Farey sequence in the appearance of subharmonic Shapiro steps

0.48 MB·English

by  Jovan Odavic

Farey sequence in the appearance of subharmonic Shapiro steps
Jovan Odavi´c1, Petar Mali2, and Jasmina Teki´c3
1 Institut fu¨r Theorie der Statistishen Physik - RWTH Aachen University,
Peter-Gru¨nberg Institut and Institute for Advanced Simulation, Forschungszentrum Ju¨lich, Germany,
2 Department of Physics, Faculty of Science, University of Novi Sad,
Trg Dositeja Obradovi´ca 4, 21000 Novi Sad, Serbia and
3”Vinˇca” Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics - 020,
University of Belgrade, PO Box 522, 11001 Belgrade, Serbia
(Dated: April27, 2015)
ThelargestLyapunovexponenthasbeenexaminedinthedynamical-modelockingphenomenaof
theac+dcdrivendissipativeFrenkel-Kontorovamodelwith deformablesubstratepotential. Dueto
5
deformation, large fractional and higher order subharmonic steps appear in the response function
1
ofthesystem. Computation of thelargest Lyapunovexponentas away toverifytheirpresenceled
0
totheobservationoftheFareysequence. Inthestandardregime,theappearanceofhalfintegerand
2
othersubharmonicstepsbetween thelarge harmonicsteps,andtheirrelativesizes follow theFarey
r construction. In the nonstandard regime, however, the halfinteger steps are larger than harmonic
p
ones, and Farey construction is only present in the appearance of higher order subharmonic steps.
A
The examination of Lyapunov exponents has also shown that regardless of the substrate potential
4 or deformation, therewas no chaos in thesystem.
2
PACSnumbers: 05.45.-a;45.05.+x;71.45.Lr;74.81.Fa

]
D
C I. INTRODUCTION CalculationofthelargestLyapunovexponenthasbeen
alreadyusedasawaytoexaminetheexistenceofsubhar-
.
n monic Shapiro steps in the standard FK model [12, 13].
li In the examination of Shapiro steps, finding the best The standard Frenkel-Kontorova (FK) model represents
n methodortooltoverifytheirpresencehasbeenthemat-
a chain of harmonically interacting particles subjected
[ ter of many studies in various physical systems. Numer-
to a sinusoidal substrate potential [10]. It describes dif-
oustheoreticalandexperimentalresultsonShapirosteps
3 ferent commensurate or incommensurate structures that
v obtained in dissipative systems such as charge- or spin- under an external driving force, show very rich dynami-
2 density waves conductors [1–4], the systems of Joseph- cal behavior. In the presence of an external ac+dc driv-
7 sonjunctionarrays[5–7]andsuperconductingnanowires
ing force, the dynamics is characterized by the appear-
6 [8, 9] have been the main impulse and motivation for
anceofthestaircasemacroscopicresponseortheShapiro
2 ourstudiesoftheac+dcdrivenoverdamped(dissipative) stepsintheresponsefunctionv¯(F¯)ofthesystem[12–14].
0
Frenkel-Kontorova(FK)model[10]. Itiswellknownthat
. ThoughthestandardFKmodelhasbeenverysuccessful
1 when these systems are subjected under an external ac
in the studies of some effects related to Shapiro steps,
0 driver,theirdynamicsischaracterizedbytheappearance
its applications is still very restricted. Namely, in the
5
ofShapirosteps. Thesestepsareduetointerferencephe-
1 standardFKmodel,thesubharmonicstepseitherdonot
nomena or dynamical mode-locking (synchronization)of
: exists in the case of commensurate structures with in-
v the internal frequency with the applied external one. If
teger values of winding number [15, 16] or their size is
i the locking appears at the integer values of the exter-
X so small that analysis of their properties is very difficult
nalfrequency,thestepsarecalledharmonicwhileforthe
r [12–14]. The absence of subharmonic steps for the com-
a locking at rational(noninteger) values of frequency they mensuratestructureswithintegervalueofwindingnum-
are called subharmonic.
