Physical Review E 63 (2001) 046211.

Physical Review E 63 (2001) 046211.
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4
2
2
λ 1,2
λ 1,2
4
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0
0
(a)
-2
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0.2
0.1
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0.6
0.4
0
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0.2
0.1
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0
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-0.2
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0
0.2
0.4
0.6
COUPLING STRENGTH ε
0.8
0
1
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õ‹[ —  E(E > :L? B  ? —  E ˜[𠏨õ‹' —  E(E =PYû
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—  B õ <=:=?î E ˜ ‹' ?+î˜(1
õ B+˜  ˜(ð  : ‹ ( ” > õ B? dš õ ?  ?Y˜ E >?†©>N<=<˜[ð  —  > ‹ E vò î ‹ >:=?  †
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B E :=?îö˜[ð  E :=‡ õ < ‰“n ò —  B?8˜[:=?î “ > E  †  ?‡> ‹ î:L?><  } B+: ò
} B>?Y˜[:¹üy>N˜':  ? ‘ : û  û ‘ > õ > ‹ ˜':=˜':  ? þ :=˜[ð  } B+: õ‹'Y“ > “ <  ‡> ‹ î:L?>N<
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0.2
0.4
0.6
COUPLING STRENGTH ε
0.8
1
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?:¹üy>N˜':  ? ÷ úgwû ð  ! D = ô ˜[ð  † ‹ : š8 ^˜(ð  †> E ð  †<=:=? 
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B  E Nô . ‘ ˜[ð  ?z:=˜ E ˜[> ‹ ˜ E ˜  :L? — ‹[ > E  > E (N( * G> > õõ‹' > — ð  E
ü W‹[‘ >?† ?>N<=< ” :=˜ ‹ : E  EpE ð> ‹[õ < ” >N˜¨˜(ð  E ” ? — ð ‹[ ?+:=üW>˜':  ?
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— ð> ‹ > — ˜ W‹ :¹ü  † “Y”
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E ˜[>˜  þ :=˜[ð D = Mh> Z ‡f:Lz? D = M D G> 7
iq
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ô B<¹<–<=:L?  :=L
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E ð Y‹ ˜[< ”ó“mnô„Y‹' ˜(ð  E ” ? — ð ‹' ?+:¹üy>N˜[:  ?z˜[ð ‹' E ð  <=† ‘ þ ð  ? “m ˜(ð
˜(ð  ! E $# EhE ˜(> ‹ ˜i˜ ô >N<=<>8?† ‹[ > — ðó˜(ð  :L†  ?Y˜': — >< E ” ? — ð ‹'ò
?:¹üy>N˜':  ? E ˜[>N˜  þ :¹˜(ð
D
=
x > wZOD G>Lx = )Zy\!D = [Zy\!D G> wZy\!D = Mh> iq
:=˜[ð  ‡ W‹ î:L?+î E ” ? — ð ‹' ?+:¹üy>N˜[:  ? þ  < Y E  ˜(ð Aõm E(E : “ :=<=:¹˜ ”
˜ ý E ˜(> “ <=: E ðñ˜(ð  †+: ‹[ — ˜[:  ? Nô :L? ô„Y‹ ‡>N˜[:  ?  þ ‘8‹ ˜[ð 
— >B E >N< ‹' <=>˜':  ? E ð+: õñ“ ˜ þ W ?ø˜[ð  E ” E ˜  ‡ E =N >?† n> 8û
˜: E B?† W‹ E ˜(>?†> “ <  ( :=?f˜[ð  :=†  ?Y˜': — >N< E ” ? — ð ‹[ ?:¹üy>N˜':  ?©˜[ð 
E W‹ :  E FN >?† IH > ‹' :L†  ?Y˜[: — >N<m>8?†f˜[ð y‹' : E ? ©õ E ò
