*qYJK"7QWO2;<7 * &"%p#"+'7'B*"+,-&'R&1JK"T +$& x¨ − λ − x2 x˙ + x = b sin ωt,

f 2"J."'? &&"J""M@+$&Tf%$&"'B_YG1D&+'B&%$&eO$l'B!2 $O.DJK_]*;%$&O= $&_ & "+p# " O7'B&* ",-&+ &1JK"T%$&&!R7FG,-+7&NO&' 1;4*" . 7;)2;6M#"+e;O7;'B"+$7;eO$&& &'B&+f+""+7;O$&7; λ U

&

Gj &'B 'P> * $hJ;2;C$&&'B &s+""+7;O$&7;4 7 b = 06 7;)1D'B"7 &+$: FlO1D&&l7+$&7+$N 7&;uJK. ' f"+ &&"J""M14 $l$&)2:&,- O$ 1;)= *"+Tx;7;'B"%$&7; λ UtW7 λ < 0 O$&&,-O2"+SN!r " "7;O$&7"]"5&O1DS.-35JL)= 6^$ZtU!jGGFlO1&&&T]!2;&,-&%$&'B"+"+$Ze"kJK%$&&"+;Oe&&S&O5$&&* ,!!> 7)2O= 35"+;Os 9*)2"5,-&&7JKO$9>t,-$&7;Os)2;"%$&Zs.&%$&&T *&&TV> s7 λ < −2 7"kJK+$)OO2;"%$l&S&T .&%$&&T *&_PT].-1D"%2 >7 −2 < λ < 0 X.&%$&&T* &_PT]FG&, .&OU B%2 0 < λ < 2 > +&+$9 "7&&"!x O2; "+$&Z "+.+$&T*_P' FG&, .&'PU B%2 >$x "+.&%$&&T *&_P' .-1%2&'PU(W7 "+7"%H&DJ" *"7"%1nO2:s7 &%= HNJK!$gλ%. >"72, 7 $& *"+,!)yS&F. 7, I ! u JK7 &&N% X& λF !> 57"%1.42:4$)O$&"l,-O= $&7&Tn 1l"+.+$&T*&FG&, .D57435JL"+$Zw.+$&T*_PTh7"kJK"%2;:&_PT I ,!2U W7r'()2_qHh1D;4*"+ !Hr. 7O2; 6^#"+n;7;O'B"+$&7a+""+7;O$&7sJ"'B&%$&7 7 .N= "+$?,NO17'B& *"+,"s,-O2"+SN!t>(,-$&&7 _P"yh7+$&' λ 7"&7O56M$&Z< 7"2K,-D I &_P"OU )1D'B"7 &+$:GF )1D&&X7 &+$7;%$&N"$&&'B&+"+"7;)$&&7;) *dOlJK"7 WO2;a7;O&;#$&7"+'P>; 1%JK"+:f&'B'B]"+JK& 3g_qHa$&*",na7"J"%2:& _qH I ,!= 2& 'B435"%$ +.-z"+$&&N)$&:JK. 'B"7 _PTr$&7t3U !t pz"'R. 7;&"+"5"$&&' = &+]" "7;O$&7; 4 {&U 0A6 5&-JK"%$&"+'B_ $&7"HeJKFGFG"7"+ I ;92:&_qHa. 7;" T "+7&&+g7--JK, x˙ = y, y˙ = λ − x2 y − x + b sin(z), z˙ = ω.

4 {U | ~ 6

JK"+: x, y, z aJK ;'B*"+, "e"+7"'B"+_P">,-$&7_P"er&7 "JK"2;6M$y7;O1'B"7!= &%$&:?FlO1&&&+ 7 &+$7;%$&N U* p# 7&,-&T&S)2K+$Y1D9* "TR7;'B"%$&7& JK'B, nJL &T? +$&"+'B_ O2;"%$&ZEJ&&O2:h243eT??7;O1&&S7;O1D &TtU w_ &+7;* 'BZ +%2"kJK&N "'07"+3g'B&w H 7 & 1D I !HES&F. 7, )= I TEw,7"%$&+$& $),E;O1_PN"'Bh&&a 1_P, w !H!7& 1D I V> ,-O= JLG*;%$&$)G H 7 &_qHf,-O2"+SN Tf+&&;4JL"+$lP*;O+$&O$&&Tf&"%p#"+l&1%J"T!= +$& V> 7*"+'QSO.4J"'<7;+'(O$7NO$:n$O2:&,-a,N)1D7 'B&*"+, "g7"%3e 'B_ $&&'B &f"+"7;)$&&7;!>K,-& JKf;O7;'B"+$7 λ 'B"+"+$g"SO2:pz"1D;4*"+ tU jGs7+. ,-"x{&U {4|s;s!2&+, +$&'B!2 $O.DJL *;O+$&O$)E&"%p#"+E&1%J"T!= +$& g&%$&7&"+_Y2 eS& FG. 7, I &_qHg1D9* "Tg H 7 &_qH]JK& 3#"T 7 FG,-7 &N &' 1;4*" c;7'B"+$7; U *`&SO2KO+$4!H B +$&"+'( JK"+'B&+$77 ."+$s7 "+3g'B_ +!H!7&!14 I t>aλa&S)2K+$ C y7 A"+3g'Q , NO1D "%= 7JK *"+,!He, )2"SNOTtU 2"kJ.&"%$]$&'B"%$& $:> * $zl$D2* "u$l. ,-&7&*"+_qH . 7;& "TgJ2;O'B!2 $O.4J_ gFlO1D_ %$&"+'B" 4 {&U | ~ 6 .&%$&&T *&_P' "%.&%$&&T!= *_P'E+!H!7& _P'sJK 35" 'r$&"*;O6^$.&%$&&T *&_P"M#"%.&+$&T* &_P"^ 7"%= JK"2:&_P" I ,!2_> G,NO1"+7JK*"+, 'C,-O2;"SN '?+&$&"+$+$&O.&"+$zJ. 'B"7!= _PTn@7+JK*"+,Tn$&&7a#$7"%H!'B"7 &'FlO1&&&' 7 &+$7;%$&&"l%$&"+'B_ULW7 !

0.05

0.04

a

l3’

l3

f

b

B 0.03

l2

b 0.02

d

C

0.01

l1’

A

c

l1

C

e

0.00 0.950

0.975

1.000

1.025

ω

) K

λ = 0.1

1.050

b

ω

1D9* " Hf;7;O'B"+$&7 &G 1 &SO2KO+$& A FlO1&&&'r7+$&7+$&"M%$&"'B_Y'B"%= "+$Z<$&7<7"kJK"%2;:&_qH I , 2KPX .+$&T*_PT C >q"kJ;2&&T C SO26M$& N

S

1.0

0.5

CN

CS CR

y

0.0

−0.5

−1.0 −1.0

−0.5

& 0

( b = 0.02, ω =31.01,

λ = 0.1 C

0.0

x

N

0.5

1.0

x, y (

C

S

K G #

CR

&

"%.&+$&T* &_PT C U W7", I n 7"JK"2:&_qH I ,!2;&#;#!2;&,-&%$&:4 x, y 6(&, )= 14O_ ;57OU {&U { U "%3e '/.g+!H!7& 14 I $&"*;"%$z.+$&T*&_PTa7 "JK"2:&_PT I ,!2 U!o r"kJ;2&&Ts7"J"%2:& _PT I , 2 C 2"%3]O$a; &&"7&H!&%$&sJK. 'B"7 &C $&7; > ,-$&7_PT<&S&7;O1&NA14'B_P, "' "+.+$&T*&_qH 'B &&&S&7;O1T+"J;2&& I , 2K C ;.&+$&T* &_PT I , 2 C U O'BO$&7' SF.!7 ,! I e@%$&!Hf 7"JK"2:O= _qH I ,!2&n7yJK& 3#"xn!2&+,-&+$E. 7;)2;6M#!HE;7'B"+$7& (b, ω) nFl , +7&&&' λ = 0.1 U jGX7%.! , "G{&U [{ 0u+$&7 &";S& FG. 7, I &;)zJL7'B'(XX14'B+$&f$ *;%$&$_\&"%p#"+e1%JK"+T+$& ω 7hFG,- 7&N &Tw'B!2 $O.DJK" b = 0.02 > * $5&$&"+$+$.&"%$g "7"+'B"+#"+6\)JO2: 2 cd ;l7UL{&U {4| U jG &+e&7JK!= ;O$XS&F. 7, I & &TzJK;+7;'B'B_ 4 7 U {&U %{ 0A6$D2;93#"_C'(,-'(92:&_P"d1;)= *"+ EJK ;'B*"+,-&T "+7"'B"+&T +&$&"+$&%$&. 6M#!H 7"JK"2:&_qH I ,!= 2& C , C , C U W7r 1D'B"+" *;Ox(t) +$&O$&_ & "+p# "n&1JK"T +$& ω 7r"%= 7"+"* "Y7; I SO2K%$&C!H!7 14 I l l "+.+$&T*&_PTR 7"JK"2:O= _PT I ,!2 C , ,!HhS&F. 7, I Ts"z7 "+$&"+7"+N"+$9UtWG"+7"%H&J?O$ !H!7&!= _qHl,-O2"+SN T ,l,NO1"+7JK*"+, 'x+ 1DG(S&F. 7, I 'BGJK.NHXJK7!.!+!H 7"kJK"%2;:&_qH I ,!2& U X1GJK;+7;'B'B_\& J>* $&e . &"%2;*" "'<!2w.!'B"+:O= p#"+"+' ω w7SO2; 35" "' ,S&F. 7, I &_P' 2!' l !2 l .&%$&&T!= *_PT I ,!2 C +SO2 35"+$Zr#"kJ;2&&_P' C UtjGe+7; I "5&y2NO6^$&Zx &+&S&_PTC"J2O=m.-1%2&&&T I , 2 4 JK A 1n'/.42:4$& !2, O$&7& I ,!2Kr7 '()= "+$f1D9* "" #{[6%>;n7w"7"HNJK" *"7"%1#S& FG. 7, I & . 6[26[+*"+1D6^$9U R

N

S

S

N

N

S

R

1

0 1

R

1

N

0 1

S

0.80

CN 0.60

xmax

CS

0.40

CR 0.20

0.00 0.980

0.990

1.000

ω

1.010

1.020

" ) (

" CN , CS , CR Z

ω

b = 0.02 & ) K

& λ x =0.1

9

Z # ;)A+"J;2;O=m.-1+2;&NOAS&F. 7, I A%2; !YC* "+1D &&" 7 "JK"2:&_qH I , 2& C C 7& HNJK $z;l &&"7&H!&%$&5JK. 'B"7 &l$&&7;!U *q" &SO2KO+$& !H!7& 1D I aFlO1&NO]$&7", $&7 &,7_P&"+$5 &&"7&H!&%$&:#$&7;z JK"X" 14O'B_P,!): >N* $&G$&&"%$&%$&."+$5,&O1D "7JK *"+,' ,-O2"+SN 'PUNW7&"+, I FlO1D&&e&7 $7"+$)gJK. 'B"+7&+e$&&7&, O14O;e;e7 U{&U %{ $ ULWG7 y"7 ""+*"%= w7;O I l l , NO1D "7 DJ*"+,"#,-O2;"SN h1D ,!O6^$f35"%$&,'Y&SO= 7;O1&'P>N'B 2 $.DJL'BJ.42; I 4 $&O2;z; d$&&7X;XFlO1D&&'r&7 $7"+$%" 6/7;)1. 35"]14 $&&* , T?S&F. 7, I 7 '("+$h, "*!.!6 JK&%$)O$&&* sS&O2:Opz. 6 "%= 2* .&U !JK"+: &'B'Bx7 $4 NO6^#"+?$&7; T EFlO1&&&' 7&%$&7; +$&" +$&"+'B_ 'B"+"+$&Zn"%#"G"+.+$&T*_PTh7"kJK"%2;:&_PT I ,!2 C U +'B$7" _P" "+7"%$&7&T ,wFlO1&&&+z7 $&7"%$)5%$&"'B_Q; +7; I "&SO2KO+$&"+T 7"%= JK"2;6M$w, ,rS& FG. 7, I &_PTx'B"%HN 1' !H!7& 1D I x*"+7"+1]A149H &CO$9UoXSO= 2K%$&: A 1D_P, f!H!7 14 I hO1D_PNO6^$g&SO2K%$&:&6\1D)H!NO$) U jGE7 +. ,-"?{&U {9vs&+$7&"+; S&F. 7, I &;O JK;+7;'B'(rJ2;A.+$&T*!= &&+ C >"+.+$&T*& C +"J;2&& C 7"kJK"%2;:&_qH I ,!2&s++$"'B_ 4 {U | ~ 65x14'B+$&Y$?O'B!2 $O.4J_`&"+pz"+ &1JK"T%$&&! 7AFG,-!= 7&&_qHC1D9* " H ω = 1.01, λ = 0.1 >B* $?$&&"%$&%$&."+$Ab "7"+'B"+z"6 )JK)2:y2 ef ;s!2,-&+$Y;O7;'B"+$7&E'B!2!$.DJL[ ?*;%$&$)x&"+pz"+ &1JK"T +$& h7 U{&U {4| U W7r'()2_qHxO'B!2 $O.4JK)Hx&"%p#"+h1%JK"+T+$& b !35"#2 l > &'B!= 'Bl7 $4 NO6^#"+#$&7; >-+.-z"+$.&"%$f"+.+$&T*&_PT 7"JK"2:&_PT I ,!2 C > 7JKOp#TZy 1z+&+$4 w7;&&"+ wg9*)2"l, &7JK;O$9U 7+$&' b ' = !2!$.DJL @%$&&+?"+.+$&T*& I ,!2K 7;O+$&"%$9UdW7A"7"+"+*" Y2 l N

S

0 1

1

R

N

R

S

1

R

1

1.0

T 0.5

CR y

0.0

−0.5

−1.0 −1.0

−0.5

0.0

x

0.5

1.0

) @ C : ( ) T

R λ = 0.1, b = 0.02, ω = 1.017

C

9

CN

S

&

0.80

CN 0.60

CS xmax

0.40

CR

0.20

0.00 0.000

0.010

0.020

b

7

b

0.030

0.040

0 7

Z :

ω = 1.01, λ = 0.1

'R *"+ "z 7& H&DJ $9> g$&7"+'/y, ,y;f7 $4 &6^z"'<J. 'B"7' $&7"#7"%1.42:4$)O$&"G+"J;2)=m.-1+2&&T S&F. 7, I n7 935JL"%$&Z 7;z7 "JK"2:&_qH r.+$&T*&_PT ?"kJ;2&&T U/W7xJL92:&"+T p#"+'0. &"2*"+ I , 2& "kJ;2&&T I , 2 C SO2C!3]"+$Zw"+.+$&CT*&_P' I ,!2;&' C U W7Y "7"+"%b= *"+E2 7 & HNJK!$s"kJ;2O=m7""22"+7;O S&F. 7, I V>@%$&xJN I ,!= 2Ky2N6M$&Z l *"%146M$9U *"%1D&"" +"J;2;&&&+ I ,!2K C 7 &JK $s, 7;O17 .-p#"+6 7 "+1D;+&f$&&7 U *RF )1D&&'A7 &+$7;%$&&" &+$)"%$&Zw$&O2:, .&%$&&T *&_PT 7"kJK"%2:_PT I ,!2 C >q$&"*6^zT<7"+3g'/.C H 7 & 1D I tU oXSO2K%$&:?+!H!7& 14 I B >(7;+O2435"+ . 6 &_dpz"n2 l ;E7 UM{&U {D| > ;O1_PN6M$e&S)2K+$:&6[JLO2;" VNU * $D2;*"l$g&SO2K%$&a14)H!N)$) A 7w&_q= HNJK"z 15&SO2+$yJLO2;" yz 1'B""+"' *;%$&$_ "+pz"+ &1%J"T%$&& ,N)1D"+7JK* ", "d,-O2"+SN &1, 6M$u'/ ,'h&S&7;O1&'P>4'B 2 $.DJLM'BO= J.42; I C;7;%$)"+$x+$&"+" $r!.92t>/* $r&S.&2&O2;"sH-7, $&"+7&' S&!= F. 7, I Tw7"J"%2:& & I ,!2K C U jGX7%.! , "G{&U { 7 "JK%$)O2"+; S& FG. 7, I &;)zJK;+7;'B'(X7"kJK"%2;:&&+ I , 2K C 7 1D'B"+" n*+$$&_ &"%p#"+]&1JK"T%$&&! ω FG,-7N!= _qH 1;4*"!!H b = 0.04, λ = 0.1 >/* $&x$&&"%$&%$&."+$ "+7"'B"%#" 6b)JO2: 2 ab y!2,-&+$ ;7;'B"%$&7s'B!2!$.DJL[ *+$$)s&"+pz"+s&1JK"T!= +$& r;g7U{&U {4| +U *c $"7N92"51D;4*"+Tt>t7; *" _qHh2 'B l l > 7"kJK"%2;:&_PT I ,!2 O2;"%$&ZC.&%$&&T* &_P5 ' 4 +!2pz;O 2!YS& FG. 7, I &!= &TJK;O7;'B'B_b;C7 U{&U { 6U l2; 7"kJK"%2:& I ,!2CJK 1E'/.42:4$&!= !2 ,!)$&&7E&+" JLs7;O&" "J I _U zNy+$)pz!HNZA'/.42:4$& 2, O$&7;E 'BJ.426 'B":pz"u"kJK I _R @%$&&'? $&"+7N)2"M1D;4*"+TtU!W7f"7"+"+*" S&!= N

S

S

R

2

S

N

2

N

N

3

0 3

1.00

CN

0.80

0.60

xmax 0.40

0.20

0.00 0.955

0.975

0.995

1.015

ω

1.035

: &

& 3 Z

:

