Slow-fast dynamics in Josephson junctions

Transcription

1 Er. Phys.. B 3, (3) DOI:./epjb/e3-5- THE EUROPEAN PHYSICAL OURNAL B Slow-fast dynamics in osephson jnctions E. Nemann a and A. Pikovsky Department of Physics, University of Potsdam, Am Neen Palais, PF 553, 5, Potsdam, Germany Received March 3 Pblished online Agst 3 c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 3 Abstract. We report on a nontrivial type of slow-fast dynamics in a osephson jnction model externally shnted by a resistor and an indctor. For large vales of the shnt indctance the slow manifold is highly folded, and different types of dynamical behavior in the fast variable are possible in dependence on the other parameters of the jnction. We discss how particlar featres of the dynamics are manifested in the crrent-voltage characteristics of the shnted jnction. PACS a Nonlinear dynamics and nonlinear dynamical systems 85.5.Cp osephson devices 7.8.-g Inhomogeneos spercondctors and spercondcting systems Introdction Resistively shnted osephson jnctions appear in many applications, as for instance in osephson jnction arrays and ladders [ 3] and in spercondcting qantm interference devices (SQUIDs) []. The simplest possible model to describe the dynamics of a osephson jnction is the resistively and capacitively shnted jnction (RCS) model where a resistor and a capacitor are shnted parallely to the ideal osephson jnction flown throgh by the spercrrent. However, if the resistive shnt de to its wiring contains a non-negligible indctance, the wellknown RCS model fails to describe the crrent-voltage characteristics of the jnction. Sch an indctance has to be acconted, e.g. in the case of the SQUID [] where a single osephson jnction is closed with a spercondcting loop. Moreover, it has been shown that measrements of the crrent-voltage characteristics of a osephson jnction not being closed by a spercondcting loop are well represented by nmerical simlations of the RCS model only below a certain temperatre [5]. For higher temperatres, anomalies in the characteristics have been fond, that can be ascribed to the finite indctance in the shnt. Conseqently, in order to remedy this lack, a resistivecapacitive-indctive jnction (RCLS) model has been sggested to describe these jnctions [5,]. Consideration of this model leads to a better agreement of the nmerical simlations with the experimentally obtained data. Only for large vales of the shnt resistance one can expect almost RCS-like crrent-voltage crves no matter which vale of the indctance is present. The dependence of the dynamics on the strength of the indctance has been considered [, 7]. It has been fond a eireen@stat.physik.ni-potsdam.de that chaotic soltions can be expected at low vales of the indctance, whereas for intermediate vales long chaotic transients precede periodic oscillations. For high indctance vales the dynamics has been fond to be dominated by relaxation oscillations [5]. Sch type of oscillations can also be observed in the resistive single jnction interferometer (SQUID) []. Nmerical simlations of the RCLS model have been sed to reprodce the qalitative featres of the crrentvoltage characteristics [7,8]. Based on this, a method has been sggested to determine the device parameters sch as the capacitance and the indctance of the jnction. Indeed, measrements of the instantaneos voltage across the jnction are impossible since the characteristic time scale is in the order of picoseconds. Only the measrements of the dc voltage in dependence on the varied net crrent are possible and can be compared with nmerically obtained crrent-voltage characteristics. In this way one can show that anomalies in the crrent-voltage crves are cased by particlar relaxation oscillations and complex voltage waveforms [8]. Frthermore, the indctively shnted jnction model forced by an external ac signal has been considered in [9]. In this article we investigate the dynamics of the RCLS model in the presence of small parameters which allow s to separate slow and fast motions. In particlar, the shnt resistance is assmed to be small. Then, the type of the dynamics is described in dependence on the parameters characterizing the capacitance and the shnt indctance, respectively. We will demonstrate that depending on these parameters one can observe different types of slow-fast dynamics. Coexistence of slow and fast motions is well known from self-sstained oscillators like the van der Pol oscillator and excitable systems like the Fitz Hgh-Nagmo model. Moreover, models