E : “ :¹<=:=˜ ” ˜  E ˜(> “ <=: E ð&˜[ð  — >B E >N< ‹[ <L>N˜[:  ? E ð+: õ&“m ˜ þ v ?Q =P
>8?† > K'B E ˜ ô^‹[ ‡ ˜(ð  †>N˜(> û
?/˜[ð  ?  ˜  >‡ õ <  þ  —  ? E :L† W‹ ˜(ð  ?  ?:=†  ?Y˜': — >N<
• ?  ? E ” E ˜  ‡ E¨þ :=˜[ð , Z . q ù >?† , X Z . q û ð Aõm E : ò
˜[: š8 ( * JB5:=î ûñÿ > 2Nô ˜(ð  † ‹ : š : E >NîY>N:L? —  ? E ˜(>?Y˜ ‘ þ ð+:=< 
λ 1,...,4
-4
-6
i(X), i(Y), i(X,Y)
0
(a)
0.4
0.2
(b)
0.2
(c)
0
0.4
0.2
(b)
0.4
0.2
(c)
0
0.1
1
10
COUPLING STRENGTH ε
100
1000
0
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: E >B+îY‡  ?Y˜  † “Y” >† ‹ : š :=?+î˜ W‹ ‡7:=? š8 < š :L?+îgF X û B5: ‹ E ˜ þ  >? ò
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ð  ˜ þ  (+( *öNô“m ˜(ð E ” E ˜  ‡ E > ‹' † Wõ : — ˜  †k:L?$B5:=î û >$P˜[ð 
—  ? E ˜[>8?Y˜ õ E :=˜': š >8?†‰ü y‹' ( *׍Nô ˜[ð  † ‹ : š8 >?† õ > ‹ ˜':L>N<=< ”
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î ( *׍ô ˜(𠏩‹[ E õ ? E 
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(
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‹ :=˜[ð‡f: —E[— >N< n t‘ þ ð+: — ð: E > —W—  ‡ õ >8? ” † “Y” > E <=:¹îYð8˜:L? — ‹' > E 
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û
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ð  ? ‘“m ˜ þ v ? . Z >?1
D = M7> :=? — ‹' > E  E£E ð> ‹(õ < ” E  ˜[ð>˜ ô„Y‹ ?  îY>N˜[: š8 (+( * G> ˜(ð  —  ?†+:¹˜[:  …
? ù iô„8‹ î  ? W‹ >N<=:=ü  † E ” ? — ð ‹[ ?+:=üW>˜':  ?‰: E >N˜ ò
˜(>N:L?  † [û ð  ! $# E"E ˜[> ‹ ˜>N˜"ü W‹[¨š >N<LB  E ô„Y‹ E ‡f><¹< . ‘ ˜[ð  ?
— 8‹(‹' — ˜[< ” ‹'  — ˜¨˜[ð  — >8B E >< ‹[ <L>N˜':  ? E “Y” ˜(ð  : ‹ :L?  } B>N<=:¹˜ ”
ùB þ ð+: — ð©ð  <L† E ‘ >Nî8>:=? ‘y ?+< ” B?Y˜':=<+˜[ð  E ” ? — ð ‹' ?+:=ü  † E ˜(>N˜ 
: E ‹[ > — ð  † g
9 ! >?†
û ð  E >‡ ©“m ð> š : 8‹hô ˜[ð  ! $# E ‘ ˆ
! %# EA— >? “m;8“ ˜[>N:L?  † ô„8‹ ˜(ð  — > E  þ :¹˜(ð Z ñ>?†
Z ÿ JB5:¹î ûw t‘ >N< E  E ˜[B†+:  †&:L? ÷ ‘ ù úSû
? 8‹ † W‹ ˜  E B‡‡> ‹ :¹ü  ˜(ð  ?YB‡ y‹ : — >< E ˜(B† ”8‘ þ  —  ? ò
9 ! !>?ˆ
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>— ?Ž:L?†+: — >N˜  E ” ? — ð ‹' ?+:=üW>N˜[:  ? P:=†  ?Y˜': — >< “Y” ˜(ð ; } B>N<=:¹˜ ”
nù >8?†Aî  ? W‹ >N<=:¹ü  † “Y” ˜(𠏨‹[ <L>N˜':  ? ù >8?† — >B E >< ‹[nò
<L>N˜[:  ? E Nô † ‹ : š8 >?† ‹[ E õ ? E  E B t“ E ” E ˜  ‡ E ‹[ <L>N˜[:  ? +ù û
ð  <L>N˜[˜ W‹ : E õm E(E : “ <  ˜ f E ˜[> “ <¹: E ð  ?+< ” :=? E ˜(>N˜  E :=? þ ð+: — ð
˜(ð  E B “ E ” E ˜  ‡ E > ‹[ —  B õ <  † ‘“ B+˜?  ˜ ”8 ˜ ô B+<=< ” E ” ? — ð ‹'ò
?:¹ü  † û
:L?†+: — >˜  (˜ ð  — Y ‹[‹[ — ˜ — >B E >N< ‹[ <L> ò
‹ : šUô > 8“ ” [˜ ð  : ‹h‹[ <L>N˜[:  ?