ω b= 0.04, λ = 0.1

C N

F. 7, I & _qHh2 T l l JK&g'/.42:4$& 2, O$&7;O2; 6^$Zy,-&'B!2"+,-O= 7N 35"+_P'B [ 'BJ.426 %$)&O $&Z S&O2;:p#" "J I _U W7"J"%2:& _PT I , 2Y$"7N"%$E.+$&T*&+$&:E 1n"+x7 935JL"%$&ZC7!$4 N6^zT J. 'B"7!= _PT$&7t> 7&"+, I e, O$&&7#; !2&+, +$&: (x, y) 7"JK%$)O2"+;#;z7 UK{&U {%2 U O2;z; /$&&7; !2KO& .!"%2* N"%$&Z#(7+$&'h;9J,7 $*&%$&tU /, OX"%= 0 3

3

1.0

0.5

y

T

0.0

−0.5

−1.0 −1.0

−0.5

0.0

0.5

x

1.0

& & @ λ = 0.1, b = 0.04, ω = 1.0357

K

l3

&

0.06

0.04

0.02

Ω−ω

0.00

−0.02

−0.04

−0.06 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05

ω

7 Ω − ω K 7 ( ω Z λ = 0.1

b = 0.02

7"++$&7 &T, FlO1&&&+X$7"+$)l;O1D_P&"+$ZgSF.!7 ,! I "Tgj"T'(O7, [ ,-"7G &;]&7 "JK"2;"%$e'B"H-!1D'+!H!7&!14 I y*"7 "+1 JLO2"+"4U * &SO2K%$&g!H!7 14 I " "7;O$&7+&&"7!p5"%$#,-O2"+SN!e;l*;%$&$" &"%p#"+5&O1%JK"+T+$& tUoMJK;,-Bz $D2* "X$z%2;. *;Og%2S&&+z&"%p#"+5&O1+= JK"+T+$& t>1JK"+:g+&S&%$&&"+;OnO$&&,-O2"+SNO$"%2:&Oa'BJL#"JK%$&7; N"+$Zh, 7 $'/.Y"+pz"+?&1JK"T%$&&!t>drO2"+$Z)2&%$&:&6 JLO2"+&TA'PU $&)2:&,- . &"2*"+"+' 7%$&7&T ,re*;O+$&O$&">L7y "7"HNJK"57 I l l &SO2+$h !H!7& 1D I Z 4 7OU{&U {D8| 6q&1S.-3]JLO"+$&ZwO$&&,-O2"+SNO$"%2:&Oh'BJL r;4*;O"+$n!2K ;7;O+$)O$:UWG7 s"S&)2:p#Tx9JK,7!$&* &+$? "+, $&7 " ,N)1D"+7JK* ", !Ha,-O2"+SNTe JK. '/e"+1D&+'B_P'B *;%$&$)'B 7JKO2= 3]6M$X7 "N)2;7&N)$&:u+",!$&7;)2;:&_P"P+$)O26^z"d+&$&"+$&%$&. 6M#" &"%p = "+Tg+!2"U WG@+$&Tg 7* "M$),-&Tf7"%3e'E%2KOS&G'BDJ;.92;7&NO_qHf,-O2"+SNT *;%$&f$), 35"G$&Z $g,n7"+3g'/. +!H!7&!14 I tU jGl7 +. ,-"z{U ~f &%$&7"; 1D&+'B&%$&:z7O1D+$&e'B"+35J.]7"kJK"+T*;%$&O= $&T $&, )2"SNOT??*+$$&&T "+pz"+x1%JK"+T+$& ++$"'B_ 4 {&U | ~ 67 U !7 1D $))2:& _PT . *;O+$&,A+&$&"+$&%$&."+$ &SO2K%$& +!= λ = 0.1 b = 0.02 H!7&!14 I t>q,-& JL?+""+7;O$&7R&"7 p#"+$ ,-O2"+SN!;x*;+$$&"s"+pz"+ @ : ) ω # 9 L K 0 ( l l # " L β,

G

7

@

0 ( ) ) @ . 3

3

3

0 3

0 3

& 1JK"T +$& ω U! 7y&7!1D& $)92:&&+y. *;+$, s1D&+ $s$s'B!2 $O.DJK_ &"%p#"+x&1JK"T%$&&! b U #&"g1D;4*"+" b = 0.02 7 λ = 0.1 $&&"%$O= +$.&"%$ &SO2+$n1D)H!NO$) U ;E 9/

E

* JK&Tf 2K"u7;+'B$7" zO2;""M !H!7& 1D I "7JK *"+,!HO= &$ , )2"SNOTa&"%p#"+TnO7'B&* ",-&Tn 2&T ;z 7'B"7 ""DO$&&'B&5+"%= "+7;O$&7; *qYJK"7QWO2;tUdWG7 "JK%$)O2"+Y& D "r&&&TR&S)2K+$Q+!= H!7&!14 I s;e!2,-&+$y. 7;O26^z!Hs7;'B"%$&7&'B!2 $O.DJL a*;O+$&O$) &"%p#"+y1%JK"+T+$& V>+O2:O1. FlO1D&&"#. 7;&"+"O>. ,-&7&* "_P"g.!7O= "+ hJ;2;s'B!2 $O.DJK_ sFlO1_0s O2&"z"DO$&&'B&"#. 7;& "" *qwJK"7 WO2;VU l2;A%2;. *;OY'()2&+ ;O7;'B"+$7;?&1S.-3]J" Y$&&'B &?+""+7;O$&7; > '()2Ts'B!2!$.DJK_[&"%p#"+Ts!2_\h*;+$$&_[& "+p# "Ts 2_\SO2 1,-&Ts,h+&SO= +$&" &T#*;O+$&O$&"d+""+7;O$&7; >)%$&"'/. 'B43e ^7;O'()$&7&O$&:, , %2KS^&O1+= '/.-#"+_PTa7'B& *"+,Ta& I !22;!$&&7VU l2; "z'B"+$&JK' .7"kJK"+ *d JK"+7yWO2;a)2;. *"_Q.!,-&7 &*" _P" . 7;"! J2;w'B!2 $O.DJK_ nF )1D_U 5O1D&&"].!7&"+"e 1D&)2;"%$h& DO$:h&S)2K+$:w1D)H!NO$) U * JK'B"7' FlO1D&&'r7&%$&7;O+$&"PJ;'B* ",-&Tf +$&"+'B_ !H!7&_P'sJK& 3#" 'r$O= &"+*;6^$5.+$&T*&Oa"%.&+$&*&Oa" DJ& 3g_P"X$&&* ,tUW7 _qHNJK" 1u&SO= 2K%$&< !H!7& 1D I <7 & HNJK!$A HY2 "yR+*"+1&&"+"O>d&1, "+$ JK7"+TF FlO1D_U 5O1D&&"y& D "wJK& .+$'BA$O2:&,- 7<&*"+: %2S&&' &"%p#"+' &O1+= JK"+T+$&t>L, JLe&1'/.-#" "'<'B 2 $.DJK_0O$&&'B&g"+"7;)$&&7;g'B93g 7"+"S7"*: 1U \u*"+$z'B!2!$.DJK&+ . 7;"!e7 &JK $ ,gSO2""M%2;93g_P'ES&!= F. 7, I & _P'0 O2"+ '[;w!2&+, +$& . 7;)2;6M#!H?;7;'B"%$&7U X1D_P, !H!7& 1D I gJK"+7-3g $l&SO2+$:G1D)H!NO$)]&SO2KO+$&: DJKO2"+ tU !H!7&!= _P'hJK&!35"!'r$&"*;6M$G" DJ& 3g_P"d$&&* ,5;XFlO1D&&T#!2;&,-&%$&t>&, N)= 1D "7 DJ*"+,' ,-O2;"SN ' C7 "JK"2:&_PT I ,!2 UdW7&_qH&JK"s 1s&S)2K)= +$14)H!N)$)Ew. &"2*"+"+' 7%$&7&T ,R *+$$&"w7& H&JK $ S&F. 7, )= I 7435JK" Q.+$&T*&A7"J"%2:& & I ,!2KC7 * "+1D &&" %2;93]= &Ts&S&Ty$&&*, x+"J;2;O=m.-1D"2UW7s"+7"%H&DJ"f!1]&SO2K%$&y1D)H!NO$) &SO2KO+$&: JL)2"!?].!"%2* ""+'0'B!2 $O.DJK_ &"%p#"+y&O1%JK"+T+$& 7 & HNJK!$ "kJ;2O=m7" "%22"+7;O?S&F. 7, I X+"J;2;h "+.&%$&&T *&_PT?.-1D"2 %2;N6M$&Z? +*"+1D6^$9(U *q_qH&JE 15&SO2+$y14)H!NO$) l.!"%2* ""+'c7;O+$7&T, x *;%$&O= $&" +&7&43]JLO"+$&Z?+. "7 ,7 $*"+,-&TS&F. 7, I "T u-JK7&N XFlP X 7 HNJK $Y743]J""s.+$&T*&C7"J"%2:& & I ,!2C 1r "+.&%$&&T *&&+ FG&, .D U

u)2 1 O2 &M"DO$&& &'B&+^. 7;& " *dXJK"+7#WO2;l7l"+S&O2:pz!H 1D9* " Hw;7;'B"%$&7;g&1DSO. 35JK"+ n7DJ $,w;)2;&* _P'R7"+1+.42:4$)O$)' s.&%$&7&T +$&"w&+&& &T &SO2K%$& !H!7& 1D I A;y!2&,-&%$& .! 7;O2; 6M= # H ;7'B"+$7&U WG7 @+$&'C2"J;.&"+$]. * $_PNO$:>;* $&57O1D'B"+7&%$&:]FlO1D&& 7+$&7+$Ny"$&&'B&+y& I !22;!$&&7h7;&a$7"'PU !H!7& _P' J&!= 35"+ ' $&&"+*;6M$7"kJK"%2;:&_P" I ,!2_U^G&O1D "7JK *"+,' ,-O2"S& ' E149H &7O!$)$45, NNO16^z"7 T JJK. *'B"+"7,_P" TC,-O@27 "S&JK * "+, 1DTC $,!&O7t6^UB$5W357"+ +$&&, _qH&'YJK&"aS7; O1a1D&'CSO2KzO+7$&"% = 1.42:4$)O$"h+"J;2)=m.-1+2&&TCS&F. 7, I A7"J"%2:& _qH I ,!2&r;s7"%1D& &' $&7"U)W7G"7 "%HNJK" 1 &SO2KO+$&G149H &O$)M &S)2K+$:^ DJKO2"+ l/. &"%2 *" "' 'B!2; $.DJK_ &"+pz"+r&1%J"T%$&& 7& H&DJ $x"J2O= 7 ""22"7 ;O S&F. 7!= , I 7"kJK"%2;:&_qH I ,!2>9* $&^7&JK!$X,l7;O1D7!. pz" 67"+1&; &+u$&7; U W7&_qH&JK"X 1u&SO2K%$&e1D)H!NO$)# . &"%2 *" "'C7;%$&7T, #*+$$&"X, N)= 1D "7 DJ*"+,"X,-O2;"SN ]&1, 6^$ '/+,'x&S&7;)1D&' X& 7"JK"2:&_PT I ,!2 $&"+7N"+$z.&+$&T* &&+$:5g!1u"+ 743]JLO"+$&Zg7!$4 N6^zTgJK. 'B"+7_PTe$&7tU

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@

;E ""F #" "02@'6* $%"(,.-0/ "%$"1324%$('61329% F.H ?&#"%"0E 28 / B /@? 92 %" $ =5 2-0HJ/" $7D%&* B / "%&% "% N* $"?0/ $ $

,

O'BO$&7' &_P .-35JK"!.!6`+!H!7& 14 I 6 ;y& &&&' $&"O>/7"kJKO= 2K)a7"+3g'C$&,-O2"S&T ,&O1DO7'B&* ", 'PUKWG7 &+$"T pz"Tn'(O$&"+'(O$&!= *"+,-&T'BJK"2:&6YJ;2;g2"J&N!]JLO&+ &7&l'B435"%$ 2;.-3e!$&:z2"J;.-= 6^z""l+$4HN+$*"+,-&" JKFGFG"7 " I )2:&"l. 7;" "LX √ 4 ~ U [{ 6 x¨ + x = (λ − x )x˙ + b cos(ω t + ϕ ) + 2Dn(t), JK" ξ(t) = √2Dn(t) gD.&+&fS&"%2;_PTap . 'Y&%$&4 &Ta $"+&&%$&:&6 D > > ω x$&&"%$&%$&&"+>('B 2 $.DJL!> ;4*;)2;:&;O F )14w S&"+17;O1D'B"+7;O b ϕ *;%$&$)f&"%p#"+e1%JK"+T+$& VU * $))s&"%p#6^6 2;.hw $"+&&%$&: pz. '(e'(92_P'Byw. *!$&_PNOyO= 2&+&&_P" &T+$Nl$&" "7;O$&7; >N 7!HNJK '?,]'BDJ"%2g$&,-O2"S&TfG&!= JK"f. 1,-&O2&+y+$DH-%$&*"+,-&+E7 I "+D >;O1_PN"+'B&w7'B& *"+,' pz. 'B&'PU ;4*"u+&&&7NV> 'B_Y7;+'(O$&7 N"+' , )2"SNO O$&&+""+7;O$&7;zSO2; 1+= ,'Bn,n7'B& *"+,'Ra #"+'7"%p#" , " 4 ~ U %{ 6qf JK" 2

0 1

1

0 1

1

x(t) = ρ(t) cos (ω1 t + ϕ(t)), x(t) ˙ = −ρ(t)ω1 sin (ω1 t + ϕ(t)),

4 ~ U~ 6

JK"f'B&"OE'B!2 $O.DJL ρ(t) E2;. *;TOx, 'B&"+ $)hFlO1D_ ϕ(t) 7"J= O26^$Ze'B"kJ;2"+l'B" 6^# 'BZeF. , I 'BfG7;O&" 6<M "7JK' UWGJK%$)O2;! &_P7;)35"! 4 ~ U ~ 6G:4 ~ U {[6 7&!1D&J ,-O2"+SNT .&+7"J" "xT1D ="2π/ω 7 J T > O2;. *;"+' %2"kJ. 6^z. 6 +$&"+'/.R+$DH-%$&*"+, H . 7;& "T J;2;w'B!2; $.DJK_ nFlO1D_Q,-O2"S&T 1

1

1

√

Z t+T1 1 n(θ) sin (ω1 θ + ϕ)dθ, ρ˙ = ρ − 2D 2π t √ Z t+T1 β 2D 1 0 ϕ˙ = ∆ − cos(ϕ − ϕ1 ) − n(θ) cos (ω1 θ + ϕ)dθ. ρ ρ 2π t ∆ = (1−ω12 )/2ω1 ∼ = 1−ω1 β = b/2ω1 n(t)

λ ρ2 − 2 8

4 ~ U |86

WG7;'B"%$&7 H-7;O, $&"+7 1."+$*+$$& . 6C7;++$&7 &T, .> ;7'B"+$7 'B!2!$.DJ.n&1JK"T%$&&!tUKW7& 1&"Jn7 "&S&7O1D&&" %2"'B_qHL>+JK"7N3]Oz!He+$&*, p .!'( > 'B93g O2;. * $:5%2"kJ. 6^z" 7 SO2 35"+_P" &_P7;O35"+ X √ Z t+T1 1 2D 1 ξ1 (t, ϕ) = n(θ) sin (ω1 θ + ϕ)dθ = − − √ n1 (t), 2π t 4ρ 2 Z t+T1 1 1 n(θ) cos (ω1 θ + ϕ)dθ = − √ n2 (t), ξ2 (t, ϕ) = 2π t 2

4 ~ U 6

K

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2

1,2

ρ˙ =

λ ρ2 − 2 8

i

j

ϕ01 )

i,j

i,j

√ D D + n1 (t), + 2 2ρω1 ω1

ρ − β sin(ϕ − √ D β 0 n2 (t). ϕ˙ = ∆ − cos(ϕ − ϕ1 ) + ρ ω1 ρ

4 ~ U 0A6

*,-qa_P;S&7g;&4%*;$&492:&!.!&Tf6 FlO+1D$)_CO&21JK6^"T +. $&6 "+7ϕ"'B"+" &7 T I ;)2"+tUoX]O2!"+$G$O2:O= Uc * JK)2:&"+T p#"+'c)243e' ϕ > & $ & L J s + " 7 " B ' + " ; O ; 7 ; y ; 7 O 1 & + $ A l F O 1 " "7;O$&7;r &"+pz"+ ϕ = 0 ϕ &1JK"T +$& tU * ~ v. $&'B"+*;"%$&Zt>(* $r;)2;&* ;O ++$"'(s.!7&"+T 'BO= 35"%$#S&_d$:#O2;. *"+;#" $&O2;:&,-#7O7'B&* ",-&' &O1%JK"+T+$&t>!#e %2;.N= 0 1

0 1

*;"O>;,-& JL5&"%p#"+" &O1%JK"+T+$&"GO2; "+$Zw "7JK *"+, ThF. , I "+Ta2;6uS&&T FG&7'B_>;;O7'B"+7h&2"J&NO$"%2:& &+$:&6 'B!.92;:&& U B%2h&_PO2!6^$Zy.&2&&! b λ, D λ >;$&eO$&,!2"!r'B!2!$.DJK_ $w""]"+&1D'/.-z"h1;4*"! a = 2√λ "+&"%2 ,tUW7""+S&7"+O 'B 2!= $.DJK _P'ByFX2;. , $.& I 'Bt>L O2;. *;"+'A. 7;& ""lFlO1D&&TeJK;O'B,ng&-JK" p 4 ~ U $86 ϕ˙ = ∆ − ∆ cos ϕ + 2B n (t), 2