2 9 The Eropean Physical ornal B I I S R S oter shnt is acconted containing the shnt resistance R S in series with the shnt indctance L. Sothismodel, in fact, represents an pgrade of the sal RCS model. The dynamics is described by the following eqations [8]: I c sin R C L I = C dv dt + V R + I c sin + I S, V = d e dt = L di S dt + I SR S. () Fig.. The schematic representation of the RCLS model. The osephson jnction is represented by the spercrrent channel I c sin, the normal resistance R, and the jnction capacitance C. The indctive shnt consists of a resistor R S in series with an indctance L. describing activator-inhibitor systems in biological isses like the spiking-brsting neronal activity [, ] show slow-fast dynamics. In these models, at certain parameter vales a limit cycle can be observed consisting of two pieces of slow motion connected with fast jmps. We will demonstrate that the motions in the RCLS model are more complex becase the fast manifold can be twodimensional. We do not consider the most general formlation of the problem, instead concentrating on the aspects of the dynamics. In particlar we assme that the magnetic field is absent, and that the resistance is linear; see [5 9] for more general formlation. The paper is organized as follows: In the following Section we introdce the model and derive the eqation of motion. The choice of parameters is discssed with respect to the possibility to observe slow-fast dynamics. Then, in Section 3 the dynamics is analysed for three types of damping. The crrent-voltage characteristics is discssed, and is analysed in its relation to the dynamics. Finally, the reslts are smmarized in Section. Model. Basic eqations The system nder consideration, the RCLS model of a osephson jnction, is sketched in Figre. The inner circit represents the simple RCS model; here the ideal osephson jnction is flown throgh by the spercrrent I C sin() withot any resistance. Since in real applications a voltage appears across the jnction which can be time dependent, the ideal jnction is shnted parallely by aresistorr and a capacitor C. Althogh dynamics of real jnctions can be reprodced in many cases by this simple model, some additional special featres of experimentally observed crrent-voltage crves appear if the shnt of the jnction contains a non-negligible indctive component. To make the electronic circit model more realistic, the Here, I is the net crrent flowing throgh the jnction and is the phase difference between the complex wave fnctions of the Cooper pairs in two pieces of spercondctor constitting the jnction, called osephson phase. The shnt crrent I S passes the shnt resistance R S and the shnt indctance L that models the wiring of the resistor. The voltage V across the jnction is determined by the second eqation of eqation () and is related to the time derivative of the osephson phase by the well-known phase-voltage relation. Ths, the expressions containing voltage V can be rewritten in terms of the osephson phase. Then, for reasons of convenience of analytical and nmerical investigation let s transform eqation () to a dimensionless form: βγ d dτ + γ d dτ +sin = i S, d dτ = αdi S dτ + i S. () Here we normalized time τ = ω S t with ω S = ei C R S / as well as crrents I S and I with respect to the maximm amplitde of the spercrrent I c : = I/I c, i S = I S /I c. Actally, the normalized net crrent can be time-dependent bt we concentrate in the following on a simpler case and restrict orselves on a constant vale of from now on. In eqation () three parameters α, β, and γ have been introdced, they are defined as follows. Parameter α = ei cl (3) represents a dimensionless indctance, becase ei c can be regarded as an intrinsic indctance L of the osephson jnction. Parameter β = ei cr C () is the Stewart-McCmber parameter ofthe circitwithot the oter shnt. It is in fact a redced capacitance and measres the damping of the system (β corresponds to high damping). The third parameter γ designates the ratio of the shnt resistance to the intrinsic one γ = R S /R. (5) As mentioned in the introdction (Sect. ), system () has already been considered in references [5 9]. However, there

