Eu ophys. Le .,58 (3), pp. 342–348 (2002)
EUROPHYSICS LETTERS 1 May 2002
Rocking bis able sys ems: Use and abuse
o linea esponse heo y
J. Casado-Pascual1,2,J.G
´
omez-O d´
o˜
nez1,M.Mo illo
1and P. H¨
anggi2
1F´ısica Te´o ica, Uni e sidad de Se illa - Apa ado de Co eos 1065
Se illa 41080, Spain
2Ins i u ¨u Physik, Uni e si ¨a Augsbu g - Uni e si ¨a ss aße 1
D-86135 Augsbu g, Ge many
( ecei ed 19 No embe 2001; accep ed in final o m 8 Feb ua y 2002)
PACS. 05.40.-a – Fluc ua ion phenomena, andom p ocesses, noise, and B ownian mo ion.
PACS. 05.20.-y – Classical s a is ical mechanics.
PACS. 02.50.Ey – S ochas ic p ocesses.
Abs ac . – The esponse o a nonlinea s ochas ic sys em d i en by an ex e nal sinusoidal
ime-dependen o ce is s udied by a a ie y o nume ical and analy ical app oxima ions. The
alidi y o linea esponse heo y is pu o a c i ical es by compa ing i s p edic ions wi h
nume ical solu ions o e an ex ended pa ame e egime o d i ing ampli udes and equencies.
The ele ance o he d i ing equency o he applicabili y o linea esponse heo y is explo ed.
The esponse o dissipa i e physical sys ems o small-ampli ude ex e nal pe u ba ions is
usually desc ibed wi h he powe ul ools o linea esponse heo y (LRT) [1], as i is gene ally
accep ed ha he effec o he pe u ba ion can be desc ibed in e ms o small de ia ions om
he beha io o he unpe u bed sys em. In pa icula , o long imes and o sys ems which
in he absence o d i ing each an equilib ium dis ibu ion, LRT p o ides an app oxima e
exp ession o he p obabili y dis ibu ion ob ained by keeping jus he linea e ms in a se ies
expansion in he ex e nal ampli ude. The pu pose o he p esen le e is o poin ou he
ele ance o pa ame e s o he han he ampli ude o he d i ing o ce, o he alidi y o LRT.
We will show ha o a pe iodic ex e nal o ce, he alidi y o LRT depends no jus on he
ampli ude o he d i ing e m bu also c ucially on i s equency.
Le us conside a sys em cha ac e ized by a single deg ee o eedom, x, whose ime e o-
lu ion is go e ned by he nonlinea Lange in equa ion (in dimensionless o m)
˙x( )=x( )−x3( )+Acos Ω +η( ),(1)
whe e Acos Ω ep esen s an ex e nal signal and η( ) is a Gaussian whi e noise wi h ze o
a e age and η( )η(s)=2Dδ( −s). The co esponding linea Fokke -Planck equa ion (FPE)
o he p obabili y densi y P(x, ) eads
∂P
∂ =∂
∂x−x+x3−Acos Ω P+D∂2P
∂x2.(2)
c
EDP Sciences
J. Casado-Pascual e al.:Rocking bis able sys ems: Use and abuse e c. 343
The unpe u bed sys em has an equilib ium dis ibu ion o he o m
Peq(x)=Nexp −U0(x)
D,(3)
whe e Nis a no maliza ion cons an and U0(x) is he unpe u bed po en ial
U0(x)=−x2
2+x4
4.(4)
This po en ial has wo minima loca ed a xm=±1 and a maximum a xM= 0, wi h a ba ie
heigh o 0.25. The po en ial U0(x)−Ax cos Ω loses i s bis able cha ac e o A≥AT=
4/27.
The analysis o he dynamics is simplified by making use o wo impo an heo ems: he H-
heo em, which ensu es he exis ence o a unique long- ime dis ibu ion unc ion P∞(x, ) [2,3],
and he Floque heo em, which gua an ees ha P∞(x, ) is pe iodic in ime wi h he same
pe iod as he ex e nal o ce [4]. Fo he sys em a hand, he symme y o U0(x) implies he
ollowing p ope ies o he long- ime unique solu ion o he FPE: P∞(−x, ;−A)=P∞(x, ;A)
and P∞(−x, ;A)=P∞(x, +T/2; A), whe e T=2π/Ω and we ha e indica ed explici ly he
dependence o P∞on A. Using he Fou ie expansion
P∞(x, ;A)=
∞

m=−∞
Hm(x;A)eimΩ ,(5)
he fi s p ope y leads o Hm(x;A)=Hm(−x;−A), while he second one implies ha
Hm(x;A)=(−1)mHm(−x;A). F om bo h o hem, we ob ain Hm(x;−A)=(−1)mHm(x;A).