0

ϕ 2

c

JK" ∆ = √β \p#7 ;\O2&+_ !H!7 14 I $&%.-$&+$& pz. '( > 2 λ B = D Q $"+&+$&: FlO1&&&+Ypz. '( ZU \X7;O&" " 4 ~ U $86w!= _P&"+$ JK8ωFGFλ. 1& _PT 7 I "O> , O$&&7_PTc'B93g 7"kJK+$)OO2; $:R, ,QS7O= . &&+,-&" J& 35"+" *;%$& I _ aa, &7JK;O$&T ϕ xDJ&'B"7 &' ,!2& "!= &' "7 DJ*"+,-&' $&"+ I ;92" U (ϕ) = −∆ · ϕ + ∆ sin ϕ 4 'PU(7 U ~ U {%6%U W7R$+.-$&%$&&q$), − arccos(∆/∆ ) + 2πk, k = 0, 1, 2, . . . , ,'B+&&"+;O 7O1D+$&:zF )1$&,-O2"+SNT e&"+pz"T +!2_<&%$)"+$ZaO= +$4Th&]&7"+'B"VULjG)2;*"pz. '(f7&JK!$e,JKFGF.-1Dn7;O1&+$hFlO1 z $&"+ I ;)2;" U (ϕ) X ϕ(t) F2.!,!$. 7 ."+$f&SO2!1D'B '/. 'B&#$" I ;)2K ϕ n&&"+7 p5O"+$2;. *;T _P" "+7"%H&JK_\ 1lJK&Tn$" I )2:&Tw'B_Q5J7 . +. 6z> , 4*,-&''B"+ :f; 2π U jG 7U ~ U ~ &, O14O_ 7")2 1D I E7;O1D &+$?FlO1#J;2;x7;O1%2*_qHx1;4*"%= Tn $"+&+$&wpz. '(fV>;&$&"+$+$&" >L7O1+2 *_qH B > O2;. *" _P"z c

ϕ

2 1

c

min k

c

c

min k

ϕ

@

1.5

U(ϕ)

0.5

−0.5

−1.5

−4 π

−2π

&

2π

0

U (ϕ)

ϕ

4π

∆ 6= 0

7 "%1.42:4$)O$&" *2" &f $"7 7&NO \ 4 ~ U $86%UlO, & JK] 1G7 +. , > 7s'(92_qHy $&"+&+$4 Hspz. '(-4 6P'B &&" ;Os7;)1D&%$&: FlO1z $&"+*" "aJ;2; $&"2:&&+E7"'B"+A+$)"%B$&ZA=SO0.02 2 1, TA, ;4*;)2;:&&'/.?1;4*" 6zU \X"%2* ""n $"+&+$&Cpz. '(s7DJ $x, . 'B"+:p#"+6 J;2 $"%2:&+$ &7"+'B"E7"S_PN ϕ aDJ&Tx 1e $&"+ I ;)2;:&_qHx' E%$)&O $&Z & JK _ "+7"+, ,5'B&" &T#7;O1&%$&#FlO1 'B"+35J.G7;O1%2* _P'B#'B"+$)%$)S&!2;:&_P'B +$&4 'B 4 B = 0.056%U \u*;%$&,A7SO2; 1D $"%2:&E&%$&4%$&NE'BO= ϕ

ϕ

400

ϕ 300 Bϕ=0.1 200

100

Bϕ=0.05

0 Bϕ=0.02 −100

0

2000

4000

6000

t 10000 9

3 ∆ = 0.06 ∆ = 0.15

c

8000

K

& "+&TE7;O1D &+$?FlO1 ϕ ∼= const "+z"5J&+$)O$&*nHN&7Op#h1D'B"+$_>JK,- 7 "JK"+"X1D;4*"+" ϕ &1D7+$)"%$5& &7"'B"+tU G"+' S&O2:Op#"u,!2& 4 7;O+$7&T!= , 86%>)*"+'y'B"%2:*" 'B_ 4 *"'y'B"+:pz"P'B!2!$.DJLu+!H!7&!1D7 . 6M#"+ +)2K86 n*"'AS&O2:pz"G $&"+ &&%$&:pz. '( >$"''B"+:pz"l%. 'B'(7&"l&7"+'/t>;5$&"+*"%= "w,-$&&7 &?FlO1_`"+"7;)$&&7;E&1JK"T%$&&!Y149H &9* "_U !2:&_PTApz. ' 7 &JK $n,rS&_P+$7&'/.s"+&7*"+&'/.y7&+$O.rS&+O26M$&&Tr&"2* _ 7;)1+= &%$&rFlO 1 4 'PU14'B+$&: ϕ(t) J;2; B = 0.16%>Ky+.-z"+$&" &'/.r 1'B""= 6[+7"JK "Ty*;O+$&O$&_ ,-O2"+SN TtU WX.&%$&:?1D9* " Y;7;O'B"+$&7 &?$&&"%$&%$&. 6^$A&SO2+$FlO1D&&x14)H!NO$) C$&%. $+$&\pz. '( UMW7" "S&7"%35"'`!1D'B" " 'B "+7"'B"+&T ϕ(t) C7"%= JK"2K)HEJK&TE$&"+ I ;92:&&TE'B_ ES.DJK"'Q.!* $&_PN)$&:y$&)2:&,-h7 7;Oz" >/1D ,!O6^# "ns7 "+1.42:4$)O$"w+, 9* , x!1aDJ&T $" I )2:&T 'B_ ±2π JK7 . %.!6 U , 4*,A7& HNJ $ r%2;. *;T _P"h'B&'B"+ $&_ &7"'B"+CA!H 'B93g * $)O$&: %$)O$& +$& *"+,<"+1D&+'B_P'BtU ,'0&S&7O1D&'P>B &&"J""w7;O1O= +$FlO1MJK&S& 2;. *;T_P' S)2;.-3]JLO 'PU !.&+$4e $"7N92a&7"+'B" t − t > 1D9* $&"2:&G7"+&& H&DJ; # Tg+7"JK ""d&7 "'/#'B"%3]J;. +, 9* ,!O'Bt>1D9* "" ϕ(t) , 2K9JK_PNO"+$&Z 1 SO2:pz&C*2KA"%14'B_qHc%2.!*T_qHQ7 7;O#"+TtU *q2"JK%$&& " I " $7;)2:&T 7"kJK"%2:&T $"&7"+'B_>/7;+7"kJK"%2"+" 2;"T %$)&'/O. = & $Z D.&+&&_P'P> 7*"+'cJK "7+ 7;+7"kJK"%2"+ 7;O+$&"%$swϕ(t) 14O, N. X σ (t) = 2B · (t − t ) > JK"y&%$&4 _PTR'B43e!$&"%2;: B ;O1_PN)= 6^$f,-&@+FlFG I " $&'Y@+FGFl"+, $& &&T JKFGF. 1a7;O1&+$hFlO1U l"+T_PT H-)= 7;,!$&"7s7 &+$)]J"+7 σ (t) O2; "+$Zy7 SO2 35"+_P'P+U *RJK"+T+$& $"%2:& O= +$?;nFl"]2;"T &w7&%$)h;SO2;6 JL6^$Z?'B"2,-&'(+p $)S&_P" !1D'B" " 1D9* " ?JK "+7V> + 1D_P"a &"JK"+"+' +$&"+'B_ s7"kJK"%2)H JK&T $" I ;)2:& &TC 'B_U l2; &_P*2"!A,-&@+FGFl I "!$)x@+FlFG"+, $&&T JKFX= F.-1DQ7;O1D &+$cFlO1x&%$&7&"+;OQ*2" A1D&+'B&+$: O$ C'B"%= $&J.C;O'B":Op#!HA,N4JK7;O$&? 7&,- 'B7 .&"%$&Z2 "T &TYσ14O& t'B&+$:&6 X U & JL],-&@FGFG I "+ $@+FGFl"+, $& &&TnJFGFG.-1h7;)1D&%$&wF )1 σ = at + b &7 "JK"2;"%$&Z, , =U * "%2* ; 7& &7 I ;)2:&r"74 $O= +$ +,!4*, nn"kJK BI .x&=7"'Ba/2 "+tUW&2"JK! BE'B93#"+$nS&_d$&:n;T-JK";a 1f7"%p#"%= s1D9JL4*s &7 "'B" x "7&eJK&%$& 35"+ r+7; I _ 4 $9U tUL149JL4*;a7;'B"+7!= D, ~ v!3> $80!3> $Nv!> v > v!E{ 6 *&O$&&"%$&+$&nX 1%2435" _P'Bn ~ .v L7;++.-35JK"!'Bt> J;2;l,-&@+FlFG I " $)^@FGFG", $&&TJKFGF.-1D 7;O1D &+$zFlO1('B43e M O2;. * $: %2;"J. 6^z""# 7SO2!35" &"l;)2!$&*"+,-&" &_P7;O35"+"&X p 4 ~ U Av 6 U 2π∆ B = π ∆ − ∆ 1 + exp − exp − , ϕ

0

2 ϕ

0

2 ϕ

2 ϕ

2

2 c

0

2

JK" x 2;. S& ;s$&"+ I ;92:&_qH 'PU X1 4 ~ U vA6f%2"kJ.&"%$9>^* $ @FGFG", $&;OJK FlF.-1D!Q7O1D+$& FlO1sS&_P%$&7C7;%$&"%$ p ∆ 2 2 U0 = 2 ∆c − ∆ − ∆ · arccos ∆c

Bϕ

Bϕ

@

G 7&%$&&'C $"+&&%$&hp . '( > H-7, $&"+7 1+.!6M#"+TZr,-&@FGFG I "+ $&' B >; $), 3#"P . &"%2;*" "'E7;%$&7T, ∆ U&j"&SDH&DJ'B$&'B"%$& $:>N* $&_P7;O35"+" 4 ~ U Av 6P"+7"%$)"+$nS&_d$:7&"J2&_P'<7 y& " &S)2K+$y !H!7&!= 14 I tUNW@%$&&T]7 *" & "d7;OS&$)"%$ ∆5→7 f∆S)2 1D, !H5,] .4261;4*"!!H >!J;2;n,-$&7_qHap#7 ;gSO2K%$&n!H!7 14 I : 4 S&"+1p .!'(A 6 &* ":]'()2K!U ∆ W7<&1JK"T%$&& c; "+"7O$&&7QD.&+&N pz. '( S&&L$9U "OUL+!H!7&!14 I !s"z'B435"%$ S&_d$:+$7&TtYU * @+$&' 2;. *;" 7!= !$&f&&7 $&:]&S]@+FlFG"+, $&&Tn!H!7 14 I t>K74)2;6M#"T Zw51D)H!NO$&" FlO1D_[;5JK&+$))$&&* J2 $&"2:&_qHs $&"+7N)2)Hs&7 "'B" tUt&@+FlFG I " $ B H-O7;, $"7 1+.&"%$w+7"J""#* %2gFlO1D&_qHnS&&"+ef"JK I .w&7"+'B"t>K]1D;4* $ 'B435"+$g%2. 3g $:g,7 $&"+7"'R@+FGFl"+, $& &&Tw!H!7 14 I tU W&,-O2;:&, .nO2OyFlO1De$&,-O2"+SNTs"++$: >L JK" = ω t + ϕ(t) JK"%$&"7 'B7N;) , +$)!$) > $&h7 I "++_ Φ(t)Φ(t) ϕ(t) H-O7;, $"7 1+. 6^ω$&Z JK' f$"'E3#"M,-&@+FGFl I "!$&&'?@+FGFl"+, $& &&TfJFGFG.-1tU /, 'ES&7;O1&'P> 7"J"%2; "+$# O2;.-p#7 .f" 7"7_P&Te,-&'B &"!$&_R ", $7;z,-O2"+SNT B ; > ,-$&7;Oa7a"+S&O2:Op#!Ha7%$&7&T ,!97 H 4 )JL92a$]7;O I _Q!H!7 149= x(t) I ? 6P 'B""+$gFG&7'/.e2;&7" I ;;!U \X7&"+" #&,,-"7;9= Wu2K, )= O2'B&+&7&N) 4 5Wl 6>$&&"%$&%$&. 6^z"" \ 4 ~ U $86>'B"+"+$ J ϕ

c

c

1

1

∂pϕ (ϕ, t) ∂pϕ (ϕ, t) ∂ (∆ − ∆c cos ϕ) pϕ (ϕ, t) − Bϕ . =− ∂t ∂ϕ ∂ϕ

4 ~ U 6

V;4*"+ gF )1D&&Tf "7"+'B"T ϕ 7f2;6uS&_qH ∆ ∆ "+&7*"+ 7;+$O.-$ S&+O26^$&Tg&"%2 *"O>!@+$&'/.g2;. *;T _PT7 I "O>-&+_PN"'B_PTg.!7O= "+"' 4 ~ U 6%> O2;"%$&Zh"%$) I &7_P'PUto^J;,->&+, )2:&, .n,-&@FGFG I "!= $&_[. 7;& " ]WG n"+7JK*"+, "f ϕ >L$ 'B43eg7;+'(O$7NO$:a7!= &"kJK"!.!6QFlO1. φ >N&7*"+ . 6cG!$&"7 N)2" [−π, π] X φ(t) = ϕ(t) − 2πn − π > JK" = Int (ϕ(t)/2π) UtWu2$&%$&: &"+74 $& &+$E7 &"J"TsF )1D_ φ &_P7)= 3]"%$&nZw , Y, X X 4 ~ U 286 p (φ, t) = p (ϕ, t) . !

c

\X7&"+"

5Wl J;2; 'B""+$n$&O$n3#"5D'B_PTx& J>V* $&n 4 ~ U 6%>V $&"%= "+7: +.-#"++$&O.&"+$Y+$) I p&(φ, ;7 t)&"r7"+pz" "U $) I &;7!.!6 !2;$&+$&:C&"+7O= $&%$& p (φ) 'B93g ;OT $&x#. *"+$&' "+7JK*"+, H +7; *_qHs.%2&T a.&2&&!w &7'B7&, R p (φ, t)dϕ = 1 ~ v. X p (−π, t) = p (π, t) φ

ϕ

n

φ

φ

φ

φ

π −π

φ

K

pφ (φ) = C exp

∆ · φ − ∆ sin φ Bϕ

Z

φ+2π φ

∆ · ψ − ∆c sin ψ exp − Bϕ

4~ U{ 6

dψ ,

JK" C ^ &7 'B7&&*;OC,-&%$) $) U* *;%$&' %2;. *;"O> ,-& JK ∆ = 0 >B$ "%$&:> , JL#&S+$&";)a*;+$$)5& I !22;!$&&7;#$&*5+&$&"+$&%$&."+$e*;+$$&" &1S.-3]J" V>N+$) I &;7 ;Og!2$&+$:#&"+74 $& &+$7"JK"+&Tg7;O1D &+$ FlO1l'B""%$e 7&+$&Th ]J X −π ≤ φ ≤ π,

1 pφ (φ) = exp 2πI0 (∆ /Bϕ )

∆ cos(φ + π/2) , Bϕ

4 ~ U {&{[6

−π ≤ φ ≤ π,

JK" I (z) ^ 'BJKFG I 7&N ;O F. , I !'q"+"2;tU l2;eS&O2:Op#&T $&"+)= &%$&npz. '( > I (∆ /B ) ≈ 1 exp [(∆ /B ) cos (φ + π/2))] ≈ 1 $), 'Y&S7;)= 1D'P>%$) I &7;O?!2$&+$:s&"7 9!$&+$& g7&%$&&'\!$&" & &+$Cp .!'( +$7"'B!$&ZR, JK&7 DJ&Tt> p (φ) = 1/2π U $)r $O.N I !$&&"%$&%$&."+$ $)= +.-$+$&6 !H!7& 1D I VPU * 7$&& O243e &' 2;. *;"u*":l%2S&&+lp .!'( $,!2&"+ gFlO1D_ φ $G"1D'/.-#"+&+#+$&4 ].&%$&&T* &&+z7;&&"+ S.DJ.-$u'()2_P'B#l'B43eMO2;93g $: cos(φ+π/2) ≈ 1−(φ+π/2) /2 > φ = −π/2 p UVWG)2;. *;"'P>V* $&a+$) I &;O7;Or!2;$O= I (∆ /B ) ≈ exp (∆ /B )/ 2π∆ /B &%$&:g&"74!$&&%$&w'B""%$D.&+&.aFG&7 '/+. X 0

0

c

ϕ

c

ϕ

φ

2

0

0

c

ϕ

c

ϕ

c

ϕ

1 pφ (φ) = p exp 2πBϕ /∆

−(φ + π/2)2 2Bϕ /∆

4~ U{ ~ 6

u '(O, +'/. 'B&' $&*,-" φ U u&7pz 14O'B"+$& _PTn'(O, +'/. ' 4.+&Nz7;O7"kJK"%= 2"+ ?7O1D+$& FlO1g JK"+$"%2:+$&O.&"+$xn14)H!NO$"eFlO1_U *07"J"%2" B → 0 !2O$&&%$&:E&"74!$&&%$&A+$)& $&Z δ-FG. , I "TV>/$&x"+$: lim p (φ) = U!WG7 "SO2:pz&Tt> #,-&"* &T $"+&+$&apz. '(#7;O7"kJK"%2;"%= δ(φ + π/2) " p (φ) $&,!2;& "%$&ZsO$D.&+&&+,-&>Le 'B""+$7h@%$&&'YH&&7pze14O'B"+$O= _PTw'(,-'/. 'PU *q&"kJK" "G$D2* &TwO$g!.92a7;%$&7T,n7&JK!$g,a 1'B""+60,-&&7JK!= ;O$_ 7;&&"+ φ a$&&"%$&%$&. 6^z"'/.h'B"+z" 6 '(,-+'/. '(f7;O7"kJK"%= 2"+ p (φ) U z"+7!w77 "JK"2"!h7;O+$&"%$f7&+$&'C7;%$&7T,tUKW7!= 'B"7 _A7;O7"kJK"%2;"T φ >O*%2;"G)2;. *" _qH#J;2;5'BJK"2 4 ~ U $86%>O7"JK"+_ ; 4 7 U ~ U 8| 6U&W7 ∆ = ±∆ %$&"'B" 4 ~ U $86S&"+1Pp .!'(X.+$&T*&"M]"%.&%$&&T!= *&"^+&+$4 e7;& &&"+ f%2 N6^$Zg*"%146M$9>!*!$&z+&$&"+$+$&O.&"+$f+7;)= I "XFlO1D&&z1D)H!NO$) UoMJK;,-## 7+.-$+$&np . '(#"7"+"+*" "l+7; I _ "G7DJ $e,n 1'B""+60HN7;,!$&"7e7;+7"J"%2"+ y "7"+'B"T ∆ = ±∆ 4 7OU ~ U A| 6%U(W7* ;E@%$&&+E+; U B%2 |∆| < ∆ >B$&E%2;. *;OT_P"w$O2*, φ 0