x> Z
% '
8
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X Z F '
‡ . qÿ F
F Z
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ù+ >NîY>N:L?  ?< ”6“mnô„Y‹' ˜[ð  E ” ? — ð ‹[ ?+:=üW>˜':  ?ט(ð ‹' E ð  <L† û ð 
> “mdšAõ <L>?>˜':  ? ô :=‡ õm E(E : “ :=<¹:=˜ ” ˜  :L? ô„W‹ > — >B E >N< ‹[nò
<L>N˜':  ? ô^‹[ ‡:L†  ?Y˜[: — >N<˜':L‡  E y‹ :  E :=?;˜[ð  E ˜(>N˜ pNô :L†  ?Y˜[: — >N<
E ” ? — ð ‹' ?+:=üW>N˜[:  ? — >? “ î  ? W‹ >N<=:¹ü  † :=?Y˜  ˜[:=‡  E W‹ :  E ‹[nò
<L>N˜  † “Y” >  ? nò ˜ òg ?  ?  ?+<=:=?  > ‹ô B? — ˜':  ?&> E : E ˜[ð  — > E Nô
˜[ð  î  ? W‹ >N<=:¹ü  † E ” ? — ð ‹[ ?+:=üW>˜':  ? û
? ˜(ð öô„ <¹<  þ :L?+î  >8‡ õ < 8‘ —  ? E :L† W‹ ˜[ð  B?:=†+: ‹[ — ˜[:  ? ò
>N<=< ” —  B õ <  ††% ' E[E < y‹ >8?†
† ( 8‹[ ?+ü E ” E ˜  ‡ E †  E(— ‹ : “ † “Y”
˜[ð k } B>N˜[:  ? E
D
4
6
COUPLING STRENGTH ε
F
ù D = M D G> 7
iq
2
ÜSÖËdß8ËÔ5¿8ÀÃ4¬W¼kÀ¬yÃ5¸Ÿ­
˜[ð  <=> ‹ î  E ˜.( *6ô ˜[𠏩‹[ E õm ? E  †  — ‹' > E  Eiþ :=˜[ðÓ:L? — ‹' > E :=?+î
.>?† “m —  ‡  E ?  îY>N˜': š >˜#. Z. q ûN ô ˜ W‹ . ZY. q k:=˜ ‹ : E  E
>?†A˜  B — ð  E ü y‹' > ‹' B?† . Z. q ÿ >?†A˜(ð  ?&:=˜ ô >N<=< E >NîY>N:L?
:L?8˜  ?  îY>N˜': šfš >N<LB  EWû8 î8>:=? ‘ ?  îY>N˜': šfš >N<LB  E ô (N( * T> †  ?  ˜(ð  E ” ? — ð ‹[ ?:¹ü  † E ˜(>N˜  EWû  þ×þ  ð> š >8?  >8‡ õ < 
Nô î  ? W‹ >N<=:=ü  † E ” ? — ð ‹' ?+:=üW>N˜[:  ? ÷Ÿù ‘ ‘ ù ú Nô ˜ þ  ?  ?+:L†  ?Y˜[: — >N<
E ” E ˜  ‡ Ev[û ð  ! D = GB5:¹î ûÿ “
: E—  ? E ˜(>?Y˜ ‘ þ ð+:¹< D T> û ð  ‡©B+˜(B>N<1! ‹[  — ˜ E ˜(ð  † vš8 < 8õ ‡  ?8˜ Nô (+( * G> ;
D = Mh> : E ü y‹'©ô„8‹ . .Rq ÿ ‘ ˜(ð  ?ó:¹˜ ‹ : E  Eþ :¹˜(ð(+( * T> > õò
õ‹[ > — ð+:=?îfü W‹[ >?†&˜(ð  ? D = M7> h‹'  — ˜ E ˜(𠏩“m ð> š : 8‹iNô
? ! * D G> û ™ :=? —  ˜(ð  ! $# E ‘ E :=‡f:=<=> ‹ < ” > E ˜[ð  : ‹ :=? E õ : ‹ >˜':  >
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˜
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N