I hen ollows immedia ely ha he odd momen s o he dis ibu ion, xn( )∞,n=1,3,...
can be w i en as Fou ie se ies con aining only odd ha monics as he e en ha monics anish
due o he symme ies abo e. Analogously, e en momen s xp( )∞,p=0,2,... con ain jus
e en ha monics in hei Fou ie se ies expansions [5]. Inse ing he Fou ie expansion, eq. (5),
in o he FPE, an infini e se o equa ions o he coefficien s Hm(x;A) is ob ained. Inspec ion
o he se indica es ha i Hm(x, A) is expanded in powe s o A, i canno con ain powe s
smalle han A|m|. F om he abo e gene al conside a ions, we ha e, in pa icula , o he fi s
wo momen s,
x( )∞=
nodd
Mn(A)einΩ =2
∞

n>0,odd |Mn(A)|cos(nΩ −φn)
=2
∞

n>0,odd |Mn(A)|(cos φncos nΩ +sinφnsin nΩ ) (6)
wi h Mn(A)=c(0)
nA|n|+c(2)
nA|n+2|+···,and
x( )2∞=
pe en
Lp(A)eipΩ (7)
wi h Lp(A)=b(0)
pA|p|+b(2)
pA|p+2|+···.
The exac analy ical exp ession o P∞(x, ) is unknown. LRT amoun s o w i ing
P∞(x, )=Peq(x)+AP(1)
1(x, ),(8)
344 EUROPHYSICS LETTERS
wi h P1(x, ) ob ained om a fi s -o de pe u ba ion analysis o he FPE (see [4] and he
Appendix o [6] o de ails(1)):
AP(1)
1(x, )=−
∞

n=1
A
λ2
n+Ω
2[λncos Ω +ΩsinΩ ]dnϕn(x),(9)
whe e ϕn(x) a e he igh eigens a es o he unpe u bed FP ope a o and λn he co espond-
ing eigen alues. The coefficien s dna e ϕn|∂/∂x|ϕ0. I ollows om eqs. (8) and (9) ha
he a e age alue x( )LRT
∞is gi en by
x( )LRT
∞=a1cos Ω −φLRT
1.(10)
The explici calcula ion o he ampli ude, a1, and phase lag, φLRT
1, equi es he knowledge o
he spec um o he unpe u bed sys em. Fo he bis able sys em a hand, no exac analy ical
exp essions o he eigen unc ions and eigen alues exis , al hough use ul app oxima e exp es-
sions a e known [7, 8]. Al e na i ely, he ampli ude and phase lag [6, 9, 10] can be ob ained
om he esponse unc ion. Using he wo-mode app oxima ion o Jung and H¨anggi [9], we
w i e
x( )LRT
∞=b1cos(Ω −β1)+b2cos(Ω −β2),(11)
whe e he fi s e m on he igh -hand side is due o he in e well hops, while he second
one desc ibes he influence o in awell dynamics. I is con enien o cas he exp ession o
x( )LRT
∞as in eq. (10), and wi hin he wo-mode app oxima ion, we ge o he ampli ude
a1=A
Dg2
1λ2
1
λ2
1+Ω
2+g2
2α2
α2+Ω
2+2g1g2λ1α(λ1α+Ω
2)
(λ2
1+Ω
2)(α2+Ω
2)
1
2
,(12)
while he phase lag o he esponse wi h espec o he inpu signal, 0 ≤φLRT
1≤π/2, is gi en
by
φLRT
1= a c an
g1λ1Ω
λ2
1+Ω2+g2αΩ
α2+Ω2
g1λ2
1
λ2
1+Ω2+g2α2
α2+Ω2
.(13)
In he abo e o mulas, λ1is gi en by [11]
λ1≈√2
π1−3
2Dexp[−1/4D],(14)
and α= 2. The weigh s, g1and g2can be ob ained om he exp essions
g2=λ1x2eq
λ1−α+x2eq −x4eq
λ1−α,(15)
g1=x2eq −g2.(16)
To leading o de in D, we can eplace λ1by λK=√2/π exp[−1/4D], g1≈1andg2≈D/α.
This is he limi conside ed in [12].
Linea esponse heo y leads o he ollowing p edic ions: he fi s momen x( )∞should
con ain a single ha monics wi h he equency o he d i ing o ce, he ou pu ampli ude should
(1)No e ha he plus signs in (A.23) o e . [6] should ead minus. This in u n yields a minus sign on he
igh -hand side in (A.28).