ϕ

Bφ →0

φ

0

φ

φ

@

5

∆=0

pφ(φ) 4

∆=0.13

3 2 1 0

∆=0.16

−3

−2

−1

0

1

2

φ

3

) φ ∈ [−π, π] @

3

9 Bϕ = 0.001 ∆ = 0.15

& +"w7;O&x&_d$))2, N6^$ %$&"+'/.Y 1w+$&4 7&&"! φ >(7DJ;A, 7;+7"kJK"%2"+6 p (φ) ,-& "*T pz7 _ '(O, +'/. 'B&'0h$&&* , " φ U B%2; > $&]$&&*, "l%.-#"%$&."+$9UoMJK,->&+,-O2:&, .f,7"%$&+$&h%= |∆| > ∆ *"%1D . pz"T $&&* , 7;φ& &&"+ ?+,-&7&%$&: φ˙ '(92K > $&yw$&"* " J;2; $&"2:&_qH 7'B"+3f. $,-&s&7"+'B" 1;4*" φ SO2 1,?, φ UjGn+7;FG,-"e14O& 'B&+$ @%$&&'/.z&$&"+$+$. 6^$l&* $5&7 1D&!$))2:&_P"P. *;%$&,VU &$&"+$+$&" > ϕ(t) .E7; 7"JK"2" p (φ) DH!7;!"+$ZC'(,-'/. '0h$&&* , "O>/SO2!1D,-&T , * "+1+= . pz"T]$&&*,-"P7;O&&&"+ VUNW7 JL)2;:&"T!p#"'r. &"%2;*" g7%$&7&T ,g%$) I != &;O7&"7;O7"kJK"%2;"" &%$&" "f7 SO2 3]O"+$&Zw,a7;& &'B"7 &'/.&U ,' &S&7O1D&'P> 'B93g5 JK"2KO$:g&_PDJh&S5O$&+.-$+$&yS&F. 7, I &&+g "7"HNJL > O 14O&+d, 9* "+$&" _P'? 1'B""+"'E+$) I &;7 &G7;O7"kJK"%2;" f7;)1+= &%$&sFlO1 h& 7"JK"2;6M#"+a7;O I _\FlO1&&&Tw!H!7 14 I s"+"7O$&&7 GD.&+&&_P'p .!'B'PU !V;OC%$) I &7 . 6 !2;$&+$&: "74 $&%$& p (φ) 'B43ex;T!$&+7"J;= 6^607O1D+$&!.!6 *;%$&$O.w,-O2;"SN T Ω X 0

0

φ

0

0

φ

φ

4 ~ U {4|86 = +7"JK! n*;%$&$)g$&, )2"SNOTnf 7+.-$+$&s&1JK"T%$&&!tUN*R&SO= ω ˙ = Ω = ω − ω1 = hφi

Z

π

∆ − ∆c cos φ pφ (φ)dφ,

JK" 2K%$& +!H!7&!14 I Cw%$&"+'B" gpz. 'B&'\"%2* ; Ω $D2;*;hO$y .42; 7"kJK+$)OO2;"%$y+&S&&Ts+7"JK T JK5*;%$&$_ H 7 &_qHx,-O2"+SN TE %$&O= 7&!.g+&S&%$&&"+&T *;O+$&O$&_Q$&" "7;O$&7; >!&_d1D&_PTFX2;. , $O.N I 'BaFl)= c

−π

K

0.2

Ω(∆)

Bϕ=0 Bϕ=0.01 Bϕ=0.1

0.1

0.0

−0.1

−0.2 −0.2

−0.1

0.0

0.1

∆

0.2

K

Ω

# 9

9

∆ = 0.15 K Ω=∆

1D_U( 7;OFl ,y14'B+$& Ω $a7;++$7&T, ∆ >V7;+*!$)_P"5J;2;s'BJK"2 4 ~ U $86> 7"JK"+_ ;w7 U ~ U U(*q"2* Ω 2K&n'B"!"+$Z ] 1D'B"+" "' 7;++$7&T,V> &S&7;O#O+: O2:g$&O2:, a$&*,-" ∆ = 0 UVWO2;. *" _P"]7!= +.-$+$&cpz. '( ,7 &_P"r7; O2K6M$&Z 'B"+35J.Y,7&Tt>d$&&"%$&%$&. 6^z"T +$&"+'B"5S&"+1 pz. '( >Kh7N 'B&T Ω = ∆ >$7;O3]O6^#"+TsO2 &" O$&+.-$+$&"]!= H!7&!14 I 4 pz$7!H-= . ,!$&7 6U /, ' &S&7;)1D&'P> ;)2 *"y2;. *;T_qHD.&%= &, Hf&1D'/.-z"Te7&JK!$z,]$&&'/.>N* $& 7"kJK !g*+$$)#$&,-O2"+SNT "M&;9JL"%$z *;O+$&O$&&Tg&1%J"T%$&& V> ;9HNJK $Z 'B"+35J.5*;O+$&O$&&Tg&1%J"T!= +$& hw*;O+$&O$&&Ty "&1'/.-#" _qHrO$&&,-O2"+SN TtULo^J;,->"2h!$&" O= &%$&:spz. '(wJK&%$)O$&*r'()2Kh1D&+'B&+$: Ω(∆) 'B"+"+$sH-O7;, $"7 . 6 J;2; !H!7& 1D I e&* $e7 1D $))2:&!.!6 O2;&*, . kU 7&%$&&'E $&"+ &&%$& pz. '(]$), O O2;&*, &+$""++*"+1D"+$9U W7< .42"&TQ7%$&7&T , "x7 "JK ! *;+$$)YO$&&,-O2;"SN TQ+&&;4JL"+$C *;%$&$&T?&O1%JK"+T+$& t>n@+$h"]1;4*;"%$9>*!$&h 'B""+$w'B"%$&w@FGFG", $&;O !H!7& 1D I Q 'B_P2"s14)H!NO$) FlO1D_ ;EJ;2!$&"%2;:&_qH $"7&)2K)H &7 "%= 'B" tUo 1D)H!NO$&"fFlO1D_0'B43e n+&&&7 $&:> "+%2x,-&@FGFG I " $w@FGFG", $&T JKFGF.-1Da7;)1D&%$&aFlO17 '("%$zJK&%$)O$&&* ]'()2_P"X1D9* " VU;G,2"%= J."+$s 1 4 ~ U Av 67 ∆ = 0 ,-&@+FlFG I " $ B .!S_PN"+$he7+$&'['B!2 $O.DJK_ &1JK"T +$& Z 4 ;7;O'B"+$&7 ∆ #. 7;O&" Z 4 ~ U $A6 26uS&""]144JL"'()2&"

@

1D9* ""x'B435"+$CS&_d$:A O2;. *" A$&)2:&,-A 7Q7"J"%2"+&' 1;4*" ∆ U ;4*"z+&&&7NV>L%. z"%$&."+$h'B!2 $O.DJK_PTs&7 &z&_P!. 35JK"+&Ty+!H!7& 14)= I nf%$&"'B" Xpz. 'B&'PU jG,-&" I >57;+'B$7'P>], , @+&O26 I & 7 .&"%$ ", $7;)2:&O 2$& &+$: 'Bz&+$g+!H!7& 1D7!.&"'B_qHeO$&&,-O2"+SN Tf 7+.-$+$&epz. '( >)2:O= 1.OJ;2;a@+$&]FlO1&. 6\'BJK"%2 : 4 ~ U $A6%U !tOFl , +7 ."'7;O+$7&T,!. ∆ a $"!= &&%$&: p . '( f7&2"JK '?@&)26 I 6 +",!$&7;)2;:&&T 2$& &+$'B# &+$ ,-O2"+SNT x(t) = cos(ω t+ϕ(t)) 7 . &"2*"+ ∆ 4 $9U "U)'B 2 $.DJK_ &1%J"T!= +$& ? 6%U "+1+.92;:9$))$&_ 7;O*"%$&& 7 &"J"_[;e7 U ~ U 0 ULW7h'()2&'1;4*" h"+, $&7" 4 ,7N)[{ 6> ,7'B"&+$7&+s, y;y*;%$&$" &1JK"T%$&&! ω ∆ 4 $"&7 "+$& *"+,n= @+$ JKO2;3g; S_d$&: = F. , I ? 6BH&&7pz5& JK; "+, $&7;92:&;O 2 fM'(O, +'/. 'B&'?;"+,-$&&7 &Teδ *;O+$&O$&" ω >&+&$&"+$+$&O.!6M5O $&, )2"%= SN 'PU ++$&4 "f'B"+35J.h+",!$&7;)2;:&_P'B '(O, +'/. '('Br7;& Ω U 7O= +$&' ∆ +", $7;)2:;OE2; $&,-O2"S&TE'B"+#"+$Z n+$&7& . δ= ,! e 7@+$&' .-p# 7- "+$Z 4 ,7 NO ~ 6%U!oXS&7;O1+.&"%$&Zpz7&, T:&"kJK"%$))2n"+, $&7 #'(,-+'/. 'B&'P>V+'B"+#"+_P' ;e&"2* . Ω O$&&+ $&"2:& *+$$&_0&1JK"T!= +$& ω U 'B"+z""g'B435"+$nS&_d$:h&*"+:a'()2> n" JKw$D2 *hO$w!.92t> "2 ∆ 6= 0 1D;4*"+" ∆ ,-& "* 4 , 7NOw8| 6U 1

1

1

20

10lgG(ω)

1

10

2 0

3 −10

−20 0.70

0.80

0.90

1.00

ω

1.10

x(t) = cos(ω1 t + ϕ(t)) ϕ(t)

0 9

∆ ) =

0.1

: K ∆

@ ω 1 = 0.9

( :

Bϕ = 0.01

G(ω : ω

$

9

$ "72@'6* $ "" ,.-7/ "%$71 24%$"' B 9 291 $05 #"&3-028:/-0/ I"-%$0* F)8:*#8 $ $ % F.H

1 2 0H

2 8 /95

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x¨1 + ω12 x1 = (λ1 − x21 )x˙1 + γ1 (x˙2 − x˙1 ) +

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ϕ˙1 = ∆c1 sin(ϕ2 − ϕ1 ) +

!

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ϕ2

c1

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c2

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ϕ1

9

0

S1(ω), dB

−10 −20 −30

2

−40 −50

1 −60 −70 0.80

0.90

1.00

1.10

1.20

ω

& . x1 (t) = cos (ω1 t + ϕ1 (t)) . ( 0 7 M B = 0.0001 B = 0.001∆c1∆ = 0 ∆ = 0.1 ω = 1

ϕ ϕ c2 # 1 K

) 9 B

1

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1

1

y˙ 1 = (m − x21 )y1 − α12 x1 .

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2 1

1

2

1

0

1

0

∗

∗

2

1

x˙ 1 = y1 , y˙ 1 = (m − x21 )y1 − α12 x1 + k(x2 − x1 ), x˙ 2 = y2 , y˙ 2 = (m − x22 )y2 − α22 x2 + k(x1 − x2 ).

4 | U ~ 6

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2

1

2

2

1

2

1

1

1

2

2

1

2

1

0.03

ls II

k 0.02

:

ls I

II

k=0.02

α 2 =1.0015

0.01

0 0.98

0.99

1

1.01

G . ) 9

α2

1.02

m =Θ0.1= α1 1: =1 1

7

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4 | U 6

x˙ = mx + y − xφ − dx3 , y˙ = −x, z˙ = φ, φ˙ = −γφ + γΦ(x) − gz.

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g = 0.263

x˙ = mx + y − xφ − dx3 + k sin(2πfe ), y˙ = −x, z˙ = φ, φ˙ = −γφ + γΦ(x) − gz.

4 | U 0A6

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x

0

i

10

y 0

5

−10 −20

S(ω), dB

0 −5 −10

−30 −40 −50 −60 −70

−15 −15

−10

−5

0

5

10

x

15

−80 0.0

0.5

1.0

& ) 9 9

. S (ω) x(t) x

1.5

2.0

ω

2.5

(x, y) α = β = 0.2 µ = 6.5

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0

0

0

0

1

1

30

ρ(Ti) 20

10

0

4

5

6

Ti

7

&

: @ y = 0 K

@

1D9* "" SN)1D&&T *;%$&$_ 7;)2:& O$O$&7;O, $&&7 R$+.-$&%$&& &"%p = "+T !2;_ S.DJK"+$ ω = 1.0683 ± 10 4 'PU 7 U U {> S 6U *q_PS&"+7"'`1D9* "" 'B!2; $.DJK_ b = 0.05 <S.DJK"+' +%2"kJK&NO$:A@+&O26 I 6 ", $7;A'Bz&%$& "7 "'B" &T x(t) 7e 1'B""+*+$$&_R&O1%JK"+T+$& ω $&"+7N)2" S (ω) U "+1.42:4$)O$_ 7;+*"+$)y+",!$&7&hJ;2 $&7 "%HE1D;4*"+T ω 1.060 ≤ ω ≤ 1.065 7"kJK+$)OO2" _0]7 U U | U G, & Jr 1eJL _qHV>B 7"JK%$)O2"+_qHA;s7OU U | >J;2 $& & $"%2: S&O2;:p#&Tw7;++$&7 &T,r*;%$&$_[&1%J"T%$&& hySNO1&&&Tn*;%$&$_0$&&'B&T +$&"+'B_RG"+, $&7"M,-O2"+SNT 4 , 7NOa%{ 67%.-$&+$. 6^$#2 ] ;)2l&O1+= JK"+T+$& ω a2; wSNO1&&&Ta*;%$&$_ ω U . 'B":Op#" "'R7;O+$7&T, y@+$ 2 rSO2!3]6^$Z 4 ,7 NO ~ 6^t>L;,-&" I >t7 HNJK $a1D)H!NO$aSNO1&&&Th*;)= +$$&_ H-$*"+,!HQO$&&,-O2;"SN T<&"+pz' +;)2&' X ω = ω 4 ,7 NO 8| 6U n&'B" $O.E1D)H!NO$)hO$&&"+*;"+$h1D;4*"+"e*+$$&_ &1JK"T +$& > "n7;& &"a HNJKTR&"%2 *" ω = 1.0683 U ,'0&S&7O1D&'P>Bxω7"%1=.421.065 :4$)O$&" 149H &O$)f+",!$&7;)2;:&;O 2 aSNO1&&&T *;O+$&O$&_ HN$*"+,!Hh$&,-O2"+SNT 'B"++$!2K+:AR%$))2K 7;& &TR*;%$&$"s&"+pz"+A1%JK"+T+$& VU 'B""+$ 'B"++$ O2;;Oe;)2 e^@+FlFG"+, $&&'?1D)H!NO$)z*+$$&_<z, 2K+*"+,-&T $&"+&7+!= H!7&!14 I x 7"JK"2:& I ,!2 U #)2:"T pz"#+%2"kJK&NO r&, O1D)2t>L* $& @+$)n;92&!r'B""%$wF. JLO'B" $))2;:&_PTxHN7;,!$&"7VU l2;r+7;92:&&+H-& > , , xJ;2?7"kJK"%2;:&&+ I ,!2K!>'B43gy&&7 $:y&Sw@FGFG", $))H 14)H!N)$)y*;)= −4

0

1

x

1

ex

1

0

0

1

1

0

0

ω1

1 2 3

−5

ω1

−10

S(ω),dB

ω0

ω0

−15 −20 −25 −30 −35 1.055

1.065

7 x(t) ω = 1.06

1

Lω0

9

ω

1

= 1.064

1.075

u ω

= 1.065

&

ω

1

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ρ(t) =

z&SN)2" "!2

4 U 6

& _P* $)" I "2&+s*%2r &&&7O$&&yw. O2 π "+&SDHNJK!= 'B> * $&S&_ JK&%$&7&!$&: ]FlO1+.wJKnO2 &a. 2Kw&7O#"+ tUW7x@+$&'c1;4*"%= " I "%2&* %2"+&Ta"+7"'B"+&T & 7"JK"2;"%$&Zn 1X.%2& a"+7"7 _P&&%$& F. , I Φ(t) U!WO2;. *" ;O$), k'AS&7;O1&'?'B &&" ;O FlO1D Φ(t) O2"+$Z O2;&ThFlO1&Tt>7"J"%2"+&TyfS&"+,-&"+*&' $&"+7N)2&" X Φ(t) ∈ [−∞, +∞] U 7&'B" $&>/ & JLr&1, "+$E"+&SDHNJK'B&%$&:E7;+'B$&7 "+$&:EFlO1.>B&7"kJK"%= 2"+ . 6bx&7*"+&' $&"+7N)2;L" X Φ(t) !2 Φ(t) U ˜ ˜ ∈ [−π, +π] ∈ [0, 2π] oX*" JK>;*!$& Φ(t) = Φ(t) >; JK" e"+,-$&&7 &" I "2&"G*2U ˜ O'BO$&7"+&" ;'Bz&7±"kJKmπ "%2"+"P'Bm+&&"+_qH]O'B!2 $O.4J_CzF )1D_ " O= 2; "+$&Zf"kJK%$&&"+_P' {4| > h LU * 'B"+$uJK;'B *"+,!H]"+7"'B"+_qH x(t) y(t) Z7" 4 $)U 8| ,6X!'BHg&+"+.-7$s"'BS&"+_d$:h_qHe+2KO2&:O 1D&&" NO=_ S&"r+JK"*7!!.!$&+:#"7"+;+O.472;_ 7"+&7"^"'B&7;"+Oz"_qHLUB"MW77 ", _PS& O = I

@

lF O1D&&TR$&7;O", $&7Q&&,7 . r"+, O$&&7T $&*,t>P,-$&7;O S&"+7"+$ZQ1D ;4*;)= 2a7;9J .&= &"+, $&&7 UWDJHNJ z'Bx"7"+'B" _P'BE'B&%.-$wS_d$&:a7&!1D&JK_P" 2./!2r"+7"'B"+_P" >V JK" w& 7-!35" _PT dx/dt dy/dt !2:&S&"+7 $.a 7 I "+ x (t) {4|!> L X x(t) x (t) x (t) H

H

H

1 xH (t) = π

Z

∞ −∞

4 U 0A6

x(θ) dθ. t−θ

7&'B"$&>;'B43g]&&"%$&wF )1.&>; O2;:1.Oh%2"kJK&NO$"%2:&+$:'B&'B"+ $&& &7"+'B" t >V$&&"%$&%$&. 6^z!HE"7 ""+*" ' $&7;"+, $&7"Tx"+, .-#"+TE!2;&,-O= +$ {4&| (w1D9JL &'c7;)2" tU W7E, .&+&*a2"+T&T?7, +'( I FlO14]]7&!1D&O2;:&_PTn'B&'B" $g&7"+'B"n&7 "JK"2;"%$&Zs, , i

Φ(t) = 2π

t − ti ± 2πi, ti+1 − ti

i = 0, 1, 2, . . . .