ber, and their presence in the case of rational (noninte-
In realistic systems due to presence of noise, impuri- ger)windingnumberwasconfirmedbythecalculationof
tiesandotherenvironmentaleffects,detectionofShapiro the largest Lyapunov exponent [13]. However, contrary
steps, particularly the subharmonic ones, is usually very to the standard case, the large subharmonic steps can
difficult. On the other hand, in theoretical works, their appear in any commensurate structure of the nonstan-
observation could also be a problem since their size is dardFK model suchas the one with the asymmetric de-
often so small that they are invisible on the regular plot formablesubstratepotential(ASDP)[17]. Thispotential
of the response function. One of the most sensitive ways belongs to the family of nonlinear periodic deformable
to verify the existence of Shapiro steps is the calculation potentials, introduced by Remoissent and Peyrard [18]
of the largest Lyapunov exponent [11]. Always when the as the way to model many specific physical situations
systemisdynamicallymode-locked,thelargestLyapunov without employing perturbation methods.
exponent has negative values [12, 13]. Therefore, an ex- In this paper, by using the largest Lyapunov expo-
amination of the largest Lyapunov exponent for some nent computation technique, we will examine the ap-
interval of driven force will precisely reveal the presence pearanceofbothharmonicandsubharmonicstepsinthe
of any harmonic or subharmonic mode-locking. FK model with asymmetric deformable substrate poten-
2
tial (ASDP). In the analysis of the largest Lyapunov ex- In the present paper, the system of coupled harmonic
ponent, we have observed one interesting property: the oscillatorsinASDPis drivenbythe dc+acforcesF(t)=
Shapiro steps and their relative sizes appear according F¯+Faccos(2πν0t). The equations of motion are
to the Farey construction only in the standard regime
∂V
when large harmonic steps are dominant in the response u˙l =ul+1+ul−1−2ul− +F(t), (2)
function. The paper is organizes as follows. The model ∂ul
and methods are introduced in Sec. II, and the results where l=−N,...,N, u is the position of lth particle, F¯
are discussed in Sec. III. Finally, Sec. IV concludes the 2 2 l
is dc force, where Fac and 2πν0 are the amplitude and
paper.
the circular frequency of the ac force,respectively. Since
the substrate potential is homogeneous (it does not de-
pend of the particles index l) relabeling of the position
II. MODEL AND METHODS
of particles will not change the properties of the config-
uration [10]. When the system is driven by an external
Weconsiderthe dissipative(overdamped)dynamicsof
ac+dc force,two different frequency scales appear in the
a series of coupled harmonic oscillators u subjected in
l system: the frequency of the external periodic force ν0,
the ASDP:
and the characteristic frequency of the motion of parti-
K (1−r2)2[1−cos(2πu)] cles over the ASDP driven by the average force F¯. The
V(u)= , (1)
(2π)2 [1+r2+2rcos(πu)]2 competition betweenthose frequency scales canresultin
the appearanceofresonance(dynamicalmode lockingor
where K is the pinning strength, and r is the shape or Shapiro steps).
deformation parameter (−1 < r < 1) which can be var- Solution of the system (2) is calledresonantif average
iedcontinuously. Bychangingtheshapeparameterr the velocity υ¯ satisfies the relation:
potentialcanbetunedinaveryfineway,fromthesimple
iω+ja
sinusoidal one for r = 0 to deformable for 0 < r < 1. In
v¯= ν0, (3)
Fig 1, the commensurate structure ω = 1/2 in ASDP is m
presentedfortwodifferentvaluesoftheshapeparameter
where triplet (i,j,m) are integers numbers. Resonant
r = 0 and r = 0.5 (for more details see [17, 19]). The
velocity is called harmonic if m = 1 and subharmonic if
m 6= 1. (In case of ω = q1 we can use v¯ = mi ων0 for
markingharmonicandsubharmonicsteps). Parametera
is the period of the potential V(u) and in the case of no
deformation a = 1, and with deformation a = 2 as can
beseeninFig1. Forarationalvalueofω =p/q(pandq
coprime integers)the triplet is not unique (this triplet is
unique only for incommensurate structures [10]). In this
paper we will consider only the commensurate structure
ω =1/2.