J. Casado-Pascual e al.:Rocking bis able sys ems: Use and abuse e c. 345
o de o ha monics
0
0.2
0.4
ampli ude
Ω=10−1
1 3 5 7 9
0
0.2
0.4
0.6
ampli ude
Ω=10−4
1 3 5 7 9
Fig. 1
0 100000 200000
-0.5
-0.3
-0.1
0.1
0.3
0.5
< x ( ) >
Fig. 2
Fig. 1 – Ampli udes o he Fou ie componen s o x( )∞ o noise s eng h D=0.1 and inpu
ampli ude A=0.2 and equencies Ω = 10−4(uppe panel) and Ω = 10−1(lowe panel).
Fig. 2 – Time e olu ion o x( ) o D=0.1, A=0.04 and Ω = 10−4as ob ained om he nume ical
solu ion o he FPE (solid line), he adiaba ic app oxima ion (do ed line), he wo-mode LRT (dashed
line) and he wo-mode LRT o leading o de in D(do -dashed line).
beha e linea ly wi h A. Ce ainly, o fini e alues o D, i he ampli ude o he d i ing o ce
is infini esimally small, he expansion p ocedu e in Ais alid and LRT applies. The poin
ha we wan o add ess he e is ha o fini e small ampli udes, A<A
T, he alue o Ω has
o be aken in o accoun when applying LRT. The uppe limi o he alues o A o which
LRT emains alid(2) depends as well on he d i ing equency.
The adiaba ic app oxima ion gi es a desc ip ion o he dynamics when Ω is small compa ed
o any o he cha ac e is ic equency o he sys em. In his app oach [4], he p obabili y
densi y is assumed o be gi en by
Pad(x, )=N( )exp−U0(x)−Ax cos(Ω )
D,(17)
whe e N( ) is he no maliza ion cons an . An analysis o he co ec ions o he ba e adiaba ic
app oxima ion has ecen ly been p esen ed by Talkne [14].
E en in he absence o d i ing, no exac explici ime-dependen analy ical solu ion o he
FPE o he model sys em a hand is known. We ha e eso ed o nume ical solu ions o
eq. (2). We ollow a echnique based on he use o he spli p opaga o me hod o Fei e
al. [15] and de ailed in [16]. F om he nume ical solu ion o he FPE we can easily ob ain he
ime dependence o x( )∞. As his is a pe iodic unc ion o ime, i s Fou ie componen s
can be ob ained by nume ical quad a u e.
In fig. 1, we show he ampli udes o he ele an Fou ie componen s o he ou pu signal
o D=0.1, A=0.2 and wo e y diffe en d i ing equencies, Ω = 10−1andΩ=10
−4.
In his figu e, as well as in he subsequen ones, we ha e aken D=0.1. This is a ypical
(2)In he linea esponse egime, he dimensionless a io A/D is assumed o obey A<D.In heopposi e
singula limi , AD, he dynamics assumes uni e sal weak noise spec al p ope ies [6, 13].
346 EUROPHYSICS LETTERS
0 200 400
-0.4
-0.2
0
0.2
0.4
< x( ) >
Fig. 3
400 425 450
-0.4
-0.2
0
0.2
0.4
< x( ) >
Fig. 4
Fig. 3 – The same as in fig. 2 bu wi h Ω = 10−1.
Fig. 4 – The same as in fig. 2 bu wi h Ω = 1.0. No ice ha , due o he la ge alue o he d i ing
equency, we only plo a ew cycles o he ou pu in he asymp o ic egime.
alue and i is adequa e o he alidi y o he wo-mode app oxima ion leading o eqs. (12),
(13). On he ho izon al axis we indica e he o de o he ha monics. I is clea ha e en o
his d i ing ampli ude, ela i ely la ge in ela ion o i s h eshold alue, he esponse o he
sys em a he la ge equency con ains essen ially he fi s ha monics. On he o he hand,
o he small d i ing equency, highe -o de ha monics a e gene a ed. This is an indica ion
o he ailu e o LRT o desc ibe he dynamics a hese low equencies, while LRT migh s ill
be a good desc ip ion o highe equencies.
In fig. 2, we depic he ime e olu ion o x( )ob ained om he nume ical solu ion o
he FPE o D=0.1, A=0.04 and Ω = 10−4. We also show he beha io s ob ained using
he adiaba ic ansa z, eq. (17), LRT wi hin he wo-mode app oxima ion, eqs. (12), (14)-(16)
and LRT o leading o de in D. The inpu signal is la gely amplified a his small equency.