4 U $A6

JK"+: i l* %2 "7"+"* "Tfu1D9JL &'s;7O2"+514u7"'/z;SO26 J" U t−t X1C2;6uS&&+c& 7"JK"2" 'B+&&"+&T FlO1_ 2"J."+$[& 7"JK"2" " Φ(t) 'B &&" &Tha+7"JK "Ty*;O+$&O$e,-O2"+SN TYX !

0

dΦ(t) , ω(t) = dt

ω =

dΦ(t) dt

= hω(t)i.

4 U vA6

JK"+:+, S&, h. . .i 'B&+.-$RO1D;4*;O$:R, ,c.&+7"J" "C 'BSO26 >M$)O,\ .&+7"J" "YR&7 "'B" t>X&+,-O2:&, .c7 I " ω(t) >M, ,[7;& 2R&SO2K4JL"+$ &T+$&&' @7+JK*+$&VU +'(O$7NOs.7"kJK"+"] &7"+'B"V>K2"+,-aO= 2;. * $: 4 U 6 Φ(t + T ) − Φ(t ) . ω = lim oX*" JK> *!$&h'B+&&"+_P"]1D;4*"+ T'B!2!$.DJK_>FlO1D_ E*;O+$&O$&_ 7 7;O1%2* _qHl&+&SN)H&&"J" 'B 2 $.DJK_E FlO1_ESO.4J;. $M7;)1+2* _UOo^J;,-> * $&S&_C'B!2!$.DJKO=mFlO1D&&"q7 "JK%$)O2"+"qJK;O'B,zH-O$&* ",-&T]++$"'B_ 'B"2^+'B_P%2>9"+&SDHNJK'B >)*!$&&S&_E+$)O$+$*"+,"qHN7;,!$&"7 +$, 4 ^,7T!= "+Tn'B"+7"X$&"O>,-$&7_P"G!14_ X;OSO26 JL"+'B_P'Bw"%2* ;'Ba)2"!'B 6 "X1D&+"%2n$]7 $&]14#&+&.e&7"kJK"%2;" wFlO1D_UjG"+&SDHNJK'B]$'B"%= $&!$&:r%2"kJ. 6^z"L" X(7 2"JKN FlO1&&&T?+!H!7& 14 I ?H-$*"+,!H $&,-O2"+SNTz&*" :^N)3e >4* $&&S_?'B&"OzFlO1DMS_q2K^& 7"JK"2"u,-&7!= 7"+, $& U-G7 $&"+7"'x,-&77"+, $& &+$e&7 "JK"2"!]'B&"TfFlO1_ X7"+3g'B" 7;)2:& O$O$&7;,!$&&7;l%2;.-3g $ &&9JK" " 7"kJK"Tg*;%$&$_ ω 4 U 6 PSN)= 1D&&Tn*;+$$&Th,-O2"+SNT ω v!> f U !

0

T →∞

0

0

W 7& 1&"J;n1D'B"!. "7 "'B" _qH 4 U |86q].!7&"+ !H 4 U ~ 6%> O2;. *'A.!7O= "+ w%$&"'B_ +%2"+7;g]& J"LX 1 1 ρ˙ = αρ − αρ cos 2Φ − z cos Φ + b cos Φ sin(ω1 t), 2 2 1 1 1 Φ˙ = 1 + α sin 2Φ + z sin Φ − b sin Φ sin(ω1 t), 2 ρ ρ z˙ = β + z(ρ cos Φ − µ).

4 U 286

tFG,- 7&N^1D;4*"+ ];7'B"+$7& α = β = 0.2 > µ = 6.5 b = 0.05 >) 7&!= $&"+7 7 .&"+'Y +$&"+'/. 4 U 286(* %2"+> 1'B" !*;%$&$O.e"+pz"+5&1%J"T%$&& U G+O2:O1. O2;. *"+_P"gJLO_P"fJ;2 Φ(t) >&_P*2'\+&$&"+$&%$&. 6M#" ω 1D9* " Y7"kJK"+T*;%$&$_ ω 4 U 6%U(jGs7OU U x7"kJK+$)OO2" _ 1D&+'BO= +$r$&Op#" TE*+$$ 4 ,7&O?[{ 6M Θ = ω /ω 4 ,7 NO ~ 6M$ *;%$&$_\+;)2K]&1JK"T Θ+$&= ω ω/ωU G,n& JKg 1 7;FG,-& >K 'B""+$'B"+$e149H &O$e+7"J"Ty*+$$&_ ω &"%p = ' +;)2' r,-&"+*&T &SO2+$A*;O+$&O$ 1.065 ≤ ω ≤ 1.078 U/W7 @%$&&' 7;+*!$)_P"1;4*"!?7 "JK"+T *;%$&$_ ω SNO1&&&Tx*;%$&$_ ω &;)= JL6M$e]7 "JK"2K)HnpzS&,n&_P*2" TtU )2:1.OAJL _P"y7;+*"+$& 'B &&" &TRFlO1D_ > &%$&7' +7;FG!= Φ(t) ,1D&+'B&+$Q'B+&&"+&TR7O1D+$& FlO1sO$&&,-O2;"SN T<R&1%J"T%$&& w14O& 'B&+$CO$y&7"+'B"VU/jGw7 U U 0a7"J+$)O2;"_ ∆Φ(t) = Φ(t) − ω t !

1

1

0

0

1

1

1

0

1

1.01

1 2

Θc,Θ0

1.00

0.99 1.060

1.065

1.070

1.075

)

. Θ = ω /ω1 ω1

9

1.080

"

ω1

1.085

Θ0 = ω0 /ω1

d

@

$), "X14& 'B&%$&t>!O2;. *"+_P"uJ;2;g$&7"He1;4*" Ta*;+$$&_ &"%p#"+5+%= ;)2 U G,l& JK^ 1 7;FG, >9^&SO2K%$& !H!7 14 I 4 , 7NOz|86t'B"+"+$X'B"++$ @FGFG",!$y14)H!N)$)hFlO1D_ +""+7;O$&7; "+2"7;s& "+p# '\+;)2' X SO26M$O= ;Ow&"2*7;O1D &+$yFlO1 "l7;%$&"+$ae&7"+'B"V>Vf&+$)"%$&Zr&S)2 1D &"+g"+7&&9*)2:&]1D;4*|∆Φ| "+ wf'B'B" $g&7"+'B" t U ^JK"+ FlO1&&&+CDO H 7 & 1D I ,-&+"7"+ $& &AHN&C& "7&_P" S&_q2K57 "J;2435"+;5#7;S$&" L U & 2],-& I " I wFlO1&&&T+!H!7& 14)= I lH-$*"+,!H5$&,-O2"S&T {4|!> &!, 2K+*"+,-&"d& 7"JK"2" "d@FGFG",!$) FlO1D&&51D)H!NO$) 4 [{ 6(JKO23e]S&_d$:g 7"&S7;O1DN]%2;"J. 6^z'R&S&7O1D&' X 4 U { 6 |mΦ (t) − nΦ (t)| ≤ K, JK" K y7; *" ;Or,-&%$) $) >14Z #Or$a;4*;)2;:&&Tr7;O1&%$&xF )1&> "2_P"e* %2K!U * O$D2*"e$y+!H!7&!14 I "7 DJ*"+,!H?, )2"%= m n I SN Tt>w2;. *;"]14)H!N)$)wFlO15H-O$&*"+, H O$&&+""+7;O$&7&>7;O1&+$:hFlO1 "5)2;"%$&ZE, +$)!$&&TtUoX; FX2;. , $. 7!.&"+$rn7"'B"+t> a" ]$,!2& " $E"7 &&;4*;)2;:&&+E1D9* " A+$7&+?&7*"+_U uS&+O26^$;OA&"%2;*; $), H $,!2&"+T "fJO2;3g;h7"+&_dp5)$&: 2π U l"+, s JK"+$:>/* $s 1.%2O= & FlO1&&&+h149H &O$) 4 U { 6 2"J."+$E7; I &;)2:&" $&Op#" "w+7"J!H *;%$&$PX 0

1

2

50

∆Φ(t) 40

1

30 20

2

10 0

3 0

2000

4000

6000

8000

t

10000

∆Φ(t) = Φ(t) − ω1 t " ) ) . 7 ω1 = 1.06 ω1 = 1.064 ω = 1.065 1

mωc1 = nωc2 ,

JK"

ωc1 =

dΦ1 (t) dt

, ωc2 =

dΦ2 (t) dt

4 U {&{[6

.

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1

b 0.05

3 0.04

1 0.03

0.02

2

0.01

0.00 1.064

1.066

1.068

1.070

1.072

1.074

1.076

1.078

ω1

( G ω , b G @ @

1

@

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#

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2

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5

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I(x) =

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1, x > 0 0, x ≤ 0.

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x(t) #

f0

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6 5 4 3 2 1 0 −1 −2 −3 −4 −5

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20000

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Sx (ω)

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#

x y 0

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0 f1=16.8 kHz f1=17.43 kHz f1=17.59 kHz

−5 −10

S(f), dB

−15 −20

f0

f1

f1

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16900

17100

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17900

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1

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f1, kHz

) Θ0 = f0 /f1 9

#

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1

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g+$&*,aD.&+&&5p . '( >7 *"' X < ξ(t) >≡ 0 < ξ(t)ξ(t + τ ) >= > #F.! , I z7;, U-W&%$&4 ;O D 149JLO"+$ $&"+ &&%$&:#pz. '( U jGn7 U U {4| > 7"J+$)O2;"_ 14O& 'B&+$?$& p#"+ ?*+$$ $w*;)=

ξ(t) δ(τ ) δ(. . .)

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1

1

Θc

10

1 2

1.002

∆Φ(t)

1

0

1.001

−10

1.000

−20

0.999

−30

0.998

−40

2

3

0.997 1.064

1.066

1.068

1.070

1.072

ω1

1.074

−50

0

20000

40000

60000

80000

t

7 " ( 0 Θ = ω /ω1 9 @ α= β= ω0.2, µ = 6.5, b =

0.03 K G

1 ) 0 D = 0.01 &D = 0.02

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L

. D = 0, 0.01, 0.02 ω1 ω = 1.0683, 1.0705, 1.0720 1

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4 U { 6

x˙1 = F(x1 , α1 ) + γg(x1 , x2 ), x˙2 = F(x2 , α2 ) + γg(x1 , x2 ).

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1,2

1

2

1

2

1

2

1

2

x˙1 y˙1 z˙1 x˙2 y˙2 z˙2

= −y1 − z1 + γ(x2 − x1 ), = x1 + αy1 , = β + z1 (x1 − µ), = −y2 − z2 + γ(x1 − x2 ), = x2 + αy2 , = β + z2 (x2 − µ).

4 U {[0A6

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2

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y2

10

10

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5

0

0

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(x1 , x2 )

−10

−5

& (y1 , y2 )

0

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10

@

−15 −15

x1

U −10

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5

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y1

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4 U {%$86

xn+1 = f (xn ) + γg(yn − xn ), yn+1 = f (yn ) + γg(xn − yn ).

\(+$&T* &&+$:

$ 7;",!$&&7 Tg++$"'B_ 4 U { $86%>O7;4J;2"+35O#!H5N7 ; $&'/. 'B&+&&S&7O1D6 4 S+", $7" 6>V@%$&&' %2;. *;O"f7"J"%2; "+$&ZyJKO.!'/ 2; . &&+,'B#U,!)14O$"%2; 'Bt> Λ x =Λy U)W&, O14O$"%2: Λ HN7;, $"7!1. 6^$G@+&O= 26 I 6<&1'/. z" Tt>2"+35O#!H# U > Λ l$&7"7)2:&_qHf,]"+'/.&U ,' &S&7O1D&'P> 7;O1D7!. pz" "z7"%3e'(# O2&T H-O$&* ",-&Tw H 7 & 1D I w 4 U {%$86 JK;O&%$&7!.&"+$Z ?1;, . K Q&< G9LD KNG#Q ( 8< M M M 8Q.O.Q D)( ;> 7"J"%2; "'B&+ &_P7;)35" "' 1 df (x) 4 U {9Av 6 . Λ = lim ln n dx B%2Y&, O14)$&"%2;: % ) $ & & $ Z R O 2 4 e 3 ! $&"%2;:&_P'P> @+$& 1D;4*;O"+$9>q*!$& 7"%= Λ JK"2:&&" 'B435"++$&>O2"%3]Oz"" U >$&"+7- "+$l.&%$&&T* &&%$&:#X$7;+&"7)2:& &' ; 7;O2"+f#$&7;O", $&7 g7],-O2;:G. +DJG'()2&'x;7 .-pz"f 'B'B"+$&7 X;4*;92:&_qHl.&2&&!!H .-H&JK $G$uN7 ; $&u'B&&S&7;O1 U ^, ,-&T!= 2S )$$&7, $&7t>-"d2"+3])#T] U 8U * 'B&'B"7'E2;. *;" @&)26 I !g1D'/.N= #"+ V>K$7;+&"7)2:& &, U >;&+_PN"+$Z N 2;!.! &&, 'By&, O1DO$&"2;!= 'BT 4 JK" N r7O1D'B"+7&%$&:yFlO1&&&+h7+$&7+$Ny;7 I ;)2:&T ++$"'B_ 6U B%2?H&$4 S&_ JK !1 !H +$) && $Z O243g $&"2:&_P'P> $y&7;!$&&" 'B&+&&S&7O1D"g"7 "+$)O"+$rS&_d$:h.&%$&&T* &_P'P=U *07"+1+.92;:9$))$&"a7& HNJK $r7;)1+= 7 .-p#"+"a7"%3e'(y O2&T +!H!7& 14 I HN& >/;O1_PN"+'B&" [I =>K8Q S I D L KNM K ; 9.Q 4 16%U!oX#&S_P*57&&435JL"%$&Z O2" "' "+7"%= HNJK &T 4 l,-&"+*_qHe&7"+'B";)/H 6B!2 %$& &T ^"7 "'B"+35"'B&%$& 4 D K D#U)DER @ QAD UM[G U)Q&B.; >8B%@ I G#Q 8I !2 ?R D K D#U)D @ QAD U)M[G { {&>k{ $& 6%U oMJK;,- 2;!.! &&, "g &, O14O$"%2 s@%$&a.&+7"JK "_P"eh)$$&7, $&7 .sH-)= 7;,!$&"7 +$,t>,-$&7_P"?"yJK;O&%$&7!.!6M$<&"H<2&, )2:& _qH 1'B""+T\ +$7 . , $. 7"5 7"JK"2:&f'B435"%$&N NU * vv!>k{ $ >k{ 2 >k{&{ LS_q2g &, O14 >;* $& "+z" J<$& >,!O, Λ +$)& $&Z O243g $&"2:&_P'P>1D'B43gR4O2;"" * "+$& &f'B435"%$&N5$&*",a&7;!$&&+g'B &&S&7;O1 t>!5, O$&&7_qHn 'B""+$ tn

tr

tn

tr

tr

tr

tr

n→∞

x=xn

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@ 1,2

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Q1,2 9 2 S1,2

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.

y

Q2

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Q

1

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. 9 &

& 9 9

9 Q x = y 1,2

)

O $O$&7;,!$&&7 "7"++$)6M$?S&_d$: O$O$&7;O, $&&7 &' JLO35" r+'B_P%2" w!2 &7; >B* $E $&&"%$&%$&."+$aS&F. 7, I h77_PN U WJK*"7 ,"'s"%#"d7O1&>9* $XO2Ol!H!7 14 I zHN&D^&1D'B43g;P$&O2:, J;2;x)2&%$&:&6 JK"+ $&* _qH?O14'BJK"+T+$. 6^z!H?++$"'PUW7x7;%$&7T,-" ;7 I ;)2:_qH] +$&"+' 7;'B"%$&7;'r+'B'B"+$7*"^JK 7&+$7;%$&& "%= 7"++$)6M$#+.-z"+$&&N)$&:>-5$&&"%$&%$&&"+> 7],-&"+*&'x1D;4*"+f;7;U'B"%$O= 7;n!1D?)2;OE H 7 & 1D I "&1'B43e;!Uo^J;,-y"2?7;)1+2* "e1D)= 'BJK"+T+$. 6^# HC%$&"'["%1D;4*!$&"%2;:&> $y7rJK&+$))$&&* x+!2:& &T?O!= 1DE;SO2;6 JL"+$Z?@+FlFG"+, $9>SO2 1,T ,xO2;&Tx!H!7 14 I tU/ox+$& $y $&'P>L* $,-O2"+SN h;7 I ;)2:& _qHh++$"'<O2;&+$:&6 &$&7- 6^$gJ7 . GJK7 .N= gG"+,-$&&7 &Tw1D9JK"7N3g,-&T τ f7"'B"+Y X x (t) = x (t + τ ) ?U $&$e@FGFG", $9> &"+7&_P"w&_PT 80. >BS_q2<;O1N LR GEI?<1JLKNM%H-O$&*"+, Tw)$$&U7, $&7$&O= O2;&* ", h@, &N)2;" $"sO$O$&7;O, $&&7!.w 7w O2&Ta+!H!7&!14 I t#U ,' &S&7O1D&'P>O14;)1%JK_PNO6^z. 6Q!H!7 14 I 6 'B93gG7;+'(O$7N)$&:z, ,f&S&SO= #"+"X & $ O2&Tg H 7 & 1D I a;l%2;. *;Te%$&"' +z%2S&&Tg*;+$$O= &Tn7;++$7&T,-&TtU d

