By integrating a set of model equations for Josephson ladder subjected to a
uniform transverse bias current we have found almost all of the kinds of
breathers described in recent experiments, and closely reproduced their
voltage-current characteristics and switching behaviour. Our main result is
that a chaotic transient occurs in the switching process. The growth of tiny
perturbations during the chaotic transient causes the new breather
configuration to be extremely sensitive to the precise history of the initial
breather and can also cause the new breather to have a new centre of symmetry.Comment: 6 pages, 4 figure

Aubry

Binder

Domı́nguez

Feodor V. Kusmartsev

Flach

Giles

Hilborn

Kardar

MacKay

Martinez

Mazo

Miroshnichenko

Richard T. Giles

Takeno

Trı́as

English

arXiv

Crossref

Chaotic transients in the switching of roto-breathers

By integrating a set of model equations for Josephson ladder subjected to a uniform transverse bias current we have found almost all of the kinds of breathers described in recent experiments, and closely reproduced their voltage-current characteristics and switching behaviour. Our main result is that a chaotic transient occurs in the switching process. The growth of tiny perturbations during the chaotic transient causes the new breather configuration to be extremely sensitive to the precise history of the initial breather and can also cause the new breather to have a new centre of symmetry

Richard Giles (1251693)

Feodor Kusmartsev (1251207)