The equations of motion (2) have been integrated by
using a fourth order Runge-Kutta method with periodic
boundary conditions. The time step used in simulations
was 0.01τ, where τ = 1 . The force is varied adiabati-
ν0
cally with the step 10−5.
We shall be focused on calculating the largest Lya-
punov exponent λ [13]. It is well known that the Lya-
punovexponentgivesaquantitativemeasureonthepres-
ence of chaos in dynamical systems [11], however,it also
proves to be extremely sensitive to the existence of both
harmonicandsubharmonicShapirosteps. Whenthesys-
tem is dynamically mode locked, i.e. on the step, the
FIG. 1: (Color online) Particles moving in asymmetric de- trajectories of particles are periodic in time which is re-
formablepotentialforω= 12,K =4,andtwodifferentvalues flected by the negativevalue of the largestLyapunovex-
of the shape parameters r = 0 and r = 0.5. Particles are ponent. Outsidethesteps,wherethereisnoonsetofdy-
represented by red dots. namicalmodelocking,thetrajectoriesarequasi-periodic
whichisconfirmedbythezerooftheLyapunovexponent
average interparticle distance ω = hul+1 − uli, or the ([11, 13]). We choose an appropriate perturbed point u′
l
so called winding number is one of the main parameters
in our computations according to:
that describes the FK model. The system exhibits com-
mensurate phase for rational values of winding number d2
u′(t )=u (t )± 0 (4)
ω, and incommensurate phase for irrational ones. l ss l ss rN
3
wheret istimewhenthesteady-statehasbeenachieved
ss
0
inoursystem,andd0 istheparameterthatexpressesthe
change in the initial positions of particles of the model.
In order to make sure that projecting is always done -1
onto the subspace dominated by the largest Lyapunov 0.5
0.4
exponent, the sign in front of the square term in Eq.4
-2 v 0.3
is randomly selected where the plus and minus sign ap- 0.2
pear with the same probability. We sample and readjust 0.1
following Sprott [20] every 25 or so time steps. In our -3 0 0.2 0.3F 0.4 0.5 0.6
calculations, we used tss = 300τ and d0 = 10−7. For 0.2 0.3 F0.4 0.5 0.6
convenience, in further text, the largest Lyapunov expo-
nent will be denoted just as the Lyapunov exponent.
0
III. RESULTS -1
0.4
0.3
Inthe presentpaper,the Lyapunovexponentis exam-
-2 v0.2
inedfordifferentdeformationsofthesubstratepotential. 0.1
In Fig. 2, the Lyapunov exponent as a function of the 00.2 0.3 0.4 0.5 0.6
F
drivingforceforthreedifferentvaluesofdeformationpa- -3
0.3 0.4 0.5 0.6
rameter is presented. The insets show the correspond- F
ing response functions v¯(F¯) (the average velocity as a
function of average driving force). As one can see, the
computed Lyapunov exponents are always λ ≤ 0, which 0
implies that with the change of deformation r we have
not introduced chaos in our system (presence of chaos -1
would result in the positive values of Lyapunov expo-
nent). Domain of F¯ in Fig. 2, for which we calculated -2
theexponent,differswithrduetothefactthatfordiffer- 0.2
ent values ofr the same steps appear indifferent regions -3 v 0.1
ofF¯ (see[17,19]). Inthe standardcaseinFig. 2(a), we 0 0.8 F 0.9
can see the large minima which correspond to harmonic -4
0.8 0.9
steps and which size changes monotonically. As defor- F
mationincreasesinFig. 2(b)and(c),the minimawhich
correspondstothelargehalfintegerandhigherordersub- FIG.2: (Color online) TheLyapunovexponentasafunction
harmonic steps appear where the changing of their size oftheaverageforceforcommensuratestructureω=1/2,K =
is not monotonic any more. 4,Fac = 0.2,ν0 = 0.2 and three different values of shape pa-
rameter (a) r = 0, (b) r = 0.28 and (c) r = 0.6. The insets
UsingtheEq.(3),theShapirostepscouldbenowiden-
show thecorresponding response functions v¯(F¯).