3 E. Nemann and A. Pikovsky: Slow-fast dynamics in osephson jnctions 95 the damping parameter has been defined in a different way: β S = ei CRSC, () containing the shnt resistance R S.Intermsofβ S = βγ, the system modeling the dynamics of the osephson jnction reads β S + γ +sin = i S, = α di S dτ + i (7) S. A disadvantage of eqation (7) is that the shnt resistance R S is now contained in two different dimensionless parameters, namely in β S and γ. Ths, from experimental point of view, a change of the shnt resistance R S leads to some shift in the parameter plane (β S,γ), i.e. to a change of both parameters β S and γ. According to or definition (), a change of R S reslts in the variation of only one parameter γ, if we assme the intrinsic resistance R to be a constant. For a possible experimental check of the reslts concerning the dynamics and crrent-voltage characteristics, the dimensionless parameters α, β, andγ can be transformed easily to corresponding vales of the shnt indctance L, the capacitance C, and shnt resistance R S of the experimental setp and vice versa. It is appropriate to write the system in terms of β (Eq. ()) not only from the experimental point of view, bt also with respect to the analysis of slow-fast dynamics of the system. For or way of defining the dimensionless parameters, it is obvios that γ shold serve as the small parameter of the system. In contrast, system (7) is nsitable for investigating slow-fast dynamics becase it is not clear how to chose a small parameter. Before analyzing system () with a restriction of parameter γ being small, let s consider a limiting case where the indctance L of the shnt is negligibly small. Then, the dynamical eqations shold redce to these of the RCS model. According to eqation (3), a negligibly small indctance L corresponds to the limit α. From the second eqation of eqation () we obtain in this case = i S,ths the dynamics is described by a single differential eqation of second order, βγ +(γ +) +sin =. (8) After backscaling time by t = τ/ω S, we obtain the differential eqation C + ( + ) +sin =, (9) ei c ei c R S R corresponding to the RCS model not acconting for the external indctivity of the jnction. It describes the dynamics of an ideal osephson jnction shnted by a capacitor C and two resistors, R S and R.. Transformation of the basic eqations The analysis of eqation () is not obvios becase it is not a system of ordinary differential eqations (ODEs) in a standard form. There are many ways to write it as a system of ODEs bt the most appropriate one for the analysis below is to introdce a spplementary variable = i S + +γ + βγ. This kind of transformation α α ensres that slow and fast manifolds are perpendiclar to each other as is demonstrated below. Moreover, parameters β and γ occr only in the first eqation of the two ODEs: ( βγ + γ + βγ ) α + +γ α +sin =, = α () ( sin ). Eqation () can be transformed into a system of three ODEs of first order that can be integrated sing standard algorithms. It describes the time evoltion of the osephson phase and its first derivative as well as the dynamics of the jst introdced new variable. In addition to the external normalized net crrent, the dynamics is characterized by the three parameters α, β, andγ. It is convenient to consider parameter γ as the small parameter being contained in the terms inclding first and second derivatives of variable in eqation () in corresponding power. Ths, the characteristic time scale the variable evolves on, is of the order γ whereas variations in variable are characterized by a time scale of the order α. Therefore, the osephson phase can be identified as the fast variable and as the slow one, provided γ α The variable is the more rapid the smaller the vale of γ is. In the limit γ thefast motion in becomes instantaneos, and we arrive at the simplified model = α +sin, = α () ( sin ), characterizing only the slow motion. Eqation () consists of an algebraic eqation describing the shape of the slow manifold = () and a differential eqation for the time evoltion on this slow manifold. The shape of the slow manifold is determined by parameter α. Forα < the fnction = () is bijective, i.e. is a fnction of, too.anexampleforthiscase is presented in Figre a. The sitation changes at α =, then is mltivaled at vales k = π +kπ, these points are the inflection points of the fnction (). For α>, the slow manifold () becomes folded. This is an interesting and nontrivial case, where one can expect slow-fast dynamics to occr. A trajectory follows the slow manifold p to its local maximm and then a fast jmp of the variable along the fast manifold to another branch of the slow manifold is expected to occr. For α>α., is mltivaled at any vale of (Fig..Aswewillseebelow,thisisimportantforcertain phenomena to be observed in the slow-fast dynamics.

