Transient chaos in weakly coupled Josephson junctions

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Transient chaos in weakly coupled Josephson junctions

B.P. Koch, B. Bruhn

To cite this version:

B.P. Koch, B. Bruhn. Transient chaos in weakly coupled Josephson junctions. Journal de Physique,

1988, 49 (1), pp.35-40. �10.1051/jphys:0198800490103500�. �jpa-00210672�

Transient chaos in weakly coupled Josephson junctions

B. P. Koch and B. Bruhn

Sektion Physik/Elektronik, Ernst-Moritz-Arndt-Universität, Greifswald 2200, Domstrasse 10a, ^D.D.R.

(Requ ^{le 23} juin 1987, accept6 ^{le 25} septembre 1987)

Résumé.

Dans cet article nous étudions les excitations périodiques ^{et le} couplage d’oscillateurs Josephson

non-linéaires. Nous utilisons la méthode de Melnikov pour démontrer l’existence de

fers à cheval » dans la

dynamique. ^Le couplage ^{de deux} systèmes réduit le seuil de chaos en comparaison ^{d’un seul} système. ^Pour

certaines valeurs des paramètres, ^les prédictions théoriques ^{sont vérifiées} numériquement.

Abstract.

This paper considers periodic excitations and coupling of nonlinear Josephson oscillators. The Melnikov method is used to prove the existence of horseshoes in the dynamics. ^The coupling ^{of two} systems yields ^a reduction of the chaos threshold in comparison ^{with the} corresponding threshold of a single system.

For some selected parameter values the theoretical predictions ^{are checked} by numerical methods.

Classification Physics ^Abstracts

03.20

74.50

1. Introduction.

Many problems ⁱⁿ physics and related sciences are

described by coupling of nonlinear oscillators. The

coupling ^{and the} nonlinearity give ^{rise to a} compli-

cated behaviour of their solutions. The most interest-

ing problem ^{is the} dependence of the chaos boundary

in the parameter space. An effective tool within the

perturbation theory, ^for analytically calculating ^the

critical values of the parameters ^at which the transi- tion to chaos occurs is the Melnikov method [1].

Originally this method was developed ^for systems ^of

two nonautonomous differential equations ^{but more} recently ^some generalizations ^have been found. One method described by Holmes and Marsden [2] ^is applicable ^to systems in which the perturbed hyper-

bolic fixed point of the Poincard map has ^an unstable manifold of dimension one and a stable manifold of codimension one. Slemrod and Marsden {3] ^have given ^some physical applications. ^Their approach

may be generalized ^to the infinite dimensional case

[2, 4].

Other generalizations ^{for finite} dimensional

dynamical systems ^{were found} by Gruendler [5],

Bruhn and Leven [6], ^Hale [7] and Palmer [8].

Later, ^Scheurle [9] has treated systems ^{with non-} periodic (but ^almost periodic) forcing ^{term. This has}

been a first step towards the application ^{of the}

Melnikov method for systems ^{with a} general pertur- bation term.

We consider in this paper ^a dynamical system

which is reducible to n weakly coupled subsystems

where each subsystem ^{has a} hyperbolic ^fixed point

and a homoclinic orbit. The complete system ^{has a} 2 n dimensional phase space in which the n dimen- sional homoclinic manifold is embedded. A pertur- bation may produce ^a transverse intersection of stable and unstable manifolds and by projection

onto the subsystems ^{one can show that horseshoes}

exist in each subsystem.

The paper is divided into ^two parts. ^{The first} part contains the basic equations ^{and the} explicit ^calcula-

tion of the Melnikov functions. In the second part the analytical predictions ^{are checked} by ^numerical

calculations for some selected parameter ^values.

2. Coupled Josephson oscillators.

We consider a dynamical system of the form

where - is the small parameter ^{and the} perturbation

functions gk(Xj, Yj, t ) ^are periodic ^{in time}

Suppose ^{that the} unperturbed ( E = 0 ) system ^is integrable and has the homoclinic (heteroclinic)

solution XOk

XOk(t - t§). ^The parameters t§ ^charac-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490103500

36

terize the specific ^initial conditions of the subsys-

tems. Using ^the independent ^first integrals Jk(XOk, Yok ) k ^{=1, ..., n, of the} subsystems ^{the Mel-}

nikov functions are given by [6] :

where

is the vector field of the perturbation. ^The phase

space of the system may be ^a higher dimensional

cylinder, ^i.e. (xk, yk) ^E (S1 ^x R)n. ^Such phase space is very typical ^for subsystems ^{with a} periodic poten- tial, e.g. for pendulum systems. ^{After the} integration

in (2.2) ^{one has to test} whether there is a transverse zero dependent ^on the t4 :

The zeros of the Melnikov functions cut out a certain

region ^{of the} parameter space. The envelope ^{of this} region ^is the O ( e ) approximation ^{to the} boundary ^of

the parameter region ⁱⁿ which the system ^has

transverse homoclinic points.