The adiaba ic esul de ia es sligh ly om he nume ical one a he peaks. The de ia ions
om he nume ical esul s a e la ge wi h he LRT desc ip ion. None heless, he wo-mode
LRT and he adiaba ic app oxima ions yield an accep able desc ip ion o he dynamics. This
is expec ed wi hin he linea esponse egime, whe e AD.
In fig. 3, we show he beha io o Ω = 10−1. I is clea ha he adiaba ic app oach yields
a signal wi h a e y la ge ampli ude and a la ge phase shi compa ed wi h he nume ics.
The wo-mode LRT s ill yields a e y accep able beha io . The same quali a i e ea u es a e
obse ed in fig. 4, whe e Ω = 1. Fo his la ge equency, he de ia ions o he wo-mode LRT
om he nume ical esul a e e y small and hey canno be no iced in he plo . In fig. 3, we
show he ull ime e olu ion including he sho ansien . In fig. 4, as we conside a la ge
equency alue, we only show a ew oscilla ions in he asymp o ic egime, so ha he de ails
o a cycle can be dis inguished. These las h ee figu es show ha o e y small A, LRT gi es
a sa is ac o y desc ip ion o he sys em esponse, wi h de ia ions om he nume ics mo e
p onounced as he ex e nal equency assumes smalle alues.
To es he alidi y o LRT as he inpu ampli ude is inc eased, we ha e ca ied ou an
ex ensi e nume ical analysis o he sys em esponse o inpu signals o inc easing ampli udes
and diffe en equencies. We e alua e he ela i e e o eampl =|Aou −a1|/Aou , be ween
he ou pu ampli ude, Aou , p o ided by he nume ics and he one ob ained wi hin LRT wi h

J. Casado-Pascual e al.:Rocking bis able sys ems: Use and abuse e c. 347
0 0.05 0.1 0.15 0.2
A
0
1
2
eampl
0
1
2
3
eampl
Fig. 5
10-4 10-3 10-2 10-1 100
Ω
0
0.5
1
1.5
Phase lag
A=0.01
A=0.05
A=0.1
A=0.2
Fig. 6
Fig. 5 – Plo s o he ela i e e o o he ou pu ampli ude, eampl =|Aou −a1|/Aou , s. inpu
ampli ude Aand se e al alues o he d i ing equency. In he uppe panel, a1is e alua ed using
he LRT wo-mode exp essions o leading o de in D, while he ull wo-mode o mulas a e used in
he lowe panel; see eqs. (12), (14)-(16) in he main ex . The noise s eng h is D=0.1and he
equencies a e: Ω = 1.0 (ci cles), Ω = 10−1(plus signs), Ω = 10−3(c osses) and Ω = 10−4( iangles).
Fig. 6 – Plo o he phase lag be ween he a e age ou pu and he d i ing o ce s. he angula
equency Ω. Wi h he solid line we depic φLRT
1, e alua ed using he LRT wo-mode exp essions o
leading o de in D(see eq. (13) in he main ex ). The symbols deno e he nume ically de e mined
alues o he phase lag Ψ (see ex ), o A=0.01 (ci cles), A=0.05 ( iangles), A=0.1 (diamonds),
and A=0.2 (c osses). The noise s eng h is se a D=0.1.
he wo-mode app oxima ion, a1in eq. (12), as a unc ion o he inpu ampli ude A. Ou
findings a e shown in fig. 5. The uppe panel shows he dependence o eampl on A o se e al
equencies, when he LRT is e alua ed o leading o de in D, while in he lowe panel, he
ull exp essions, eqs. (12), (14)-(16) ha e been used. Fo ela i ely high equencies, Ω = 1.0
(ci cles), and Ω = 10−1(plus signs), he e o emains small and is p ac ically cons an ,
e en o inpu ampli udes which a e a he la ge compa ed o i s h eshold alue. On he
o he hand, o small alues o Ω, Ω = 10−3(c osses)andΩ=10
−4( iangles), he e o
inc eases d as ically wi h he inpu ampli ude. In pa icula , he explici ela i e e o s a
D=0.1, i.e. (e1,e2,e3,e4), co esponding o he d i ing equencies (Ω1=10
−4,Ω
2=10
−3,
Ω3=10
−1,Ω
4=1.0), espec i ely, ead o A=0.01: (0.028, 0.028, 0.056, 0.063); o
A=0.1: (0.249, 0.249, 0.072, 0.0659); and o A=0.2: (2.539, 0.797, 0.090, 0.074). Thus,
he ou pu ampli ude p edic ed by LRT a hese small ex e nal equencies is e y much in
e o , e en hough, o he same ex e nal ampli udes and mode a e- o-la ge equencies, LRT
p edic ions a e s ill adequa e.