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#

+ $&" _qH?H-O7;, $"7%$&,C+$""+C!H!7&%$& 14O'BJK"T%$&. 6M#!H HNO= $& *"+,!H O$&&+""+7;O$&7&Udo^J 1h!HA1&O2; 6^$ & 7"JK"2 $: 7; I . *;%$&$O=mF )1D&& 14)H!N)$) > JK7 . "yJ; &+$7 . 6^$A14;)1%JK_PNO6^z. 6 !2 O2; . 6 +!H!7&!14 I 6zU * 1D'B;Oe*;%$&$O= FlO1&NO!H!7 14 I H-O$&* ", !Hn$&"+"7O$&O= 7& >-$)O,35"^, ,ee7;+'B$7" ;O 7;O""^&_P .-3]J";) !H!7& 1D I V> 2"+%= ,-eJK;+&+$7 ."+$&Z?5&'Bz:&6 $& p#"+ sH-7, $&"+7_qHE*;%$&7 $ 4 SNO1D&_qH !2s+7"J!Hs*+$$a14O'BJK"T%$&. 6M#!HxO$&&+""+7;O$&7& 6 Θ NU *c&+&&T &SO2+$Y !H!7& 1D I R_PO2 "+$Z7;"%$&& Θ = 1 U(7'B"a$& > Fl)= 1D&_PTh149H &O$'B43ge& 7"JK"2 $: z&'BO#:&60,-&@FGFG I "+ $) @FGFG", $&T JKFGF.-1D 7O1D+$& FlO1 B >,-$&7_PTEJLO"+$s I ", .x+, 7&+$?2"+TO= ]7+$)zJK "7+h7;)1D&%$&nFlO1G&5&7"+'B"VU *<2;. *;"G+$7&+&fFlO1D&& 149H &O$)e& JKO2;35"+hS&_d$:e7;O&"w .426z>K* $&g!14 el&_PO2 ""+'R.&%2;&& 4 U { 6%8U * *2" _qHe@+,-"+7'B"!$))H!1+=m1Dl7; *" &Tg$&*+$&g&_P*2"%= TEJ&S& $:&Z .42"&y1;4*"! , @FGFG I "+ $)s@FGFG", $&&T JK FlF.-1D 7;O1&%$&fF )1P"q.DJL"%$&ZV>NDJ;,-&7"%1D,- 4 ;X ",-O2;:&,-l &7N JK,-& 6.!'B"+:O= p5O"+$&Zh;57;O I "l&SO2KO+$&a149H &O$) U & JLlJ2; I ", %$&"+"n !H!7&+$&n+O2:1+. 6^$5H-7, $&"+7%$&,V> $& ;g2"J. 6M#"T X σ 4 U { 6 , ν= JK" x (t) x (t) zJKO$& _P" JK2(σ 'B*+"+σ,"M) "+7"'B"+_P"XO$&&+""+7;O$&7&> σ sJK+"7+C"7"+'B" _qH x (t) x (t) > σ sJK "7+ C!H σ +. 'B'B_U oX*"+& JK ν ∈ [0.5; 1] > 7 *"' ν = 1 +&$&"+$&%$&."+$ %2;. *;6 )2= &T +!H!7&!14 I t> y7 O2 &T "+1D&+'B&%$&Y 7 I "+ x (t) x (t) O2.!*"' U o^J;,-JK;O5HN7;, $"7 +$& ,!5"M1&O2; "+$ *"+$,-lJK;)= &+$7&N)ν$&=:e 0.5wFlO1&&&+]1D)H!NO$) >a14O;O1%J_PN6^z"Tw!H!7 14 I tU * 80.#7 "J;2KO"+$Z H-O7;, $"7%$&, >l1&O2; 6^#O[&S&;O7 .-3e&O$&:<" $&)2:&,-GO2 . 6 >#1D;O1JK_PN6Mz. 6R H 7 & 1D I 6<1D'BJK"T +$&O.!6M#!H H-O$&* ", !Hy++$"' X 4 U %{ 286 κ = min G(τ ), JK" G(τ ) eF. , I wJK&S t> ∆Φ

2 x1 +x2 2 2 x1 x2

1

2 x1

2

2 x2

1

2

2 x1 +x2

1

2

τ

2

G (τ ) =

2 x2 (t + τ ) − x1 (t) q

, x21 (t) x22 (t)

4 U ~f 6

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2

X$&&S_

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4 U ~ {[6

hx1 (t)x2 (t + τ )i − hx1 (t)ihx2 (t + τ )i . hx21 (t)i − hx1 (t)i2 hx22 (t + τ )i − hx2 (t + τ )i2

*q "!2H!7*& 1Dη =max S.DJK"%$#7;O&; "kJK I "u %2;. *;"^14;)1%JK_PNO6^#"+T R (τ ) I 4 t>)7;O1. 'B"+"+$&ZV>)2&T#+!H!7& 14 I 6 +$&7 "'B $Z],z .426 7n$"7"l,-O2"+SN 'B x (t) x (t) +$)O$+$*"+,-&Tx14 'B&!1DtU $&"+" :g+!H!7& &+$nH-O $&* ", !Hw, )2"SNOTa'B43e5 I "!$&:]a57;' = , )Hs+",!$&7;)2;:&&+wJ;H&JL U @+$&T I "%2:6 'B93g +O2:O1D&N)$&: F. , I 6 ,-&"+7"!$&&%$&: N| >k{&{D|& 4 U ~h~ 6 |S (ω)| r (ω) = p , S (ω)S (ω) JK" S , S u ", $7_ 'B# &+$zFX2;. , $.& I T x (t)− x x (t)− x S τ

x1 x2

1

2

x1 x2

x1 x2

x1

x2

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x1

x2

1

1

2

2

x1 x2

r ≡ 0 >9^M%2;. *;"B2 "TT#14O'B&O 1D x (t) x (t) 'B""+' r ≡ 1 U X$&&S_ &&"++$&n"14O&Z . 6 $f*;%$&$_ ,-O2* "+$&" . 6 H-7, $&"+7%$&, .> 'B43e A7;O'()$&7&O$&:R+7"JK ""E1;4*" "?,-&@FGFG I " $)Y,-&+"7"+ $& &+$ +%2"kJ.&"+'B&'R*;%$&$&'!$&"7 N)2" r U jG^7 U U {4vMu, 4*"++$&"d7'B"+7;u7 &"J"_ 14O& 'B&+$57;+'B$&7 "_qH &_dpz"HN7;,!$&"7 +$,y+$""+n!H!7&%$&nHN$*"+,!Hn,-O2"+SN TnO$];)= 7;'B"%$&7;C!1Dt>P O2;. *" _P"rJ;2<%$&"'B_ J.NH !14 _qH & I 22; $&7& +2"7;gl "S&O2;:p#&Tn*;%$&$&Tw7;%$&7T,-&YT X x1 x2

1

2

x1 x2

x˙ 1 = −y1 − z1 + γ(x2 − x1 ), y˙ 1 = x1 + αy1 , z˙1 = β + z1 (x1 − µ), x˙ 2 = −(1 − ∆)y2 − z2 + γ(x1 − x2 ), y˙ 2 = (1 − ∆)x2 + αy2 , z˙2 = β + z2 (x2 − µ),

4 U ~ |86

JK" ∆ ];7'B"+$7t> 1D9JL6M#Te7;%$&7T, .g;7 I )2:&_qHeO$&&+""+7;O$&7& γ ;7;'B"%$&7n!1DtU G,y'B43ge&-JK"+$: 157FG,-&>V 7"JK%$)O2"+_qHr7U U {9v!>K7;O1%2*!= _P"(H-7;O, $&"+7+$,fO=m7;O1D &'/.l7"D+7 . 6^$X;^"+7"+"*"+" 7 I l l t> 1

2

#

Θ(γ) 1.04

l1

1.03 1.02 1.01 1.00 0.99 0.00

0.05

0.10

0.15

γ

ν(γ)

κ(γ)

0.9 0.8

l2

l1

l1

l2

0.05

0.10

l1

l2

1.0

0.7 0.6

0.5

0.5 0.4 0.00

0.05

0.10

0.15

γ

0

γ

0.15

γ

0.15

η(γ)

r(γ)

1.0

0.8

0.8

0.6

0.6

l1

0.4

0.4

l2

0.2

0.2 0

0.05

0.10

0.15

γ

0

0.05

0.10

L 7 7 γ

. L @ α = 0.2 Θ = Ω 2 /Ω 1

= β µ

= 6.5 ∆

= 0.02

7

B ∆Φ κ

ν

# ( @

l , l

η

r

1

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0

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430!U 0A6

9

w+&&"+_P" 'B!2; $.DJLX5FlO1Du7 I "X&7"kJK"%26^$Z]&O$&&"%$&+$&" , , S&+O26M$&&"G1D9* ""lw7+. 'B"+ $e,-&'B 2",- &TyF. , I w(t) X 430!U $86 w(t) = R(t) exp[iΦ(t)]. W7'B"+ 5,-& I " I 6R;92 $& *"+, l+;)2KX , 4 0!U 6%>9O2.!* 'E2"J;.!6M#"+" \ J;2;w;)2; $&* ",-&+ ;)2 w(t)X

w˙ = α w −

β (3 R2 w + w 3 ) + ψ(t) + A exp(iΩt), 4

430!U vA6

JK" ψ(t) = ξ(t) + i η(t) nO;)2 $*"+,Tsp .!'P> η(t) a7 "&S&7O1D&&" 2:O= S&"+7 $) $ ξ(t) U X1y. 7;& " 4 0!U vA652"+,-?14OD)$&: \ J;2;R'B+&&"+_qH 'B!2; $.DJK_\aFlO1D_VX β R˙ = α R − R3 [1 + cos2 (φ + Ωt)] + A cos φ + ξ1 (t), 2 A β 1 φ˙ = −Ω − sin φ − R2 sin[2(φ + Ωt)] + ξ2 (t), R 4 R

JK" φ(t) = Φ(t)−Ω t "++$&:G'B+&&"+;Oz7;O1&+$:GF )1&U %$&&* ,5pz. '( &7 "JK"2;6M$&Zy2"J;.!6M#'Bh&_P7;O35"+ 'BYX ξ1 (t) = ξ(t) cos Φ + η(t) sin Φ, ξ2 (t) = η(t) cos Φ − ξ(t) sin Φ.

430!U 6 ξ1,2 (t)

430!U 286

* $&7&"q. 7;O&" "d 4 0!U 6%>)&+_PN6^z""P@+&O26 I 67;)1D&%$&]FlO1&>) HNJK n+$7 . , $. 7" ]$&&"%$&%$&. 6^z'[.!7&"+"'QJ2;x!H!7 1DNh+"%= "+7;O$&7; q* eJK"7wWO2; Xp .!'B'PUKoX$D2;*"G14, 26u*;O"+$&Zw5$&&'P> * $&f]@+$&' . 7;& "V>)"++$"+$&" >O$&+.-$+$&O.&"+$G* 2"t>9$7;O3]O6^# T ;92*"(7;O+$7&T!= ,f*;%$&$"U1q* 'B"++$&l"+l7%. $+$.&"%$5$&O2:, G*!2;" Ω U $&G)2;"%$&Zf"+z"

JK' Fl, $&7&'P>!&-JK"+$"%2:&%$&. 6M#'S#$&%.-$&+$&JK"+$"7'B 7&&&+ &7"+'B" &g'(+p $)SN]g++$"'B"U -0 &

- & / , & . , !tO& 'B&+$:#7;O1&+$FlO1^$#&7 "'B" t>!_P*%2;";)z7'B, )H ,-& I "+ I ]O;)2 $*"+,-&+)2K >97"kJK+$)OO2"GM7 LU 0!U |!> qJ;2; 7;O1%2*!= _qHa1;4*" Tw!$&" & &+$ypz. '( UjG,!2n, 7&_qHeJK"+$f7;O1&%$&:f'B"+35J. 'B &&" &TC*;+$$&TA7 I "+D x Ω =U * JK> * $&x+.-#"++$.&"%$ & $'()2:O= _PTr. 7&&"+: pz. '( >t7 x,-$&7&' 7;O1&%$&:nF )157;&a!.92;6zU 7&'B"z@+$&> J;2;5JL &Tf $&"+ &&%$&pz. '(lf1D9JL &TeO'B!2 $O.4J_R&1%J"T%$&& eFlO14 &, O1_PN"+$Zh1D)H!N4*" &Tw]$&"+*" "z&7 "'B" w;OSO26 JK"+ tU

@ : #

φ (t) 100

0.05

<ω>

80

D=1.05

1

0.04

60

2

40

0.03

20

φ( t )

<ω>

D=0.80

0

3

0.02

−20 −40

D=0.44

4

0.01

−60 −80 −100

0

100

200

300

400

500

t/T 0

t/T0 600

0.00

0.1

0.3

0.5

0.7

0.9

1.1

D

1.3

1.5

D

7

( 0 A = 3 # ) φ " ) #

3 K

: • .

( A = 0 A = 1 A = 2 A=3 α = 5 β = 1 Ω = 0.01

uo $&,!2;&" n$f& $& '()2:&].!7 && p .!'(57 &J $g,a7;O1&+$wFlO1G 4)2" 6\S&&"+]F )1&> *!$&]&"J"+$],a&%$&4 &'/. $D2 *&'/. O$5 .42;a;, 2O= .]14O& 'B&+$"T Φ(t) U !t& 'B&%$& 7 "JK H *;%$&$9> &7"kJK"%2; "'B_qH R FG&7'/.42" hωi = lim >$X $&"+ &&%$&gp . '(X7&"kJK" _A; dt 7O8U 0!U | > SMJ;2;z7;O1%2*_qHz1D;4*"+Tf'B!2; $.DJK_C+;)2K A UW7"kJK+$)OO2" _P" 7"%1.42:4$)O$&_Q; 2;-JK#+& JK"%$&"2:&+$. 6^$g&Sz@+FGFl"+, $&"^14)H!NO$"X7 "JK"+T *;%$&O= $&_ _qHNJK&++;)2K ω 7#"+, O$&&7's& $& '()2:&'h. 7&&" pz. '( LU $&$ @FGFG",!$5S&_q2a& "7&_P"M.&%$)&O2;"az7;OS&$&" J;2;2;. *;Oe+$&DHN+$*"+,-& $&7 +"7; 'B $$) U B+. C , 0 U | > S?!2 26u%$&77!.&"+$Y, ,<;)2* "y&7 &? H 7 & 1D I t>d$)O, s7;%p#7 ""]&S)2K+$s H 7 & 1D I xz7+$&'c'B 2 $.DJK_0'BJ.42; I tYU * $+.-$&%$&&"G&O1%JK"+T+$& 4 A = 06B7"kJK ! *;%$&$)#'B& $&& #7;%$&"%$5z+&$)= &"%$&+$&Ez14,-& &'c7;O'B"7 U W781D;4*"+ A ≥ 1 ;+7;FG,-"#14O& = 'B&%$& ω $ D 9 O2; "+$&Zr2KS&1D'B"+$_PTr 1D+S t 7 A = 2 %$)&&!$&Z *"%$&,-C1D'B"+$&T i+O2;&*, ;iO> JK" ω "y14 $C$ D U #)2:& "T pzT<7&+$ 'B!2; $.DJK_ A = 3 "JK"%$e,n7;%p#7"+6[1D& _ H 7 & 1D I tU G,?,!)1D_PN)2;&:y_dp#">&;9JK"+"*;O+$&O$r&7 &&43]JK"+$Z y,7T!= "+TC'B"+7" '()2_P'[* %2&'0S&"sFlO1U/W&@+$&'/.?7N J. e14)H!N)$&&'07"kJK"+T *;%$&$_t*!$& %2.!*T_PTx7 I " 1 T →∞ T

T d Φ(t) 0 dt

9

"+7",!2;6u*"+T x(t) @FGFG", $&n H 7 & 1&NE5&"+pz"TE"7 JK*"+,-&T 2&TtU O* $)"'R,-&@+FGFl I "!$@FGFG",!$&& &TaJK FlF.-1D YX Deff =