4 9 The Eropean Physical ornal B Fig.. Shape of the slow manifold = () in dependence on the vale of the redced indctance parameter α of the osephson jnction. For α =.5 < the fnction is bijective, and for α => the fnction () is folded. 3 Dynamics and crrent-voltage characteristics Before discssing the different featres of slow-fast motion let s describe a range of parameters we are going to consider in this section. As already mentioned, parameter γ is assmed to be small, i.e. γ. With regard to parameter α, we consider the interesting case α>α., where the motion on slow and fast time scales can be observed. The Stewart-McCmber parameter β determines the strength of damping. We will discss three types of damping in detail, namely the high damping limit (β ), intermediate damping (β >), and the low damping limit (β ). The crrent-voltage characteristics reflecting the dependence of the dynamics on the normalized net crrent is analysed for these three types of damping, and the dynamical featres are discssed. 3. High damping limit In the high damping limit β, eqation () redces to a set of two first-order differential eqations: γ = +γ α sin, = α ( sin ). () Fig. 3. Types of dynamics in the high damping limit. For α =.9 < the trajectory follows the slow manifold, whereas for α => slow-fast dynamics can be observed. Ths, the system of the overdamped osephson jnction (Eq. ()) is of the same type as typical systems demonstrating slow-fast dynamics (e.g. the FitzHgh- Nagmo model) since it is characterized by the small parameter at the first derivative of one variable ot of two. Let s consider the soltion of () for two vales of α, α =.9and α =, respectively. The strctre of the slow manifold for these vales of α differs as pointed ot in the previos section. Therefore, the phase portraits (, )presented in Figres 3a, b show different types of dynamics. In the case α< the trajectory follows the slow manifold (Fig. 3, the motion on = () is niform. For α> (Fig. 3, de to the folded strctre of the slow manifold, the slow-fast dynamics is observed. In this case the motion takes place on two different well separated time scales. Slow motion along stable branches of the slow manifold alternates periodically with fast jmps in. Note that the fast motion is one-dimensional and tangential to the direction of slow motion at local maxima of fnction = (). For large α, the jmp length in the fast variable is approximately π whereas changes only slightly dring motion on the slow manifold. In order to estimate the crrent-voltage characteristics, the mean voltage has to be compted in dependence on the net crrent. Figre shows the crrentvoltage crves for two different vales of α, α =and α =, respectively. No matter which vale of α is accepted, for < the mean voltage across the jnction is zero ( = ). Ths, a spercrrent flows throgh the jnction withot any resistance. Dynamically it means that the osephson phase approaches a fixed point. The

5 E. Nemann and A. Pikovsky: Slow-fast dynamics in osephson jnctions 97 <>...8. α= α= Fig.. The crrent-voltage characteristics in the high damping limit for α =andα =. For <themean voltage across the jnction is zero, for > the jnction is in the resistive state, where the resistance depends on α. The dashed vertical line marks the vale =.5 for which the time dependence of is presented in Figres 5a, b. For several vales of, the semi-analytically estimated vales of the mean voltage are marked by diamonds for α =andα =. fixed point coordinates (, ū) aregivenby =arcsin and ū = +γ α +. In contrast, for >, the jnction is in the resistive state, i.e. a nonzero mean voltage appears across the jnction. Ths, the characteristics consists of two branches, the spercrrent branch and the resistive one. For two vales of α, α =andα =, the time dependence of the osephson phase is presented in Figres 5a, b at the same vale of the net crrent =.5. Both dependences reflect the periodic repetition of the fast jmps in by approximately π, bt the mean slope and, ths, the mean voltage across the jnction differ. At the same vale of one observes that for large α, themean voltage is small. The mean voltage can be compted as well semi-analytically as is shown below. First, or goal is to estimate the dration of the slow motion. To this end, we assme that the fast jmps are instantaneos, i.e. γ =. Then, eqation () redces to eqation (), and the expression for the