Now this formalism is applied ^{to a} nonautonomous

system ^{of two} coupled pendulum equations. ^{Such a}

system may be realized by ^two Josephson point junctions ^{which are connected} by ^an Ohmic resistor.

The external driving ^{term is} given by ^the alternating

currents 11 ^and 12. Figure ^{1 shows the} equivalent

circuit diagram. C1, R1, I,, ^and C2, R2, Ic2, respect-

ively, ^are junction capacitances, resistances and critical currents. The tunnel current lei ^sin (xi) ^and

Fig. ^1.

Equivalent ^circuit diagram ^{of the} coupled Josephson junctions.

the junction voltage (h /4 7Te) Xj ^{( j}

1, 2 ) ^{can be}

obtained from the Josephson phases xj (t ), ^{where h is}

the Planck constant and e the elementary charge.

With the help ^{of the Kirchhoff} rule and Ohm’s law

one finds :

or with the dimensionless time

Now the dot indicates the derivation with respect ^to

T, and the dimensionless parameters ^are given by

From the definitions (2.6) follows that in any ^case

,8 1 :::-- « 1 and 8 2 :::-- a 2. A change ^{of the direction of} II ^or 12 (see Fig. 1) yields ^a change ^{of the} sign ^{in the} coupling ^{terms of} equations (2.5). ^{In the} following

we use the symbol t instead of the variable T. The

driving ^currents may be composed ^{of a} direct and an

alternating part, ^i.e.

with the parameters iol, i02, ill, i 12 and J1. The

special ^case of identical Josephson junctions ^is

contained in (2.5) ^with the choice 8 =1, /31 = f3 2

and _a,

_a2-

Following the paper by Huberman, Crutchfield and Packard [10] ^a number of papers have been

published ^{which deal with} the chaotic behaviour of the single Josephson oscillator (cf. Refs of Kautz and Macfarlane [11]). ^Also the Melnikov method has been applied [12-16] ^{to the} single Josephson junction. ^For coupled junctions ^and their chaotic behaviour there are only ^{a few} contributions [17-19].

The Melnikov prediction ^{for the} single junction gives ^{a threshold} somewhat lower than that found by

direct computation [12]. ^{That is because the} compli-

cated invariant set whose existence is proved by ^the

Melnikov theory [16] ^{is not an attractor and as a rule}

one observes transient chaotic behaviour which

frequently ^{is followed} by periodic ^{motion. A conse-}

quent ^numerical check of the accuracy of the Melni- kov method should show the intersections of the concerned manifolds. The results of such calculations

are given in section 3 for coupled junctions.

An application ^{of the} Melnikov method to the system (2.5) requires ^that dissipation, coupling ^and driving ^{terms be small} quantities, ^i.e.

where 0 -- e .-c 1 and the equations (2.5) ^become (as

a dynamical system ^{of first} order)

The unperturbed system ^has hyperbolic points ^at 101

xo2 ^{= :t 7T,} Y01

Yo2

0 and the associated heteroclinic solution is given by

We assume that the corresponding phase space is the

cylinder (S 1 x R )2, ^i.e. the solutions (2.8) ^become

homoclinic orbits. The first integrals of the unper- turbed system ^are

From the theory ^of ordinary differential equations [20] and invariant manifolds [21] follows that the

perturbed system (2.7) ^{has a} unique hyperbolic

solution of period ² 7T / n ^{in the} neighbourhood ^of

the unperturbed ^fixed point solution for all sufficient-

ly ^{small E. The} associated Poincard map possesses ^a

hyperbolic ^fixed point ^{with two} positive ^{and two} negative eigenvalues. ^This hyperbolic point ^{has a}

two dimensional stable and a two dimensional unst- able manifold. The functions (2.2) are a measure of the distance between the stable and unstable man-

ifold in terms of the first integrals. Using ^{the vector}

field of the perturbation

one obtains the two Melnikov functions

Inserting ^the unperturbed separatrix ^solution (2.8),

one finds the following ^cases dependent ^{on whether}

the upper ^{or the lower} separatrix ^{is used.}

Case 1: system 1 - upper separatrix, system 2 - upper separatrix,

Case 2 : system 1 - upper separatrix, system 2 - lower separatrix,

Case 3 : system 1 - lower separatrix,

system 2 - upper separatrix,

Case 4 : system ^{1 - lower} separatrix,

system 2 - lower separatrix,

38

where the function b (to - to, ^{6) is defined} by :

This integral ^{has the} properties

and

Figure ² shows the functional dependence ^of b (s ) - b (0, s ).