4 0!U { 6

1 d 2 hφ (t)i − hφ(t)i2 . 2 dt

$&"*"+$G14X, 4*"++$& +!H!7&!14 I YX-*"'x'B":Op#"P,-&@FGFG I " $ @FX= FG", $&T<JKFGF.-1DV> $"' SO2:pz"x 7JKO2;3g $&"2:&+$&:Y $&"+7N)2;&Y&7 "%= 'B" <1D)H!NO$) FlO1&U "%1.42:4$)O$_ *%2;"_qHR&_P*2" TQ7 "JK%$)O2"+_b; 7OU 0!U UNG,]& JKG 1M+7;FG,-&>N 7&%$&&' 'B!2!$.DJK_&O1%JK"+T+$& f14& != 'B&%$& D $X $&"+&+$5pz. '(u&+" S&)2"" *"+$,-^JK"'B& +$&7 7 . 6^$G;92*" 'B '/.!'(!U/O2* "+$&" _P"1D;4*"+ D y&S)2K+$? !H!7& 1D I +&!= JK"%$&"%2;:&+$. 6^$nf$&'P>K* $&g@FGFG", $g14)H!NO$)g*;+$$&_\wFlO1D_ 'B""+$'B"+$e; &7"+'B";9HV>N 7"&_dp#6^z!Hf"+7J "+pz"Tf!2;_YS&O2;""P*"+'E 10 7O1&U ,' &S&7O1D&'P>;)2* "P&_P .-35JK" &Tg@+FGFl"+, $& &&Tf H 7 & 1D I f;& &&&' $&"l.+$) &O2"+]J;2wS+$)S!2:&f& I 22; $&7; +U $$e@+FGFl"+, $g74O2!= "+$Z< 149H &O$&"y+7"J"T<*;%$&$_ "7"+,!26u* "T<& "+p# ' 7'B& *"+,' ;)2;&' 'B'B 1D I a,-&@+FlFG I " $) JK FlF.-1D t?U *dO3g $&'B"%$& $:>;* $& JK&S&O2"+" pz. '(ll +$&"+'/.f7&JK!$z,]. &7N JK*N 6Q"+" F )1D&&T5JK ;)= 'B, X. &"2*"+" $"+&+$& p .!'(w&_d1D_PNO"+$y. 'B" :p#"+"fJK FlF.-1D 7;O1&%$&hFlO1U Deff

eff

eff

3

0

10

Deff

−1

10

−2

Deff

10

−3

10

A=3.0 A=2.0 A=1.0

−4

10

−5

10 10

−6

0.0

0.2

0.4

0.6

0.8

1.0

D

1.2

1.4

1.6

1.8

D

α = 5 β = 1 Ω0 = 0.01

K9

%" &% "(')' H 7 $ 8K832

9

$ &&*

*#8:/&, 2 * 8 "D

B

2@'=*

$%"(,.-0/ " $71324 $"'68K-%$75#5 3-02

O'BO$&7' 7"+1+.92;:9$))$&_ @,- "7'B"+ $))2:&Y2"JKN Q J;. I 7O= N &T p . 'B&'y !H!7& 1D I 5M+$DH-%$&*"+,-&'r$&7 "+7" 'B!$$)XJzJ"T!= +$&"+' 2KS&&+ "7 JK*"+,-&C ;)2 U 7+""+7 'B $$) 7"kJK+$))2;"%$ S&&T JK+$)O$&*R7 &+$&"?@%2;", $7&" .&%$&7&T +$& x149JK_P'`7&' 7SNO$_PN K U B%2 ;)2C 7"&_dp#"+$E&7 &&&"g1;4*" " x(t) > K > $&7 +"7a"7 "%HNJK!$] +&+$9 " x(t) ≡ 1 U W7 x(t) 6 −K > $7+"7 "+7", 26M= *;"%$&Z]X+&+$4" x(t) ≡ −1 U ,'yS&7;O1&'P>O DJ5J"T%$&&"+'E"+pz"+Gp .N= 'B&&z+)2K >!'B&"_P"^1D;4*"+ ,-$&7&+#S.DJ.-$5%2.!*T_P' &S&7O1D&' 7"+&_dp5)$&:lG'BDJ;.92;6&7P7;S&O$&_PNO t>+;)2g;&_qHNJK"d$7+"7 S.N= JK"%$G7"J+$)O2 $&:lS&&T#2;. *;T_PT5$"%2"+7;FG_PTf7 I "+>&7 "JK%$)O2"+_PT ;57O U 0!U {&> S U hO$"'(O$*"+, . 60'BDJ"%2:zJK'B, $&7 +"7;fJ JK"T%$&& "' pz. '(]a7'B& *"+, f+;)2K]'B43e5 7"JK%$)& $:e2"J. 6M#'S&7;O1&' X 430!U {&[{ 6 x(t) = sgn[Kx(t) − Dξ(t) − A sin(Ωt)], JK" K 7&z+7;SN)$&_PN w$7+"7 > D a $"+&+$&: p .!'( ξ(t) > A 'B!2; $.DJL]w*+$$)g"+pz"Tw7'B& *"+, Ty!2;_U Ω "%1.42:4$)O$_ @,-+"7 'B" $))2;:&_qHE 1D'B"+7" Try*2" _P"57;* "+$&_07 "J;= "+T?*;%$&$_ "7"+,!26u* "T n$&7+"+7" 'B $O$)r 7&"kJK"_ w7;S&$ " U ' *CJLN92Zgp .N= jGl$7+"7 'B!$$)#u &7&+&'?7SNO$_PN 'B&&TY ;)2Rh*;%$&$&TR7 "+14 f = 100 ?, I ∆UY="7150 DJ*"+,T< ;)2R *;%$&$&T f = 100 I U W&_qH&DJ&'/.s+;92;.&>t 7"JK%$)O2; 6^z"'/.rS&&Ty$&"2"+7;FG_PTx7 I " I Fl7 &&_PN"+'B&'/.f#,-&'B:&6M$&"+7">; 1'B"7N!2:5+7"JK! *;O+$&O$)] "7"+,!26M= *"+T f = lim U "%1.42:4$)O$_ 1'B"7"+Ty &, O14 _ π 1 P ;57ONU 0!U 0!> !U;W7w2KS&M'A+;9t2"14−& t'B&%$&:f7"kJK"+Th*;%$&$_\$] $"!= &&%$&Yp .!'(r%2;"J."+$?1D,-& . ^77" .& > @+,-& " I )2:& &1D7+$)OC 7&%$&&' $"+&&%$&Yp . '( U . &"2*"+"+' 'B!2!$.DJK_ +;92Kx@%$)r1D)= &+'B&%$&:]+$) && $Za,!4*"++$&" f &YT X!4O2; "+$Z &SO2KO+$&:#1;4*" Ta!= $&"+ &&%$&Yp .!'(!>Bx,-$&7&T +7"JK! C*;%$&$)x7;, $*"+,A" 'B" "+$Z e7 &+$&'cpz. '(hEh7 "JK"2K)HE&+7"+pz&+$ @,-"+7'B"+ $)s&+$)"%$&Z 7;O&&T *;%$&$"G+;)2K U jGSO26 JLO"+$&Zn@+FlFG"+, $514)H!N)$)#7"kJK"Tn*;%$&$_Q "7"+,!26M= *"+TtU W&$&7N f 1D'B"+7" +7"JK "T*;O+$&O$&_AJ;2;f7;)1D_qHf1D9* "T'B!2 $O.DJK_ *;O+$&O$&_ +;)2K!> 'B43g )2;. * $&: SO2K%$&+!H!7&!14 I R;E!2;&,-O= +$Q7;'B"%$&7& $&"+&+$:Ap .!'( Y'B 2 $.DJL ;)2 k>d ,-$&7_qH c

0

M →∞

M k=1

k+1

k

9

3.0

<ω>

A=0.01 A=0.05 A=0.1 A=0.15

2.5

<ω>

2.0

1.5

1.0

0.5

0.0 0.02

0.04

0.06 D

0.08

D0.10

: K K

: )

A =0 4 A = 60 2 L A = 100 ? & K = 0.2 K Ω = 0.5

7 "JK !n*;+$$)]7;&;#*+$$&"l+;)2K!UoXSO2K%$&n!H!7 14 I t>; &'B!= ;6M#"w 1D_P, u7 O2:)JL!> &, O14 _ E7OU 0 U $ U G,A& JKE 1y7 +. , > +.-z"+$.&"%$f&7&+&&&"d1;4*" "X'B 2 $.DJK_R+)2K A >N;4*;)g ,-$&7O= e;SO26 JK"+$Zr@FGFG", $ 1D)H!NO$)e+7"JK "Tr*;%$&$_U W7nJK&%$& 3#"r7O= G"7 DJ*"+,T +;92;4* ;"+$z@FGFG", $&l. 7;O2 $&:l+$DH-%$&*"+,-&T th

L 7 0 f = 100 ? f = 250 4 f = 500 2

0

0

0

#0

7

9

JK'B,-&Tx"7"HNJK& U 7 &+$&'Q*+$$&_ +)2K @FGFG", $a+!H!7& 14 I .NH .4J;p5"%$&ZY XNO= "+7&_qHL>)+.-3]6M$&Z]&SO2+$# H 7 & 1D I t>&O= O$&&7_qHL>9. &"%= 2* N6M$&Zw&7&+&&_P"G1D;4*"+ w'B!2; $.DJK_\+;92K 4 'PU;7 UN0!U $86%U jG 7OU 0!U v &, O1D;?!2;$&+$&: &"+74 $&+$Q7"'B"+R7"S_PN p(t) $&7 +"7;ryJK&'[ 1'B"+$)%$)S&!2:_qHC&%$&4 T 7 7;O1%2* _qH . 7&& H pz. '( UtWG7 y%2KOS&&'Rpz. 'B " 4 V = 35 ' * 6 p(t) JK"+7-3g $a"+,-O2:, a , ; &7"+'B";9HV>,7;)$&_qHEO2;&&"f"+7JLw+)2K U *0SO2K%$& +!H!7& 14 I 4 ' * 6+7"J""7"'/x;)H&43]JK"+ x$&7 "+7;ynDJ&' 1g+$&4 T V&;=9JLO70"+$5^O2&&T "7JL#e!2$&+$:]&"74!$&&%$&a&7"+'B")HN435JK"%= C+DJ"7N3e!$x 7,-E_P7;O35"+_PTC, 7 t = T /2 U/WG7 CSO2:pz&'0p . 'B" 4 ",!2'6^* *"6 > &t ">'B&SO2K&Oh+$&'B "+:!pzH!"e7& ) 214& &I t_ >B7 ""7JK D"+JK" @U &!t7"+a'/t"+> 7 "&JASDHNJK% $&'B"+'(&" J;V2; ="7 115 .&+"&"+$a'B&,7O$& "7"+,!26u* $&:&Zx 1zJK&+&%$&4 s]J7 . &"#hO= )2;"%$&Za,V>&O$&&"%$&+$. 6^zTh'()2;_P'C7"'B"+;' "7 ",!26^*" TtUKW,t> +O= N

N

0

N

@ K A = 100 K f = 100 0

&

9

$&"+$+$&O.!6M#TCO2;&&"f"+7JLw+)2K >7;O1'B_PN"%$&Z x,-&"+7" $&%$&: &_qH&DJ&g+;)2K]7;O17 .-p5"%$&ZVU W7&&"kJK"'n7&"+"(O2;. *"+_qHl@+, +"7 'B" $)92:&_qHz7"+1+.42:4$)O$&&^JL!= _P'B *%2;"X'BJK"27&& VU @+$&T I "2:&6C&&+O2:1+.&"+'BZz. 7;& ""+' 'BJK"%2; 4 0!U {[{ 6%> 149JLOf2"J;.!6M#"l1D;4*"+ n;7'B"+$7&/ X > &_PS&"= 7"+' z& J"u@,-+&"+ I ;)2;:&5,-&77"27Nzp .!'( uK&7"+='B"0.2 "+' ξ(t) ,-&77"2;!= I τ = 10 !$&" & &+$:&6 D >!& _PNO"'B_PTa7 I "+&'AoX7 pz$"T \/2"+"%S&1".4, 2 :4U$)O$_c7;+*"+$&57 "JK"+T *;%$&$_Q, ,gFG. , $"+&+$&apz.N= I '( >B7 &&"J"_qHAr. 7;& "6 430!U {&[{ 6>/&, O1D_ x7 U 0!U 0!> Sr JK"+'B&!= +$77 . 6M$5&&"kJK" ">- JK"+ $& *&"M 1D'B"+7" &'/.fG@,-"+7'B"+ $&" 4 7O1U 0!U 0 > 8 6U jGw7 =U 0 U > w &, O14h@&)26 I !?'B &&" &T?7;)1D&%$& FlO1 φ $7+"7;y OH&DJ&C ;)2?J;2<%2;. *;O A = 0.1 UPW7<& $'()2:& &' . 7&"spz. '( 4 D = 0.066 7;)1D&%$&:]FlO1X+$)"%$&Zw&+$4Tn#$"*"+"G&7"+'B"a;OSO26 JK"= V%U *q"(&S)2K+$G!H!7 14 I , 4 D = 0.04 > D = 0.086K, 4*,lF )1D_x&7;)1%JK S&O2;""G*;%$&_P">K* $e7 &JK $g,wS&_P%$&7TaJKFGF.-1DwFlO1D_Qt>;%2"kJK&N)$&"%2;:&> ,n$&%. $+$&6 +!H!7&!14 I tU * %2;. *;O"n'(92&T 'B!2; $.DJK_ +)2K 4 6 )2" "14)H!NO$)r7 "J;= "+T *+$$&_ $&%. $+$.&"%$9U "%1.42:4$)O$_ 7;A* "+≤$) A'B&"T 7O1D+$&
c

th

2

10

60

φ

Deff

D=0.07 40

0

10

20 −2

10 D eff

φ

D=0.06 0

−4

10

−20

D=0.05 A=0.1 A=0.15 A=0.05

−6

10

−40

−60

−8

0

1000

2000 t

t

10

0.02

0.04

0.06 D

0.08

D0.10

# & &

& K 7 A.= 0.1

Ω = 0.5

) D eff

K = 0.2 Ω = 0.5

"%1.42:4$)O$_

;7 +*"+$)y,-&@+FGFl I "!$)hJFGFG.-1?7;O1&%$& FlO1 &, O14 _ ;z7 UP0!U > S U !tO& 'B&+$: D (D) H-7;O, $&"+7 1."+$Zy'B '/.!'B'P>!* $z)2;!= "+$Zh2"J+$&"'R@+FlFG"+, $)g%$&DH-%$&* ",-&Tr+!H!7&!14 I tU WO2;. *"+_P"eJL _P" & J"+$&"2:&%$&. 6^$ y$&&'P>* $&r& "+p# T "7 DJ*"%= , Ts+;)2yJK+$)O$&*&TrO'B!2 $O.4J_ +!H!7&!1D7 ."+$n+$DH-%$&*"+, . 6 JK!= ;'B , .#"+7"%H&JK&U $O2"+"P+&7&935JL"%$&Z51D)H!NO$&'x'B+&&"+&T]7;)1+= &%$&wF )1 a+7"JK "Ty*;O+$&O$&_U eff

* ,

"%&% "(')' $ 132 $0H"02@'6* 8 /&, 29*#8 $7D&&* B 2@' $ "" ,.-7/ "%$71 24%$"' I"-0/4&&*#* / I%3-& B ? D&#" $ 29/8 " $ D&&* B $", * $0* 8 &HJ2,

G ,rS&_q2a$'B"*"+h& &"JK"+t>& 7"JK"2" _PT ,!2#H-O$&* ", !HE!= +$"'R]7"%1.42:4$)O$&"z,7 1Df7"D)2; 1."+$ 7"%3e 'Y"+"7 DJ*"+,!Hh"7 ",!26^*"%= T5'B"%3]J;. _ JK"%2"+_P'B]JK'B435"++$N'BzH-O$&* ",-&+z)$$&7, $&7; UA*E!= 2;.h"%2 "T &+$ +$&"+'B_ *;%$&$) @%$&!Hr "7"+,!26u* "TE,-& $&7 O27 ."+$Z?" &"%p# '[p . 'B&'P>/a.! 7;O2; 6^z'[;7;'B"%$&7'PU G,? &, O14)2;?2"J&N)= V>;f$)O,!Hn +$&"+'()Hw&1'B93g57")2 1&NO$:e7"%3e 'AFlO1&&&Ta+$DH-%$&*"= ,-&Tg H 7 & 1D I tU l2;]!22;6u+$7; I 7;+'B$7'?& "+p# 6u6 g1D'B . 6 !H!7& 1D I n7 I "+&5"+7", 26u*"+Tn#7N JK"u%$&"'CMJK"%$&"+7'B7 &N!= &T H-O$&* ",-&TaJK;'B , 7 T 82 >k{ ~ &| U )xt* & - oXS&7;O$'BZn,wO$&&S&7O35" 6 430 U { ~ 6 x = (ax − x ) exp(−x /b) + A sin(Ωn) z7 "+3g'B"^ "7"+'B"+3]"+'B&+$xiH-O& H-Ni+7 a > a = 2.839 . . . +'B$&7 ' JK'B, .w%$&"+'B_ g$&%. $+$&"5&"%p#"+p .!'(5J;2;nJ&+$)O$&* SO2:pz!H 1D9* "T'B!2 $O.DJK_ &"%p#"+TY+!2_ A U(WG7 C@%$&&' 7 I "+n"7 ",!26^*" T 7 I )2:&x"2"+"t>("+r+$)O$+$, S.DJK"+$x%. z"%$&&"+ 1D&+"+$&:x$ ;7'B"+$7; a U )2:1.Oe'B"%$&JeJ.NHf&%$&4 Tt>!_P*%2;'?@O26 I 6Q 2$O= &%$&s&7 "'B"h7"S_PN p(τ ) e$JK"2:&e1!$&&Ty $&"+ I ;)2;:&&Ty 'B" 7 N7 ; I e;7;'B"%$&7; a U "+1+.92;:9$))$&_c 7"JK%$)O2"+_c 7O/U 0 U 2le, 4*"%$&&"+!= a&O$&&7N6M$aJK_P"O> O2;. *"+_P"]J2;s$&7 "+7; 'B $$)Z 4 7O]U 0!U Lv 6%UtW7 & $'()2:&'R1D;4*"+ a = 8.34 7;O7"kJK"%2;""]'B"+"+$n"J+$&" _PTED.%= &S&7;O1_PT , ]'(,- '/.!'B' 7 > * $h+&$&"+$&%$&."+$r7;O&"%$&. 7 "JK"+T]*;+$$&_C"+7",!2;6u*"+T]'B"+35Jτ.=H-0.5 O$&*"+, 'Bg)$$&7, $&7;'B]*;%$&O= $&"l&"%p#"+Th+!2_U qFGFG", $ 1D)H!NO$) 7"kJK"+TQ*;O+$&O$&_b "7"+,!26u* "T &"%p# ' +;)2' J;2; 7O1+2 *_qH 1;4*" T ;O7;'B"+$7; A !226^+$77 ."+$C7O U 0!U { > !U n+1

n

3 n

2 n

∗

9 0.04 p (τ) 0.03

a = 8.14

0.02 0.01 0.00

0

1

2

3

4

τ

5

0.06

p (τ)

a = 8.34

0.04

0.02

0.00

0

1

2

3

4

τ

5

0.06

p (τ)

a = 8.48

0.04

0.02

0.00

0

1

2

3

4

τ

5

9 K

@ " 0 0 7 . a L τ

.