slow manifold (first eqation of ()) is differentiated with respect to time, and is sbstitted to obtain = sin +αcos (3) Integration of this eqation yields T = ( + α cos )d sin ( ) sec(/)(cos(/) sin(/)) atan = α ln( sin ). () t 8 Fig. 5. Time dependence of the osephson phase at γ =., =.5, and α =, and α =. The mean slope and ths the mean voltage across the jnction depends on α. To estimate the period T we have to find the limits of integration and being the limits of the phase interval within which the slow motion takes place. For explanation of the estimation of this interval see Figres a, b where the trajectory in the phase space (, ) issketchedfor α =andα =, respectively. The periods of slow motion end at those vales of for which the fnction () =/α+sin() has local maxima. st at these vales n =πn + arccos( /α) thefast jmps in start. In Figres a, b the first two fast jmps are presented, starting at c = = arccos( /α) and =π + arccos( /α), respectively. The first fast jmp starting from ( c, c = ( c )) ends at (, c ) on the next branch of the slow manifold. Ths, the sbseqent period of slow motion along () is limited by (, c ) and (,( )). The still needed estimation of leads to a transcendent eqation, c = /α +sin() which cannot be solved analytically. However, we can find its root nmerically. Now, having the vales and for α = and α =, the dration T of slow motion can be compted according () for each α. This has been done for several vales of the net crrent (, ] for both vales of α. Then, the mean voltage can be estimated by t = π T (5)

6 98 The Eropean Physical ornal B slow motion 8 c fast jmp c slow motion fast jmp c c Fig.. Visalization of slow and fast motion in the phase space (, ). For α = the phase interval (, ) of slow motion is larger than for α =. The fast jmp in the osephson phase can be approximated by π the better the larger α is. becase the period in is eqal to π. Usingthedata of period T compted for both vales of α, the corresponding vales of the voltage have been estimated according to expression (5) and marked in Figre by diamonds. The semi-analytically estimated vales of the mean voltage confirm those obtained by integrating system (). Note, that the deviation from the prely nmerically obtained resistive branch of the characteristics becomes larger for increased. The assmption that the fast jmp happens instantaneosly is more reliable the larger α is. Therefore, the deviation is smaller for α = than for α =. 3. Intermediate damping For intermediate damping (β ) the dynamics of the osephson jnction is described by eqation (). Again, one can expect to observe slow-fast dynamics since parameter γ is assmed to be small. However, the jmp in the fast variable depends now on the damping parameter β. We do not only observe jmps by approximately π (Fig. 7, bt also by mltiples of π (Figs. 7b, c) at α =. Let s have a closer look onto the strctre of the slow manifold (Fig. 8). It consists of stable and nstable branches. The trajectory follows the slow manifold along the stable branch p to its first local maximm, denoted by c. Then the first fast jmp wold occr in the di c) 3 5 Fig. 7. Different jmp length in the osephson phase in dependence on the strength of damping. One observes at β = jmps by π, β = jmps by π and c) β = 3 jmps by π. The other parameters are α =, γ = and =.5. rectionmarkedbythearrowwhich crosses three stable branches. Depending on the vale of β, the jmp length can be approximately π, π or π corresponding to the first, second and third crossing of stable branches, respectively. It is obvios that for large vales of α, theslow manifold is stronger folded and more crossings can occr. Hence, jmps by higher mltiples of π can be expected to happen. On the other hand, for α<α., when the slow manifold is not folded strong enogh and the arrow wold not cross any stable branch, we cold observe only jmps by π in as in the high damping limit. In that