The a e age ou pu lags behind he inpu signal wi h a phase shi be ween 0 and π/2.
The alue o he phase shi p edic ed by LRT, φLRT
1gi en by eqs. (10), (13), is independen
o he d i ing ampli ude, bu depends on Dand Ω. I s a s a 0 o e y small equencies,
hen eaches a local maximum, and ends o i s limi ing alue π/2 o e y la ge equencies.
I s beha io o he small- o-mode a e equencies conside ed he e (Ω <1) is depic ed wi h
he solid line in fig. 6 o D=0.1. On he o he hand, he phase lag o he nume ical esul ,
Ψ, depends on D,Aand Ω. The Ψ alues plo ed ha e been calcula ed om he diffe ence
be ween he ins an o imes wi hin a pe iod, a which he d i ing signal and he pe iodic
ou pu , x( )∞, c oss signs, i.e., he co esponding phase delay in c ossing ze o. In fig. 6, we
348 EUROPHYSICS LETTERS
plo he alues o Ψ o se e al alues o he d i ing ampli ude and equency. Fo e y small
equencies, he ou pu is almos in phase wi h he inpu o all he ampli udes conside ed.
As he equency inc eases, de ia ions be ween he nume ical p edic ions and LRT esul s a e
mani es ed, being la ge o la ge d i ing ampli udes.
In conclusion, ou analysis clea ly indica es he influence o he d i ing equency Ω on he
alidi y o he LRT p edic ions o he ampli ude, phase and numbe o highe ha monics o
he esponse o he sys em o sub h eshold inpu signals. As he d i ing equency assumes
sufficien ly small alues, he ou pu ampli ude significan ly de ia es om i s linea beha io
p edic ed by he wo-mode app oxima ion LRT, e en hough he d i ing ampli ude migh s ill
be qui e small in o de o p ese e he bis able cha ac e o he unpe u bed po en ial. E en
o sub h eshold inpu s, highe -o de ha monics migh con ibu e o he sys em esponse o
small d i ing equencies, con a y o he p edic ions o LRT. Al hough he global beha io o
he phase lag indica ed by LRT is quali a i ely co ec , as expec ed, i s quan i a i e p edic ions
a e no eliable as he inpu ampli ude inc eases.
∗∗∗
Suppo by he Di ecci´on Gene al de Ense˜nanza Supe io o Spain (P ojec No. PB98-
1120), he Jun a de Andaluc´ıa (JC-P, JG-O, MM) and he Deu sche Fo schungsgemeinscha
HA1517/13-4 (PH) is g a e ully acknowledged.
REFERENCES
[1] Kubo R.,J. Phys. Soc. Jpn.,12 (1957) 570.
[2] Lebowi z J. L. and Be gmann P. G.,Ann. Phys. (Leipzig),1(1957) 1.
[3] Risken H.,The Fokke -Planck Equa ion (Sp inge -Ve lag, Be lin) 1984, p. 135.
[4] Jung P. and H¨
anggi P.,Phys. Re . A,44 (1991) 8032.
[5] H¨
anggi P., Jung P., Ze be C. and Moss F.,J. S a . Phys.,70 (1993) 25.
[6] Gammai oni L., H¨
anggi P., Jung P. and Ma chesoni F.,Re . Mod. Phys.,70 (1998) 223.
[7] H¨
anggi P. and Thomas H.,Phys. Rep.,88 (1982) 207.
[8] Dykman M. I., Haken H., Gang Hu, Luchinsky D. G., Mannella R., McClin ock
P. V. E., Ning C. Z., S ein N. D. and S ocks N. G.,Phys. Le . A,180 (1993) 332.
[9] Jung P. and H¨
anggi P.,Z. Phys. B,90 (1993) 255.
[10] G´
omez-O d´
o˜
nez J. and Mo illo M.,Phys. Re . E,49 (1994) 4919.
[11] H¨
anggi P., Talkne P. and Bo ko ec M.,Re . Mod. Phys.,62 (1990) 251.
[12] Gang Hu, Haken H. and Ning C. Z.,Phys. Le . A,172 (1992) 21.
[13] Shneidman V. A., Jung P. and H¨
anggi P.,Eu ophys. Le .,26 (1994) 571.
[14] Talkne P.,New J. Phys.,1(1999) 4.1.
[15] Fei M. D., Fleck J. A. j . and S eige A.,J. Compu . Phys.,47 (1982) 412.
[16] G´
omez-O d´
o˜
nez J. and Mo illo M.,Physica A,183 (1992) 490.