7 &%$&&'Q'B!2; $.DJK_0+;)2K1%JK"+T+$& sSO2K%$&: !H!7& 1D I r. &"%2;*!= N"%$&ZV>P,!O,RR%2"kJK&N)2; 93g JL)$&:U 1r7 U 0!U { > S JK> * $& JLO;OY&SO= 2K%$&:w7"J+$)O2"+$h&S&Ts$& * . 6 1& .s+!H!7&!14 I t>, ,rxa%2;. *;" $&7 +"7; 'B $O$)7 4 7WU 0!U 0L6%ULlO,s& JK !157O(U 0!U { > S>L 'B""+$a'B"++$n &7& !H!7& 1D I V#U ,' &S&7O1D&'P> @FGFG", $f&"%p#"+Tw+!H!7& 14 I h*;O+$&O$&_ "+7",!2;6u*"+T eJK"+$"7'B 7&&&TxH-$*"+,-&T +$&"+'B"g. &"+7" h7"+!= +$77 ."+$Z &, O1_PN"+$ZC,!4*"++$&" @, &N)2;" $_P' @+FlFG"+, $. J. I != 7&&&Tapz. 'B&'A+$4HN+$*"+,-&Tr H 7 & 1D I tU qFGFG", $g14)H!NO$)g7"kJK"Ts*+$$&_["+7",!2;6u*"+TsO2"+$Zh. "7)2:O= _P' 7 9 O2; "+$]"+S4zpz7&,-&' ,!2KO"^J;'B* ", !H ++$"'CX7"+3g'B&' "+7"'B"%3]"'B+$&VU +%2"kJ.&"+'?G,!4*"++$&" 7'B"+7;l "%2 "T . 6 %$&"+'/. .& > 7")2 1+.!6Mz. 6[7"+3g'EJK ;'B*"+,-&Ta"+7"'B"%3]"'B+$& { ~ L U G"+: X.N#!= _P&"+$Zy +$&"+'B&Tw. 7;O&" YT X

<Ω> 0.03 <Ω>

0.10

AA 0.08

0.02 0.06

0.04

0.01

A = 0.01

Ω=0.0153

A=0.01

2.82

Ω=0.0153

0.00 2.76

2.78

A=0.005

A=0.025 A = 0.025 0.00 2.81

0.02

A = 0.005

2.83

2.84

2.85

a

a2.86

2.80

2.82

2.84

a2.86

a

9 L " # L 7 9 9 a9

430 U {4|A6

x˙ = α[y − h(x)], y˙ = x − y + z, z˙ = −βy + F (t),

JK" h(x) = m x+0.5(m −m )(|x+1|−|x−1|) X,!.&*O= 2"+T;OGH-7, $&"+7!= +$, n++$"'B_> ;O7;'B"+$7_ ,-$&7&TrFG,- 7&N _VX m = −1/7, m = 2/7 "+pz n"+7JK*"+, Oh!2K!U F (t) = A cos (Ωt) G,s&, O1Daa7;S&O$& " { ~f >V7 I "+]"7"+,!26^*" T H-O$&*"+,-&TES&!= +$)OS&!2:& &T<%$&"'B" 4 0!U {D8| 6f'B435"%$ S&_d$&: -J. I 7&&pz. 'B&' 2S& 'B"+$: *%$&yJK;O'B*"+, . 6b7 7J.&U(W7 'B"' 'B"+$DJCJK.NH +&+$4TVU !tFG,!= 7 .&"+' 1D;4*"+"h7;'B"%$&7; β = 14.286 U z;O'B, x++$"'B_ 430!U {48| 6zSO.4J"+$ 14O&"%$&:s$y&"%2 *_ ;7'B"+$7; α >w$),!35"$y'B!2 $O.DJK_ A ?*+$$&_ "+pz"+e ;)2 U *<$&%. $+$&"#&O1%JK"+T+$& (A = 0) 7"+3g'C"7"+'B"+3]9= Ω "'B+$&Y7"D)2; 1."+$&Z<x +$&"+'B: " 430 U {4A| 65 7 α ≈ 8.8 U l2; α > 8.8 7 "JK ! *;%$&$)a"+7",!2;6u*"+T 4 ;)2;&#*;%$&$_ 7;'B"+7DA 6^'B&$& 1D7;O+$)"%$ . &"%2;*" "' α U *q_PS"7"+' 1D9* ""w'B!2; $.DJK_ A = 0.1 > ,-& JKx7", I ! +$&"+'B_\;]&"+pz""l&1JK"T +$&" 7 I ;)2:e"2"+T; U jGn7 U 0!U {&{g 7"JK%$)O2"+_ 7"+1+.42:4$)O$&_ 7;+*"+$)h 2$& &+$ 7; 7"JK"2"%= &7"'B"+a&1D7;O$) p(τ ) J2; 7;O1%2* _qH 1D;4*"+Tn;7;O'B"+$&7 α { ~ 0. U jG)= SO26 JK"+$ZY.DJK&!$&"%2;:&&"n HNJK%$&&EJK_P'Bt>(7 &"J"_P'BA;r7 U 0 U v J;2;g$&7 "+7; 'B $O$)]g;z7 PU 0!U 2!U!W7e&7"kJK"%2"+&' 1;4*" ;7;'B"%$O= 7; α = 8.8325 7;O7"kJK"%2;"" p(τ ) 'B""%$w 7;, $*"+,x"JK +$&"_PTED.&%= 1

0

1

0

1

9 p (τ)

p (τ)

p (τ)

α = 8.8, Ω = 0.03

τ α = 8.8175, Ω = 0.045

τ α = 8.822, Ω = 0.05

τ

. ( 0

: α : 3 Ω τ

T = 2π/Ω

&S&7;O1_PTn,nl'(O, +'/. 'B&'YSO2 1 τ = 1 >* $&f$&&"%$&%$&."+$a7;"%$&. 7 "JK"+Th*;O+$&O$&_ "+7", 26u*"+Tya*;O+$&O$&_ & "+p# "Th+!2_U " +$&7 7 .&"%$&Z @FGFG",!$ &_P .-35JK" &Ts H 7 & 1D I t>L7 O2; 6^# TZsg1D)H!NO$"#7"kJK"+T *;%$&$_\"7 ",!26^*" Ty"+7JK*"+, 'R+;)2&'PU jGs7 U 0 U { ~ 7 "JK%$)O2"+_ &SO2K%$&C H 7 & 1D I t> O2.!* "_P"wr7"%= 1.42:4$)O$"?FG 1D *"+, @+, +"7 'B" $)!>X,-$&&7 _P" , 4*"%$&&"+ ;&'B ;6^$ i%!1D_P, u7 O2:)JK;iO>V, ,ss 2;. *;"#$7+"7; 'B $O$)7 4 7O(U 0!U $86%UVou$D2* +$&4!$ 2 pz: G$&'P>N* $&l 7&+$&'?*;O+$&O$&_+;92Kl. 7&":l&7l+!H!7O= 1D I r1%J":n7;O, $&* ", "] 1'B" "+$&ZV> pz7n&S)2K+$"Tr !H!7&!= 14 I e. &"2*&"+$ZtU $ $D2*!&S.%2O2" _ "%2 "T_P'Ba+&&T%$&N'B +%2"kJ.&"+'B&Ts%$&"+'B_ 4 0!U {48| 6qh " , 6^$ZyDO'B&Tw%.-#&%$&s;OSO26 JL"+'B& @FGFG",!$) YU 'q)2""XJK"+$)92:&_P"57;O*"%$&_[&-JK"+$"%2:&%$&. 6M$w]$&'P>L* $@FGFG",!$. !H!7& 1D I E7"kJK"+TE*;O+$&O$&_ "+7", 26u*"+TE %$&"'(9 H 430!U { ~ 6^ 4 0!U {48| 6 $&"*;O"+$f@FGFG", $]FlO1&&&T +!H!7& 14 I t>;O2;&+$:&6\@, &N)2;" $_PTh7;O%=

#

a

Ω Ω Ω

α

: 9 9 Ω

'BO$&7"+_P'2;. *;O' $&7 +"7; 'B $O$)g "+7"JK"+'BFG7&&&+g I !2 2;!= $&7; U - , ! &

- *0, 4*"%$&&"e7'B"+7;nO14'B &TE+!H!7& 14)= I 7 I "+5"7"+,!26^*" Tn7+'B$&7 ' JK ;'B, .g +$&"+'B_RJ.NH+ 1D!= _qHn'BJK"2"T 7" I { ~ $& X

*q_PS&"+7"'

x˙1 y˙1 z˙1 x˙2 y˙2 z˙2

= σ(y1 − x − 1) + γ(x2 − x1 ), = r 1 x1 − x 1 z 1 − y 1 , = x1 y1 − z1 b, = σ(y2 − x2 ) + γ(x1 − x2 ), = r 2 x − 2 − x 2 z2 − y 2 , = x2 y2 − z2 b.

4 0!U { 6

1 ;4*" ;7'B"+$7& σ = 10 > r = 28.8 > r = 28 > b = 8/3 > &O$O= &"%$&+$. 6^z" 2;. *;6 >!,-& JL# , O3]J&T 1uJK%$&"+'A%. z"%$&."+$eO$$7;, $&7 l&7"+ I { ~ v. U+ $&$ O$$7;, $&7re;7 I ;)2:&Ty++$"'B"5'B43e g7;O'()$&7!= NO$:s, , &S&&SO#" _PT S&%$)S&!2:_PT & I 22; $&7t>/h, O$&&7'0 'B""+$r'B"++$ "+7"%.927_PTA7 I "a"+7", 26u*"+T+s7"kJK"+TY*;%$&$&Tt>B,-& $7O2 7 .&"= 'B&TC.! 7;O2; 6^z' ;7;'B"%$&7' r { ~f U *q"JK"+"y+ 1 (γ > 0) JO2;3g &_d1_PNO$&: 1'B""+ R+7"J!HR*+$$ "7 ",!26^*" TQ ,!)3]JKTY 1s DJ!= +$"'n7 &JK $&:g,a@FGFG", $O.wO14'B &Tw+!H!7& 14 I tU 1

2

9 1.8 1.6 1.4 1.2

1

1

1.0

0.8

22

0.6 0.4 0.2 0.0 0.0

@

&

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

γ

γ10.0

˙f ¸ ˙f ¸ ("

1 7 2 σ = 10 r =

# 7

28.8 r2 = 28 b = 8/3

1

G ,]&, O1D)2f7;* "+$&_>-7 &&"J"_P"^G7SO2; 35" 5J;'B, ]JK.NH]+O= +$4TV>/7 1;4*" !H ,-&@FGFG I "+ $)s!1D γ > 5 7"kJK" *+$$&_ "= 7"+,!26u* "T f f 7, $& *"+,R&;9JL6M$ 4 7OU 0!U {D|86%U 'q)2""a$&&+> *2" &"'BJK"27&&"5JK;'B , 'B+&&"+&T?7O1D+$& F )1e7 I "+& J$&"7JK 2#@FGFG",!$z1D'BTg+!H!7&!14 I Y X! 7 γ > 5 7;O1+= x (t) x (t) &%$&:nFlO1]SO2 1, n,r .426 ; &7 "'B")HL> +.-#"++$&"h7"_dp56M#!HE7 "J;= "+"#&7"+'/h"+7", 26u*"+T { ~ $& U ,'Y&S&7;O1&'P>7y+!2:& &TyO 1Dh'B"+35J. JK. '/aH-O$&*"+, 'BsS&+$)OS&!2:& _P'By& I !22;!$&&7;O'By'B""%$'B"++$&g@FGFG", $ 1D'B&Tn !H!7& 1D I h*;+$$g"7"+,!26^*" TtU 1

1

2

2

8:/, 29*#8 $7D%&* B $%"LF)4 $7-7/ 92 "%"7E

$%"",)-0/ "%$"1324%$(' F.H /9H I0/- ' / B

2@'=* %

B

2 B

oMJK&Tg 1^NO3g_qH5H-7, $&"+7%$&,a@FGFG",!$)z H 7 & 1D I O2;"%$&ZeO= & _dpz""x+$""+<.! &7N JK&* N!<+;)2K 7"+1+.42:4$)O$&"x H 7 & 1D I tU z"T%$&O$&"%2;:&> . 3#"w;x, 9* "+$&" &' . 7&&"n'B43e E I " $:> ;7 'B"7t> @FGFG",!$E1D)H!NO$)s*;%$&$_ 7C H 7 & 1D I A"7 DJ*"+,!HY$&,-O2"+SN)= Ta& "+p# 'AO7'B&* ", 'Y+;92&'PU z5;%$. !2"+ n@FGFG",!$)f+!H!7O= 1D I n5 +$&"+'B" SO26 JLO6^$&Z JK.NH!*;%$&$_P"z,-O2;"SN t>S&7;O1&' , O$&O= 7_qHEO2"+$Z JK. 'B"7 _PT $&&7tU * 7"%3e 'B" H 7 & 1D I 7")2 1+.&"+$ZC7"%=

9

3g'0.+$&T*_qHC"7 DJ*"+,!HA,-O2"+SNTV>B, O$&&7'/. $&"*;"%$ 7"JK"2:O= _PT I ,!2UtW7"kJK"%2;:&_PT I ,!2x 'B""+$a'B"%$&7 *"+, .s7;O1'B"7 &+$:n7& . 6 "kJK!= I ">&P$&7 uJK&T,-"U $&"+"5. &7N-JK&*"+&%$&] JLO&'y2;. *;"d'B43gMO= +$)O& $&:sn$&&"%$&%$&&". 'B"+:p#"+"7O1D'B"+7&%$& 7"kJK"%2;:&&+y'B435"%$&N 5FlO1D&&'?7 &+$7;%$&&"G +$&"+'B_>;,-$&&7 &"&S.%2&)2"5@+FGFl"+, $&'A+!H!7O= 1D I V=U $&O$s35"e@+FlFG"+, $s'B"+"+$r'B"%$&s? 7 H 7 & 1D I ,N)1D"+7!= JK*"+, H xHN$*"+,!HCO$&&,-O2;"SN TtU $&DH-%$&* ", OA !H!7& 1D I $), 3#"PJK)2;3el7 &JK $:z,].!"%2* "6Q+$""+e. &7N J&*" &+$ 7"+3g'( ,-O2"+SNTVU l2;nJ$&"7N35JK"!h$),-&+e&_P&JL]&S&7O$&'BZh,w@+ $&7 &y, , ; S&O2"+"g&SO#"TsH-O7;, $"7%$&,-" %$&" "?2;. *;T &+$ 7 I "+ U #"+T+$&!= $&"2:& >M%$&DH-O+$& *"+,!)\ H 7 & 1D I c'B""%$Y'B"++$& >M, , S&_q2 ,!)14 &_dpz">J2;y2;. *;T_qHy7 I "+& "+7",!2;6u*"+TtU B2s7&"J2&e.-$&"7!= 35JK" "hx&&_dpz"R%$&"+"R &7N JK, 7 Y !H!7& 1D I V>q$&E@ $7& 7 I "+D 7 A$+.-$&%$&& @+FGFl"+, $) %$&DH-%$&* ",-&T<+!H!7&!14 I AJKO2;3]= ;zS_d$&:#S&)2:p#"uz&"2* "X@!$&7& a7 I "+#z7"%3e'B"^ H 7 & 1D I tU q $7& n, , 'B"+7;#+$""+w2;. *;T &+$w7 I "+D JKO23e;l.!'B"+:p#O$&:&Zn 7"%1.42:4$)O$&"@+FGFl"+, $) +!H!7& 14 I tUW&, O3#"'P>N* $ @%$&GJ"T%$&& $"%2:#$),tU w"+7&Tt>4& _PNO6^#"+Tz"+7"JK9*!. 2;. *;T &^+;)2K *"7 "+1 S&+$)OS&!2:&!.!6 +$&"+'/.&>/'B435"%$y2;.-3e $:y ", $7?.&%2;&&_qHE@!$&7& T " &; U "7;O7NH! .&2&& _qHh@+ $&7 &T ";fH-O7;, $"7 1+.&"%$a, 77"%2 I h&"Hn&_P+pz!HwO= 7N JK,-&fnf 7"JK"2"z2;.-3e!$g'B"+7&Ta. &7N JK*" &+$y7 I " U u)2 1l5$&"+7'B;)Ha$"&7 h Fl7'( I $7"S."+$ &"JK"+ a+'B&O2:& )2Fl $) >&O$&&"%$&+$. 6^z"r%$&DH-%$&* ",-&T JK ;'B,-"e++$"'B_U l2;xS&!= +$)OS&!2:& _qH?%$&DH-%$&* ", !H +$&"+'["+$"%$&&"+_P'[O2; "+$Z?S&7_PT )2= Fl& $9> &%$&4 zTw!1^JK.NH +'B&O2;&> 7'B"+7t>i i%riO{-iO>!,-$&&7 _P"$&&"+*;)= a6^$lJK&;&9 HN*43];OJr"6<JK +&%+2$"k"JK'B&_N)$&2"%"2;N:&G&!%2$&]:h+ 27;rN%2&$lSN>tO7:&+"$&74 !U&#"WX.&"f%$& : 1 in =S. i, , U. .o. S, iO= 1D9* ' $& & $"%2: . 6 *;+$$.A&O$&&7"+ A@+$&?2&NxsJK&TA&2"%= JK&&O$&"2:&&%$&7 4 $9U "U"74 $&%$&: 6d*"+7"+1 U B%2a%2"kJK&NO$"%2:&+$:g+O= JK"+7-3g $r"+7JK* ",!.!6`,-&'B &"!$.&>/$yp(i&7"+)'B" _P" , 77"%2 I SO.4J;. $y$O= 7;O35O$&:&ZAx&_P&,-&TA+$7 . , $. 77 &N &+$ F. , I 7;+7"kJK"%2"+ Y%2& J;2 _ n U ^;)2&' 1%JK"+:g%2;.-3g $e77 "JK"2" "z&7"+'B"n7"+S&_PN!tU l2;<,-O2* "+$&" &Tc I ", Q%$&" " .! &7N JK&* "+$&Q@+$!H +$7 . , $. 7 +O2:1+.&"%$&Z]@!$&7& "+& { ~ 2& >47'B"+ "+'(O ,z7;+7"kJK"%2"+6A%2& J;2 _ n 4 n= %2;& 6 X X 4 0!U [{ 0A6 H =− p(i ) log p(i ). n

n

n

n

(in )∈{0,1}n

2

n

1

n

9

; O1D_P&6^$Z n=mSO2&,-&&_P'BE@ $7&!'B?E $&"+77"%$&7!.!6M$&ZY,!O, 7 "J;= !GFG&7'( I t>D "&SDHNJK '(OXJ;2;7 "JK+,!)14!z4 O2" G%2N (i , . . . , i ) J;2 _ n U \(%2;&&_P"O>X!2cJK'B*"+," {4| >N{D| { >u@!$&7& &J $ZcJ;2; n = , , 1, 2, . . . X 4 0!U %{ $86 p(i |i ) log p(i |i )i , h =H − H = h− Hn

1

n

n+1

n+1 n

n

n+1 n

2

(in )

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