7 E. Nemann and A. Pikovsky: Slow-fast dynamics in osephson jnctions 99 c saddle-node bif. nstable branches P c S S ~π ~ π ~ π stable branches - c 3 Fig. 8. Strctre of the slow manifold. Stable branches are presented by solid lines, nstable branches by dashed lines. The arrow shows the direction of an instantaneos jmp whose length depends on the vale of damping parameter β. P c S c Fig. 9. Schematic representation of the nstable trajectory of the semi-stable point P c =( c, c, = ) (solid line) for two different vales of β. This trajectory can go to one or another stable fixed point depending on its position relative to the stable manifold of the saddle point S (dashed line). As a reslt, fast motion can lead to a jmp of by π or π. case, the jmp dynamics wold be qalitatively independent of the vale of damping parameter β. Provided that the slow manifold is folded strong enogh to make slow-fast dynamics possible, what type of mechanism is responsible for the jmp length? In order to answer this qestion we consider the fast motion of the first jmp taking place in the sbspace ( = c,, ) in more detail. A schematic representation of the topology arond the point P c =( c, c, = ) from where the first fast jmp starts is presented in Figre 9. At P c a saddle-node bifrcation occrs, i.e. stable and nstable fixed points collide there, and then both disappear (typically a stable point is a focs, bt in the vicinity of bifrcation point it becomes a node). In the intersection plane ( c,, ), the point P c is a semi-stable point, characterized by only one nstable trajectory. As observed, the different jmp length depends on the vale of β. In Figre 9 two different trajectories (at two different vales of β) starting from P c and seperatrices (dashed crves) of the saddle point S are sketched. This saddle point S is jst the intersection point of the plane ( c,, ) with the first nstable branch next to P c (see Fig. 8). Obviosly, the jmp length depends on whether the nstable trajectory is above or below the stable manifold of saddle point S. For small β it is below, and the jmp length is approximately π. For larger β, the trajectory moves above the stable manifold of S,thsthe jmp length changes and becomes larger than π. The same mechanism jst described nderlies the behavior of trajectories near the second saddle point S even thogh it is omitted in Figre 9 for clarity of representation. To prove these theoretical considerations, let s decople the fast jmp dynamics from the slow motion by observing the system () only on the fast time scale. The corresponding transformation to the time scale t = γτ and sbseqent consideration of the dynamics in the limit γ case that only variations of the fast variable are visible while the slow variable is qasiconstant. Ths, the time evoltion of the fast variable is given by β + + α +sin = c, () where c is jst the vale on the slow manifold from where the first fast jmp occrs. The dynamics now starts at corresponding vale c with velocity = (it is the initial voltage across the jnction). Hence, the first fast jmp starting from P c can be simlated nmerically. The crossing points of the plane ( c,, ) (presented by the arrow in Fig. 8) with stable and nstable branches of the slow manifold mark positions of foci and saddles in the (, ) phase plane. By forward and backward integration of eqation () in time, for saddle locations S, S as initial vales of and initial zero voltage =,thestable and nstable manifolds of the saddles S and S in the (, ) plane have been compted (see Fig. ). Ths, we find different connections between the foci and saddles. Then, simlating the dynamics with the initial vales ( c, = ), the trajectory representing the jmp dynamics has been generated. So, one obtains the phase portraits of the fast motion in the (, ) phase plane in dependence on the damping parameter β (Figs. a d). We observe that the fast dynamics is two-dimensional in contrast to that in the high damping limit. Moreover, de to the variation of β, the topology of the connections between foci and saddles changes; the stable and nstable manifolds wind arond the saddles and foci and come nearer to each other as β is increased, i.e. for lowering the dissipation (compare Figs. a d). This topology in fact cases the different jmp lengths ( π, π, π). For the jst considered limiting case γ =,thejmp dynamics is well defined. The regions of phase jmps of different length in the (β,) parameter plane are estimated in dependence on β, and presented in Figre a. The borders between different regions do not depend on the net crrent becase the whole dynamics is independent of (see Eq. ()). They are parallel to the -axes. This sitation changes if we do not restrict or observation onto the fast dynamics bt accont for the inflence of the slowly varying variable (γ )onthefast