tified. It is well known that in the standard FK model
(r = 0) with integer value of winding number, there
would be no subharmonic mode locking [10], and con-
sequently, no steps between harmonic ones on the plot
1 4 3 5 2
of response function v¯(F¯). On the other hand, when , , , , . (5)
1 3 2 3 1
ω = 1/2, only halfinteger step 3/2 which appears be-
tween the first and the second harmonic could be visible This sequence of numbers represents the Farey se-
[13]. However, computation of the Lyapunov exponent quence well known in number theory [21, 22].
betweenfirstandthe secondharmonicstepsrevealother The Farey sequence FN of order N is an ascending
subharmonic steps as can be seen in Fig. 3. The areas sequence of irreducible fractions between 0 and 1, whose
under the minima correspond to the size of the steps, denominatorsareless orequalthen N [21, 22]. The first
i.e. for larger step, the area under the minimum will few would be:
be larger. If we examine the subharmonic steps in Fig.
3, we could see that the first largest fractional step be- F1 = 10,11
n o
tween the step 1 and the step 2 is the step 3/2. Then, F2 = 01,12,11
the largest step between the steps 1 and 3/2 would be n o
the step 4/3 while the largest one between the steps 3/2 F3 = 01,13,12,23,11 (6)
n o
and 2 is 5/3. Therefore, according to the appearance of F4 = 01,14,13,12,23,34,11
fractionalsteps betweenthe first1/1andthe second2/1 n o
harmonics we may write the following sequence: F5 = 01,15,14,13,25,12,35,32,34,45,11
n o
4
0.05
steps 1 and 2 is halfinteger step 3/2. From the set the-
0 ory[22]weknowthatbetweenanytworationalfractions
-0.05 lie countable many, ℵ0 rational fractions and therefore,
4/3 5/3
3/2 countable many possible Shapiro steps between any two
-0.1
harmonic steps in our model.
-0.15 0.2 If the potential gets deformed, the large halfinteger
-0.2 stepandhigherordersubharmonicsteps appear[17,19].
1 v 0.15 Contraryto the case r =0 in Fig. 3, now for r =0.01 in
-0.25
Fig. 5,thelarge4/3and5/3stepsareclearlyvisible. The
0.1
-0.3 0.3 0.305 0.31 0.315 2
F
0.25
-0.35
0.3 0.302 0.304 0.306 0.308 0.31 0.312 0.314
F
0
FavIeGr.ag3e: d(rCivoilnogr foonrlcienef)orLrya=pu0n(otvheexrepsotnoefntpaarsamaeftuenrcstaiorne aosf -0.25 6/55/4 7/49/5
inFig. 2). Theinsetshowstheresponsefunctionv¯(F¯)drawn
for the same interval of force. This result is obtained in [13]. -0.5 0.2
0.15
-0.75
Therefore, if we have two rational fractions in Farey se- 0.1
quence p (p, q arecoprime integers)and p′ (p′,q′ areco- -1 0.3 0.31 0.32
q q′ 0.3 0.31 0.32
prime integers), the rational fraction which lies between
them and has the smallest denominator is
FIG. 5: (Color online) Lyapunov exponent as a function of
p′′ p p′ p+p′ average driving force for r=0.01 (the rest of parameters are
= ⊕ = (7)
q′′ q q′ q+q′ as in Fig. 2). The inset shows the response function v¯(F¯)
drawn for thesame intervalof force.
where p′′, q′′ are coprime integers. This statement is
triviallyextendedtothecaseofintervalbetween1and2, higher order subharmonic steps, such as 4/3 and 5/3 (to
andfurtheron(Theorems28and29in[22]). Thelargest the left and to the right), are appearing in a symmetric
step between p and p′, if exists, will be step p ⊕ p′. manner with respect to the step 3/2.
q q′ q q′
The Farey sequence could be easily understood from the With the further increase of deformation r, the step
diagram in Fig. 4 widths increase faster on the right side from halfinteger
step 3/2 than on the left one as can be seen in Fig. 6.