Loughborough University Institutional Repository

arXiv:cond-mat/0101439 v1   29 Jan 2001Chaotic transients in the switching of roto-breathersRichard T Giles and Feodor V KusmartsevDepartment of Physics, Loughborough University, Loughborough, LE11 3TU, U.K.April 12, 2006By integrating a set of model equations for Joseph-son ladders subjected to a uniform transverse biascurrent we have found almost all of the kindsof breathers described in recent experiments, andclosely reproduced their voltage-current characteris-tics and switching behaviour. Our main result is thata chaotic transient occurs in the switching process.The growth of tiny perturbations during the chaotictransient causes the new breather configuration to beextremely sensitive to the precise history of the ini-tial breather and can also cause the new breather tohave a new centre of symmetry.The recent discovery[1, 2, 3] of roto-breathers inJosephson ladder arrays (Fig. 1) driven by a trans-verse dc bias current have shown not only that theselocalised excitations exist but that they exhibit re-markable behaviour.The Josephson ladder has been studied theoreti-cally for many years. In the absence of a driving cur-rent the interaction between vortices has been foundto be exponential and this leads to the vortex densityexhibiting a devil’s staircase as the magnetic field isincreased[4, 5]. Quantum fluctuations[6], meta-stablestates[7] and inductance effects[8] have been studiedas has the interaction of vortices with a transverse dcbias current[9, 10].A roto-breather[11] is a stable group of verticaljunctions rotating together (θ′j − θj ≈ ωt, see Fig. 1).Such solutions have been studied theoretically[12] inladders driven by a uniform transverse ac bias cur-rent and are also stable[13] for the case of a dcbias current. Rotation at a single vertical junctionwas studied numerically[14, 15] and two experimentalgroups[1, 2] have now independently confirmed theexistence of such solutions and discovered further un-predicted behaviour. In an annular ladder[1] (Fig. 1a)and in a linear ladder[2, 3] (Fig. 1b) breathers wereobserved with various numbers of “vertical” (or ra-dial) junctions rotating. The voltage drop associatedwith rotating vertical junctions causes some of thehorizontal junctions to rotate and various configura-tions were commonly observed (Fig. 2).However the most interesting and puzzling featurewas the switching between breathers as the bias cur-rent was varied. Once a breather has been initialisedit is stable, but if the bias current is slowly decreasedthen, at some critical value of the current, the config-uration of rotating junctions may change thus form-ing a new breather. Furthermore, it appears fromthe data that the new breather may have a differentcentre of symmetry from the old breather despite thesymmetry of the ladder itself and the symmetry of thedriving currents about a single vertical junction. Wepresent here the first numerical studies of the switch-ing behaviour of roto-breathers and show that thiseffect arises from the exponential growth of even in-finitesimal perturbations during a chaotic transientwhich is associated with the switching process.Aiming to describe the experiments as accuratelyas possible, we first built up a model in which thecurrent I through a junction is determined by thewell-established RSJ model equationIIc=d2ϕdt2+ αdϕdt+ sinϕ (1)where Ic is the critical current and ϕ = ∆θ −2piΦ0∫A.dl where ∆θ is the change in superconduct-1ing order parameter θ across the junction and A isthe vector potential. The “vertical” (i.e. radial)junctions may differ in area from the “horizontal”junctions by an anisotropy parameter η = Ich/Icv =Ch/Cv = Rv/Rh where Ich (Icv), Ch (Cv) and Rh(Rv) are, respectively, the critical current, capaci-tance and resistance of a horizontal (vertical) junc-tion. From Fig. 1a we see that there are three un-knowns per plaquette: θj , θ′j and fj . To solve forthese unknowns we construct three equations per pla-quette as follows. The first two equations are ob-tained from current conservation at the top (inner)and bottom (outer) rails. The third equation is ob-tained by making the approximation that the inducedflux fj − fa (where fa is the applied flux, assumedconstant) is produced solely by the currents flowingaround the immediate perimeter of the jth plaquette(Fig. 1c):fj − faΦ0=βL8pi(Ivj+1 − Ihj − Ivj + I′hj)(2)where βL is an inductance parameter and we havechosen Icv = 1. Eq. (2) introduces a significant dif-ference from previous models[14, 15] which assumethat the induced field is proportional to the loop (or“mesh”) current circulating the plaquette. For squareplaquettes, Eq. (2) represents an improvement sinceit properly takes into