8 3 The Eropean Physical ornal B π π π π π c) β π π π π π 8 8 β c) complex cycles π π π π d). 8 8 β Fig.. Regions of phase jmps by mltiples of π in the (β,) parameter plane for γ =,γ =,andc)γ = Fig.. The phase portraits in the (, ) phase plane in dependence on the damping parameter β, the observed jmp length at β =is π, atβ =is π, atc) β = 3 is π and at d) β = 5 is π. Thestable manifolds (thick lines) and nstable manifolds (thin lines) represent the different connections between the saddles and foci. The trajectory representing the jmp dynamics is shown by dashed thick line. motion. Since stable and nstable manifolds of saddles S and S are close to each other for large β (Figs. b d), one can gess that the jmp dynamics may depend sensitively on a nonzero small parameter γ.moreover,for γ> the regions of different jmp length are expected to depend on becase inflences the slow variable (see Eq. ()). Yet, for γ =, we still get a similar pictre (Fig. in the (β,) parameter plane althogh the lines separating different regions depend now on as expected. The sensitivity of the dynamics with respect to the velocity of the slow motion (determined by γ) becomes more visible in Figre c for larger γ =.A strong dependence of the shape and location of the borders

9 E. Nemann and A. Pikovsky: Slow-fast dynamics in osephson jnctions Fig.. Illstration of the sensitivity of the dynamics with respect to the velocity of the slow motion defined by parameter γ. Forα =, β =7and =.5 one can find jmps by π at γ = and by π at γ =.(Theslow manifold is presented by a dashed line.) <> <> β=8 β=5 β=.5 β= Fig. 3. The crrent-voltage characteristics for α = and γ =,γ =. At certain vales of damping parameter β one observes sdden jmps in the mean voltage. 3.3 Low damping limit between different regions on can be observed. Moreover, the region complex cycles in the right hand part of the plot contains sch parameter points for which jmps of different lengths have been fond to alternate. We illstrate the different jmp behavior in dependence on γ in Figre. For γ = we have fond fast jmps by approximately π (Fig. whereas for γ = the jmp length is abot π (Fig.. The other parameters are kept constant (β =7, =.5), and denote a parameter point that for γ = and γ = lies in different regions of phase jmps (see Figs. b, c). Ths, for γ >, one can hardly predict the transitions between different regions of phase jmps. The crrent-voltage characteristics at intermediate vales of damping parameter β show a featre cased jst by sensitivity of the dynamics with respect to parameter γ. For certain vales of β we observe sdden jmps in the crrent-voltage crves (see Figs. 3a,. This is de to the -dependence of jmp regions of different length. If, by variation of, one crosses a crve separating regions of different jmp lengths in the (β,) parameter plane (Figs. b, c), the jmp length changes by approximately π and a sdden jmp in the mean voltage occrs, correspondingly. In the low damping limit (β ) one can actally find the same featres as for intermediate vales of the damping parameter, like jmps of mltiples of π, and also complex cycles. For β an additional featre in the crrent-voltage characteristics, namely hysteresis, can be observed [8]. In Figres a, b the crrent-voltage crves are presented for β = and β = 5, respectively. For β = 5 (Fig. the hysteresis is more prononced than for the smaller vale β = (Fig.. Hysteresis can be observed if there exists bistability. Here, either the trajectory approaches a fixed point soltion or a limit cycle in dependence on the initial condition. Another featre, typical for large vales of β, istheappearance of complex cycles. The reason is that the motion does not evolve on well separated time scales anymore. This can be explained as follows: If we only consider the jmp dynamics withot evoltion of the slow variable, then already for intermediate vales of β, the oscillations rather slowly relax to the fixed point soltion, as can be observed from Figres b d. The time needed for the trajectory to become finally attracted to the fixed point increases with β. So, in the low damping limit, the relaxation time is non-negligible anymore and presents another time scale. It may happen that for large β these relaxation oscillations are not yet finished when the next fast jmp in occrs. As a reslt, the time scales of slow and fast motion are not strictly separated anymore. This fact is a