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>
8
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÷Ÿùyÿcú ‹'  ”
—
š ‹ E‘
 — E 2ô  ? y‹ >< ò
:=ü  † E ” ? — ð ‹' ?+:¹üy>N˜[:  ? þ  — >8??  ˜  õm — ˜i˜(𠏏 } B>N<=:¹˜ ” nù 0‘
ð  þ vš8W‹W‘ ˜[ð  î  ? W‹ >N<=:¹ü  † E ” ? — ð ‹' ?+:¹üy>N˜[:  ?©: E > —v—  ‡ õ >?+:  †
þ :=˜[ð D = Mh> ‹ : E :=?+î:L?Y˜ š >N<LB  E
ð 
˜':  ?  ô
0.6
0
i(X|Y), i(Y|X)
i(X|Y), i(Y|X)
0
0.1
(a)
-2
-4
i(X), i(Y), i(X,Y)
λ 1,...,4
0
-2
(c)
0.2
(a)
UX
100
0
-100
(b)
UY
UX
(b)
UY
100
0
-100
100
0
-100
i(X), i(Y), i(X,Y)
(a)
i(X), i(Y), i(X,Y)
100
0
-100
0.1
(c)
0.2
0.1
i(X|Y), i(Y|X)
i(X|Y), i(Y|X)
0.06
0.04
0.02
(d)
(d)
0.08
0.06
0.04
0
0
10
20
TIME [sec.]
30
40
0
10
ÜSÖË NËÔ5¿YÀ2Ã4¬W¼kÀ¬yøŸ­
ÃSÀ0Åy¼kÀ0­d½tË
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Û $ #à ª„Ô 0¶hª„ÄY¬WÃS¿YÀtÄfȜ¸Ÿ­YÀn¶[³ 'ª Í ¶hª„ÄY¬WÃS¿8¾P¬W­YľPÄ8»v½4½ÀtĩȜ¸Ÿ­YÀt¶
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Ȝ¸Ÿ­YÀn¶¬W­YÄ 'ª Í Ô 0¶kªLºL®ÈœÈ5Ȝ¸¹­8Àn¶[ËzÔ5¿8À©ÕÕÖ*ªKÉY¯¬n¸Ÿ­ÓÂ+»W½À0­¾
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· Á(»W­ÊyÀ0¯S½À0¯'¶[ËñÔ5¿8ÀzÍÜg
­¬n½'Ã(Ë
20
TIME [sec.]
30
40
¸ŸÅ8Ë2ÞN³ÉY®½º=»y¯¬W­z¸Ÿ­d½À0¯S¸¹Á[½'¬vȖÕÕ5Ö
õ <  E  :=üWB ‹[ E ‘ <L>N˜ y‹ ‡ ”8 — <  ?+: — ò > E ˜[>N˜[: —fE  :=üWB ‹[ E > õõ > ‹[ † û
 —  ?Y˜'< ” ˜ þ  <  ?+î ò ˜ W‹ ‡ š :L† Wn**E- ‡  ?:¹˜ 8‹ :=?+î E  E(E :  ? E
þ W‹[zõmW‹'ô„8‹ ‡  † ‘ ˜[ð  m‹ E ˜  ?  E ð  þ  †ø: — ˜[><  ? E  ˜¨:L?ñ˜[ð 
< nô ˜2˜  ‡ õY‹ ><)< 8“m‘ ˜[ð  E  —  ?†Ó‡  ?:¹˜ 8‹ :=?+î “Y” E(— >< õ; <  — ò
˜ ‹' †  Ehùû ” > ‹ E <=>˜ W‹‹'Wš ><  †p‡  E ˜[< ” î  ? W‹ >N<=:¹ü  † E õ :  :=?î
þ :=˜[ð#> E <¹:=î8ðY˜  —  E(E :L?ñ˜(ð A‹ :¹îYðY˜©˜  ‡ õY‹' —v— : õ :=˜[><< 8“m û
?Y˜ W‹ : — ˜[>N<@ * E ð  þ  †Ñî8<=B —  E  ð ”õm ‡  ˜[> “m <=: E ‡ :L?ט[ð 
< nô ˜A˜  ‡ õm8‹[ —v— : õ :=˜[><< 8“m y
û õ > ‹ ˜ Nô ˜[ð  ‡  E ˜ ‹[ —  ?Y˜
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