account the flux produced bythe vertical conductors, although it is still not a fullnearest-neighbour mutual inductance calculation[16].Using Landau gauge we then have the followingthree coupled differential equations for each plaque-tte:{ddt2+ αddt}(−θ−j−1 + 4θ−j − θ−j+1)= 2Ij + sin θ−j−1−4 sin θ−j + sin θ−j+1 +8piβL(fj−1 − fj)(3){ddt2+ αddt}(χ+j − χ+j−1)=sinχ+j−1 cosχ−j−1 − sinχ+j cosχ−j (4){ddt2+ αddt}(θ−j+1 − θ−j + 2ηχ−j)=sin θ−j − sin θ−j+1 +8piβL(fj − fa)− 2η sinχ−j(5)where θ−j = θ′j − θj , θ+j = θ′j + θj , χ−j =12(θ−j −θ−j+1 − 2pifj), χ+j =12(θ+j − θ+j+1 − 2piF/N) and Nis the number of plaquettes. Putting F = 0, this isa sufficient set of equations for describing the linearladder (with the imposition of the appropriate bound-ary conditions). For the case of the annular ladderwe also need one further equation for the loop flux F(proportional to the total current circulating aroundthe annulus):d2Fdt2+ αdFdt=N2(Fa − F )2ηBL+N2piN∑j=1sinχ+j cosχ−j(6)where Fa is the applied flux, assumed constant, andBL is the annulus inductance parameter. Note thatthe equations remain invariant if all θ+j are rotatedby the same amount, an amount which can also bevaried with time. It is only differences in θ+ from onesite to another which are important. Note also thatthe equations remain invariant if θ−j and/or θ+j arerotated by 4pi (not 2pi).In the RSJ model, on which our equations arebased, each ideal Josephson junction has in parallelwith it a capacitance and a resistance, or in otherwords a frequency filter. The cut-off frequency is1/(RC) = α. Thus voltage oscillations of frequencyω ≫ α tend to be suppressed, being shorted out bythe capacitance. In the experiments, and in our sim-ulations, the rotating junctions have dθ/dt ≫ α andso we can expect voltage oscillations to be suppressedby a large factor. We might therefore expect that allobserved attractors should be characterised by hav-ing all rates of rotation, currents and fluxes at moreor less fixed values. Many junctions will not rotateat all, in which case Eq.(1) gives us I = Ic sinφ. Forthose that rotate Eq.(1) gives us I = αdφ/dt, i.e. thesupercurrent and capacitative current more or lesscancel one another. We call this the dc approx-imation . Using this approximation one can solvefor the distribution of currents, voltages and fluxesfor any configuration of rotating junctions. The ob-served roto-breathers are in fact quite well describedby this approximation.2We now focus on determining whether or not ourmodel exhibits the interesting switching and symme-try breaking behaviour observed in the data of Binderet al[1, 3]. Attempting to mimic the experiments, weuse a random number generator to give θj , θ′j and fjextremely small, but non-zero, initial values at t = 0.If instead, the initial state were taken as having allθj = θ′j = 0 then we could neglect Eq.(4) and allsolutions would maintain the symmetry χ+j−1 = χ+jfor all j; however it turns out[14] that this symmetricstate can be unstable to small perturbations. LetIj ={IB + I∆ if j = 0IB otherwise(7)where IB is called the bias current. Again followingthe experiment, we slowly increase I∆ while keepingIB = 0 until rotation starts at site j = 0. I∆ is thenslowly decreased while at the same time increasingIB to keep I0 constant. When IB has reached thedesired value it is then held fixed while I∆ is slowlyreduced to zero. At this point, provided IB is nottoo large or small, we have just one rotating verti-cal junction (at j = 0), the horizontal junctions ro-tating in the I -configuration (Fig. 2). We call thisan I -breather with NR = 1 rotating vertical junc-tions. In Fig. 2a the I -breather appears to have top-bottom symmetry, but in fact this symmetry is bro-ken since θ+ (which is more or less static) actuallyvaries from site to site. Also the currents circulat-ing the two plaquettes on either side of the rotat-ing junction cause a static flux pattern rather like avortex-antivortex pair, the value of the flux in eachplaquette being what one would deduce in the dc ap-proximation. This must cause a compressive force onthe breather and an attractive force with the end ofany nearby breather of the same kind.Finally IB is slowly reduced while keeping I∆ = 0.The voltage current characteristics of the breathersproduced as IB is ramped down are shown in Fig. 3.We show here only the results for an annular lad-der with N = 8 vertical junctions and the parametervalues otherwise chosen to mimic the annular ladderexperiment[1, 17]. We have also put BL = 0 sincewe find that although the inner region of the lad-der tends to pick up trapped flux (a vortex) in thebreather switching process, this has little influenceon the voltage-current characteristics for this range ofparameter