10 3 The Eropean Physical ornal B.5 <> <> <>.5 Hysteresis Hysteresis λ I II III Fig. 5. The crrent-voltage characteristics in the low damping limit at α =, γ =. and β =. The maximm Lyapnov exponent in dependence on the net crrent. The arrows mark the vales of for which the phase portraits are presented in Figre Fig.. The crrent-voltage characteristics in the low damping limit at α =,γ=. and β = and β = 5. The hysteresis is more prononced for larger vale of β. reason for the appearance of complex cycles, and even of chaos in the atonomos osephson jnction. Figre 5a shows the crrent-voltage crve for β =. Note that the hysteresis is not as mch prononced as for β = and β = 5 (Fig. a,. The maximm Lyapnov exponent has been compted in dependence on in the same crrent interval, and its plot is presented in Figre 5b. In the crrent interval where the osephson phase converges to a fixed point, the maximm Lyapnov exponent is less than zero. At = the dynamical behavior changes de to a bifrcation giving rise to a limit cycle characterized by zero maximm Lyapnov exponent. However, for certain vales of the maximm Lyapnov exponent becomes positive, ths chaotic behavior occrs there. Examples for nontrivial dynamical behavior are presented in Figre. Conclsion We have considered a osephson jnction model acconting a nonzero shnt indctance in series with the shnt resistance. This RCLS model is nearer to physical reality than the simpler RCS model. If the ratio of shnt resistance and intrinsic one serves as a small parameter, the osephson jnction model can be regarded as a nontrivial example for slow-fast dynamics. The strength of folding of the slow manifold depends on the indctance of the Fig.. Representative crves of nontrivial dynamical behavior for α =, γ =. and β = and vales of sketched in Figre 5b. One finds at I) =. a limit cycle, II) =.3 chaotic behavior, and III) =.9 a limit cycle.

11 E. Nemann and A. Pikovsky: Slow-fast dynamics in osephson jnctions 33 shnt. If the shnt indctance is larger than the intrinsic one, then the slow manifold is folded and has many branches instead of the sal two as for instance in the van der Pol oscillator and the FitzHgh-Nagmo model. We have analysed the dynamics for high, intermediate and low damping, as well as the crrent-voltage characteristics for all three types of damping. For each type of damping the characteristics contains a spercrrent branch (for a net crrent smaller than a critical vale) and a resistive one. In the high damping limit, one obtains a system of two first-order differential eqations with a small parameter. Here the fast motion is characterized by jmps in the osephson phase by approximately π. For intermediate and low damping, even jmps by higher mltiples can occr. Then, the fast motion is two-dimensional and there are different connections between saddles and foci. De to the topology of these trajectories, which come closer to each other as the dissipation is lowered, the dynamics is sensitive to the velocity of the slow motion. The main manifestation of this sensitivity is the presence of vertical jmps in the crrent-voltage characteristics (Fig. 3). For small dissipation, complex cycles have been observed, characterized by alternating jmps of different length. In the low damping limit, one can even observe chaos in the atonomos osephson jnction. The reason for the appearance of complex cycles and chaos is that a separation of time scales is not valid any more. De to the possibility to transform the dimensionless parameters to corresponding vales characterizing the osephson jnction, it is possible to check for the appearance of the reported phenomena in experiments. We thank S. Kznetsov, A. Shilnikov, and A. Ustinov for valable discssions. References. M. Kardar, Phys. Rev. B 33, 35 (98)..Kim,W.G.Choe,S.Kim,H..Lee,Phys.Rev.B9, 59 (99) 3. H. Eikmans,.E. van Himbergen, Phys. Rev. B, 897 (99). K.K. Likharev, Dynamics of osephson nctions and Circits (Gordon and Breach Pblishers, NY, 98) 5. C.B. Whan, C.. Lobb, M.G. Forrester,. Appl. Phys. 77, 38 (995). C.B. Whan, C.. Lobb, IEEE Trans. Appl. Spercond. 5, 39 (995) 7. A.B. Cawthorne, C.B. Whan, C.. Lobb, IEEE Trans. Appl. Spercond. 7, 355 (997) 8. A.B. Cawthorne, C.B. Whan, C.. Lobb,. Appl. Phys. 8, (998) 9. S.K. Dana, D.C. Sengpta, K.D. Edoh, IEEE Trans. Circits-I 8, 99 (). N.F. Rlkov, Phys. Rev. E 5, 9 (). D.E. Postnov, O.V. Sosnovtseva, S.Y. Malova, E. Mosekilde, Phys. Rev. E 7, 5 (3)