0
-0.5
0.2
-1
0.15
0.1
-1.5
0.29 0.3 0.31 0.32 0.33
0.29 0.3 0.31 0.32 0.33
FIG.4: (Coloronline)SectionoftheFareyconstruction(rep-
resentedasarootedbinarytreegraph)a)from0to1,b)from
FIG. 6: (Color online) Lyapunov exponent as a function of
1 to 2 according toEq. 7.
average driving force for r=0.05 (the rest of parameters are
as in Fig. 2). The inset shows the response function v¯(F¯)
For example, in the case of the FK model with the in- drawn for thesame intervalof force.
teger value of winding number, there is no subharmonic
mode locking, which implies there is only the Farey se- It was shown in our previous work [19], that size of
quence of order one. However, if the winding number halfinteger and subharmonic Shapiro steps increase with
is rational noninteger such as the case ω = 1/2 in Fig. deformation and after reaching their maxima for some
3, one can see that the largest step between harmonic valueofr,decreasetozero. IfwecalculatetheLyapunov
5
exponent for r = 0.28, which is the value of r for which deformable potentials [23], such as variable, double bar-
halfinteger step reaches its maximum, we obtain results rier and double well potential, and we have been always
presented in Fig. 7. At this value of deformation some able to observe the appearance of steps in accordance
withtheFareyconstruction[11]. Therefore,fortwosteps
p and p′,thenextlargeststepbetweenthemwillbe p+p′
q q′ q+q′
where denominator determines the size of steps in terms
that the size of steps decreases as the denominator in-
9/5
7/4 creases. It is important to note that Farey construction
4/3
5/3 tellustheorderandtherelativesizesofstepsbutitdoes
3/2 nottellustheactualstepwidthorwhytheyappear[11].
0.3
1 It is well known that the size of harmonic and halfin-
0.2 teger steps are correlated, whereby the larger the size
v of harmonic the smaller the one of halfinteger step and
0.1
vice versa [19, 24, 25]. In some cases, depending on the
00.36 0.38 F 0.4 0.42 system parameters,the size of halfinteger steps could be
0.36 0.38 F 0.4 0.42 even largerthan the size of harmonic ones [19]. The size
of halfinteger and other subharmonic steps strongly af-
fectsthebehaviorofthesystem,andaccordingtothatin
FIG. 7: (Color online) Lyapunov exponent as a function of
previous works [19, 25], the three different types of sys-
average driving force for r=0.28 (therest of parameters are
as in Fig. 2). The inset shows the response function v¯(F¯) tembehaviorhavebeenclassified: thestandardbehavior
drawn for thesame intervalof force. for small halfinteger steps, the behavior for intermediate
halfintegerstepsandthebehaviorinthepresenceoflarge
higher subharmonic steps already start to disappear. halfinteger steps.
If we have two harmonic steps, according to Farey se-
Atlargedeformationofthepotential,thesizeofhalfin-
teger steps decreases, and the most of higher order sub- quencethenextlargeststepwhichappearsbetweenthem
isthehalfintegerstep,butthisisnotthecasefornonstan-
harmonic steps have completely vanished [19]. This is
dard behavior [19, 25], since halfinteger steps are larger
confirmed by the results in Fig. 8, where the Lyapunov
than harmonic ones. In such case, could we still have
exponentforr =0.5hasbeencalculated. Disappearance
the presence of Farey sequence? In Fig. 9, the response
functioninthecaseoflargehalfintegerstepsispresented.
It is obvious that in the nonstandard case, the relative
0
0.15 3/2
4/3
-1 5/4
6/5
0.3 7/6
00.2.25 0.1 1
v0.15 4/5
-2 0.1 v 3/4
0.05 2/3
00.64 0.65F 0.66 0.05 1/2
0.64 0.645 0.65 0.655 0.66 0.665 1/3
1/4
F
0
FIG. 8: (Color online) Lyapunov exponent as a function of 0.05 0.1 0.15 0.2
average driving force for r = 0.5 (the rest of parameters are F
as in Fig. 2). The inset shows the response function v¯(F¯)
drawn for thesame intervalof force.