values. Note that while I and T -breathersoccur, there are no Z -breathers. This is because inthe annular ladder the two ends of a breather areconnected by a quiescent region in which all super-conducting order parameters must be static or rotatetogether in synchrony (apart from small vibrations),and so Z -breathers cannot exist in the absence ofother breathers.All of the breathers are well characterised by a con-stant resistance R (i.e. V = 〈dθ−/dt〉 = IBR) andthe value of R agrees very closely with the value ex-pected in the dc approximation. Furthermore we seethat at a critical rotation frequency the single-site(NR = 1) I -breather generally broadens into a multi-site I -breather, but may also convert to a single-siteor multi-site T -breather. In agreement with the ex-periments, this occurs when the rotation rate of thehorizontal junctions is close to the minimum rotationrate that could be supported if the junctions wereisolated and driven by a constant current (cf. themiminum rotation rate of an underdamped pendu-lum driven by a constant torque). This critical fre-quency is more or less independent of α (for smallα). The multi-site I -breathers switch to T -breathersat the same critical frequency. In the switching, twoof the four rotating horizontal junctions stop whilethe other two double in frequency (to maintain Kir-choff’s Law). As IB is reduced further the rotationrate of the junctions falls again until the same criti-cal rotation rate is reached at which point all motionceases. We obtain qualitatively similar results bothfor longer (and shorter) ladders and for linear ladderswith similar parameter values. The main differencein linear ladders is that Z -breathers are producedalso. These observations agree very well with the be-haviour reported for a linear ladder with similar pa-rameter values[3]. The annular ladder experiment[1]did not find any T -breathers but his may very wellbe due to a construction fault[18].Ref. [3] shows that the linear ladder can supportat least several types of breathers. Nearly all of theconfigurations found experimentally arose naturallyin our simulations. Furthermore we found that in amulti-site breather, it appears that any two adjacent3vertical junctions can rotate in phase or 180◦ out ofphase. It is clear that a great many breather config-urations are possible.The main result of the simulations relates to theswitching to multi-site breathers. Although thesingle-site I -breather possesses left-right symmetryabout j = 0, the multi-site breather produced in theswitching process frequently has a different centre ofsymmetry. Any breather with an even number NRof rotating junctions cannot have its centre of sym-metry at j = 0. Such breathers are produced evenin the annular ladder (where translational invarianceis exact) and even if we set all variables to zero ini-tially, in which case the dynamical equations, initialconditions and injected currents all have exact left-right symmetry about site j = 0. The origin of thesymmetry breaking lies in the occurrence of chaoticdynamics during the switching. Further evidence ofchaotic behaviour is provided by the fact that eventhe smallest change in initial conditions or of any pa-rameter usually results in the production of a com-pletely different kind of multi-site breather. Both ofthese effects (i.e. symmetry breaking and extremesensitivity) were noticed in the experiments[17].To study the origin of these effects we have lookedmore closely at what happens when the single-site I -breather becomes unstable at V = 4.3 and switchingtakes place (see Fig. 3). Approaching the switchinginstability by reducing IB very slowly, we find that atIB = 0.4524 the breather appears to be stable and pe-riodic with no positive Lyapunov exponent. Howeverwhen the current is lowered to IB = 0.45235 the mo-tion becomes weakly chaotic with a maximum Lya-punov exponent (Fig. 4a) of 5×10−5; but the chaoticmotion is itself unstable and switching to a new sta-ble breather eventually takes place after 460 000 timeunits. Thus the switching between breathers occursvia a chaotic transient[19]. Clearly we have herethe origin of the observed symmetry breaking andextreme sensitivity. The nature of the resulting fi-nal breather is extremely sensitive to the smallestchange in the simulation (as one might expect giventhe chaotic nature of the motion) but the time beforeswitching occurs is more or less fixed. If, before wait-ing for switching to occur, the bias current is slowlylowered further to 0.45234 then the maximum Lya-punov exponent increases to 2 × 10−4 and switchingoccurs after only 65 000 time units (Fig. 4b).The only clear failures in the predictive power ofthe model occurred near the superconducting