FIG.9: Theaveragevelocity asafunction ofaveragedriving
force for Fac =0.55, and r =0.2 (the rest of parameters are
of steps is also clearly visible in Fig. 2. asinFig. 2). Thenumbersmarkhalfintegerandsubharmonic
Deformationofthepotentialobviouslystronglyaffects steps.
the steps as we can see in Fig. 6-8. It appears that with
the increase of the deformation r, the right side of the sizes of harmonic and halfinteger steps do not follow the
Farey construction is heavily favored over the left one. Fareyconstruction,andgoingfromharmonictohalfinte-
In particular,we observethat at eachlevel ofour binary ger steps, the size of step does not decreases as denomi-
tree graph (Farey construction in Fig. 4) the child node natorincreases,onthecontrary,thehalfintegersteps1/2
(or step) that takes preference is the one on the right. and3/2arelargerthanharmonicones1/1and2/1. How-
Thismeansthatwiththeincreaseofthedeformationthe ever,the higherordersubharmonicsteps betweenhalfin-
stepsthatarepresentandbecomeincreasinglydominant teger and harmonicsteps still appear accordingto Farey
are 3/2,5/3,7/4,9/5. constructionandtheir sizes decreaseas the denominator
We haveanalyzedalsothe systemswithothertypesof increases.
6
CalculationoftheLyapunovexponentgivesapossibil- the system. Computations of Lyapunov exponent have
itynotonlytodetectallresonancesintheresponsefunc- been performed for different system parameters, and re-
tion, but also to detect the presence of chaos. In all our gardless of the deformation, no chaos has been observed
simulations performed on the ac+dc driven overdamped in the behavior of the system. Absence of chaos in the
FK model we did not observed any chaos. Presence of presence of deformable potentials certainly requires fur-
deformable substrate potentials and differentlevel of de- ther investigation. This problem and the possibility of
formations did not introduced chaotic behavior into the chaotic behavior in other situations such as presence of
system. Contrary to our case, chaos has been observed noise will be subject of future examinations.
in the spatiotemporal dynamics of moving kinks in the
Presentedresultscouldbeimportantforthestudiesof
damped dc driven FK model where the resonances ap-
Shapirostepsinallsystemsthatarecloselyrelatedtothe
pear due to competitions between the moving kinks and
dissipative dynamics of the FK model. In experimental
their radiated phase modes [26]. Also, structured chaos
and theoretical works performed in charge density wave
hasbeenobservedinaJosephsonjunctionsystemswhere
systems and the systems of Josephson junction arrays,
chaoticregionsappearbetweenthesubharmonicShapiro
measuringof differential resistance is usually used to de-
steps at certain values of system parameters [27].
tect subharmonic steps. If we look for example at the
results obtained in sliding charge-density wave systems
[2], the systems ofJosephsonjunctionarrays[6,7]orsu-
IV. CONCLUSION
perconductionnanowires[8],wecanobservethepresence
ofFareyconstructionintheappearanceofShapirosteps.
In this paper, we have presented detailed analysis of
Our analysis shows that Farey construction can not be
the Shapiro steps in the ac+dc driven dissipative FK
always generally applied when it comes to relative sizes
model by using the Lyapunov computation technique.
oftheobservedsteps. Sincetheappearanceandoriginof
TheobtainedresultsshowthepresenceofFareysequence
the subharmonic Shapiro steps are still a matter of de-
in the appearance of subharmonic steps. The steps and
bates, we hope that these results could bring an insight
their relative sizes follow exactly the Farey construction
into understanding of these physical phenomena.
only in the standard regime when harmonic steps are
Acknowledgments
the largest one. However, in the nonstandard regime,
the halfinteger steps are larger than harmonic ones, and
Farey sequence appears only in the order and relative We wish to express our gratitude to Prof. P. J.
sizes of higher order subharmonic Shapiro steps. Lya- Mart´ınez for helpful discussions. This work was sup-
punovexponentanalysisiscertainlyoneofthebestways portedbytheSerbianMinistryofEducationandScience
to get anaccurateansweraboutthe presence ofchaosin under Contracts No. OI-171009and No. III-45010 .
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