gap fre-quency where the RSJ model is bound to fail. Inthis region the experiments found that breather resis-tances changed and that T -breathers could be con-verted to I -breathers; neither of these effects werereproduced by the simulations.We conclude that our simulations exhibit mostof the main features of the roto-breathers recentlyobserved in Josephson ladder arrays. In additionto reproducing the experiments we have found thatswitching occurs via a chaotic transient, and this maybe a general feature of the switching of roto-breathersin Josephson arrays. This chaotic transient has somepractical consequences. First of all it leads to theextreme sensitivity and symmetry breaking alreadyobserved in the experiments. Secondly, since it hashas a very broad Fourier spectrum, characteristic ofchaotic motion, we can predict the existence of asource of noise associated with switching. The du-ration of the noise (i.e. the transient) should dependon how quickly the bias current is ramped (Fig. 4).We hope that future experiments will discover thenoise and confirm this effect.We are grateful to A.V. Ustinov and P. Binderfor kindly giving us their data and explaining itprior to publication. We are also grateful for dis-cussions with M.V. Fistul, S. Flach, A. Osbaldes-tin and J.H. Samson and for the hospitality of theMax-Planck-Institut, Dresden and the Universita¨tErlangen-Nu¨rnberg.References[1] P. Binder et al., Phys. Rev. Lett. 84, 745 (2000).[2] E. Tr´ıas, J. Mazo, and T. Orlando, Phys. Rev.Lett. 84, 741 (2000).[3] P. Binder, D. Abraimov, and A. Ustinov, Phys.Rev. E 62, 2858 (2000).[4] M. Kardar, Phys. Rev. B 30, 6368 (1984).4[5] R. Giles and F. Kusmartsev, J. Low Temp. Phys.117, 623 (1999).[6] M. Kardar, Phys. Rev. B 33, 3125 (1986).[7] J. Mazo, F. Falo, and L. Flor´ıa, Phys. Rev. B52, 10433 (1995).[8] J. Mazo and J. Ciria, Phys. Rev. B 54, 16068(1996).[9] S. Kim, Phys. Lett. A 235, 408 (1997).[10] M. Barahona, S. Strogatz, and T. Orlando,Phys. Rev. B 57, 1181 (1998), and referencestherein.[11] S. Takeno and M. Peyrard, Physica D 92, 140(1996).[12] P. Martinez, L. Flor´ıa, F. Falo, and J. Mazo,Europhys. Lett. 45, 444 (1999).[13] R. MacKay and J.-A. Sepulchre, Physica D 119,148 (1998).[14] J. Mazo, E. Tr´ıas, and T. Orlando, Phys. Rev.B 59, 13604 (1999).[15] S. Flach and M. Spicci, J. Phys.: Condens. Mat-ter 11, 321 (1999).[16] D. Domı´nguez and J. Jose´, Phys. Rev. B 53,11692 (1996).[17] P. Binder, private communication.[18] See footnote 14 of Ref. [3].[19] R. Hilborn, Chaos and Nonlinear Dynamics(Oxford, UK, 1994).θjθ′j-1θ′j+1θ′j+2θ′j-2θ′jθj+1θj+2 θj-1θj-2fj-1fj-2fj-3fjfj+1fj+2IjIj+1Ij+2Ij-1Ij-2(a)IB(b)IvjIhj(c)I′hjIvj+1Figure 1: (a) An annular ladder subjected to trans-verse bias currents Ij . Each circle represents a su-perconducting island. Each link between islands rep-resents a Josephson junction. θ is the phase of thesuperconducting order parameter and f is the fluxthreading a plaquette. (b) A linear ladder. (c) Ex-planation of the notation used in Eq. (2).(a) (b) (c)Figure 2: Three configurations for the rotation ofhorizontal junctions around a breather with NR = 1vertical rotating junctions: (a) the I -breather, (b)the T -breather, and (c) the Z -breather.50.50.60.70.80.91 2 3 4 5 6 T3chaotic transientSwitching viaT1T2T4T6I1I4I6Two critical frequenciesV =d 0dtR1Figure 3: Resistance αR (where R = V/IB) plot-ted against voltage V =〈dθ−0 /dt〉. The resultswere obtained for an annular ladder with N = 8vertical junctions, α = 0.07, η = 0.44, βL = 4.8,BL = Fa = fa = 0. The middle vertical junction wasexcited in the manner described in the text and thenthe uniform bias current IB was ramped down. Thedata come from many runs with different starting val-ues of IB . The labels T2, I1 etc. refer to breathertypes, e.g. “I1” means the one-site I -breather.1e–091e–081e–071e–061e–05.1e–3.1e–2.1e–1.1.1e2.1e310 20 30 40 50 60 701e–091e–081e–071e–061e–05.1e–3.1e–2.1e–1.1.1e2.1e3100 200 300 400 5001(a) (b)10 60 310 9 0=j 0perturbed  0jt=103t=1031Figure 4: Exponential growth of perturbations dur-ing the chaotic transient which occurs when thesingle-site I -breather becomes unstable at V =〈dθ−/dt〉 = 4.3. (a) The initial state (t = 0) wasprepared with IB = 0.45235. A perturbed initialstate was then produced by introducing small ran-dom changes to all state variables. After an initialperiod ∆θ−0 starts to grow exponentially with time.Switching to a new stable breather takes place att = 460, 000 and produces the sudden rise in ∆θ−0at that time. (b) Same except the initial state wasprepared with IB = 0.45234.6

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Chaotic transients in the switching of roto-breathers

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