Fig. ^2.

Representation of the function b (0, 8 ).

3. Discussion and numerical experiments.

The four cases (2.11)-(2.14) differ from each other

by ^the sign ^{of the} single ^{terms and this} yields

different conditions for the homoclinic bifurcations.

According ^to equation (2.3) ^{we must secure that for}

a given ^{set of} parameters there exist values to ^and t2 ^{so that}

and

The cases 1 and 4 reveal an effective decrease of

damping ^{as a} result of the coupling. The decrease

has a maximum for to = to, ^which yields the bifurca- tion conditions (we consider the case iol = i02

⁰ only)

The equality ^is valid for f2t’ = nt 2

7r/2 ^±

2 n7T (n = 0,1, ... ). ^For larger values of ill i and

i 12 ^{there exist} transverse zeros of the Melnikov functions and hence intersections of stable and unstable manifolds. The cases 1 and 4 have a particu-

lar meaning ^{in that} the smallest amplitudes ill, i 12 ^are necessary ^to produce ^the homoclinic bifur-

cation, ^i.e. by increasing ^the amplitudes ^{these cases}

are the first to be generated. ^{In order to test the} validity ^{of the} bifurcation condition (3.1), ^{we have}

calculated parts of the unstable and stable manifolds for two sets of parameters. For the first example ^the parameters ^are selected in such a way that the

coupling between the junctions ^{is zero and the}

bifurcation condition is not fulfilled.

Fig. ^3.

Parts of the manifolds of the fixed points ^near (±7r,0) ^{for the} single ^systems (parameters : 8 = 2,

Figure ^{3 shows} parts of the unstable manifolds of

both single oscillators, and there is no transverse

intersection of stable and unstable manifolds. In the

second example ^the junctions ^are coupled

(a 1 = a 2 = 0.25) and the other parameters ^{are the}

same as in the first example. Using these values of the parameters ^of damping, coupling ^and frequency,

the bifurcation conditions yield ill - 1.42, i 12

^1.45

for the critical currents, i.e. for ill = l 12

^{2.0 the}

Melnikov theory predicts transverse intersections.

Therefore the coupling ^{of the two} systems ^reduces the chaos threshold as compared ^{to the} correspond- ing threshold of the decoupled single systems.

The numerical proof ^{of this statement is not so} simple ^as in the first example. The associated Poincard map Pg ^{is four} dimensional, ^{i.e. one has} P,: (S1 ^x R )2 , (S1 ^x R )2. In the first step ^{of nu-} merical treatment the fixed point, ^its eigenvalues

and eigenvectors ^are determined. In the neighbour-

hood of the fixed point ^{the unstable} eigenvectors

form the unstable manifold. One finds only ^small

deviations from the eigenvalues ^{of the} unperturbed system 1, B/S- ^{and from} the associated eigenvectors (1, 1, 0, 0) ^and (0, ^0,1, J 8). ^{The main} problem ^{is to}

select the most suitable path ^on the unstable man-

ifold. The quantities to and to parametrize ^the unperturbed manifolds, ^{i.e. one can find a homoc-}

linic orbit with to = t6 in ^{this case} (see the discussion in connection with the birfucation conditions). ^Be-

cause with E

0 the perturbed ^{manifolds tend to the} unperturbed ^{manifolds one finds a curve element in}

the neighbourhood of the fixed point, ^{which is}

associated with the unperturbed ^{curve element}

characterized by to

t6. ^{This curve element on the}

unstable manifold is developed by ^{a numerical}

Fig. ^4.

Projection ^{of the curve on the unstable man-}

ifold of the fixed point ^near (- 7T, 0, - ?r, 0 ) ^{onto the} subsystems (parameters : 5=2, f2 = 1. 5,

0.1,

iteration of the Poincard map. Figure 4 shows the result of this iteration, in which the curve is projected

onto the subsystems. ^{In the} neighbourhood ^{of the} hyperbolic ^fixed point there exist transverse homoc- linic points and this is of course a confirmation of the fact that coupling reduces effective damping ^{and also}

the chaos threshold. A change of the direction of

12 (see Fig. 1) yields ^a change ^{of the} sign ^of

a and a 2, but the reduction of the chaos threshold remains the same (see (2.11)-(2.14)). ^{In the case of}

two coupled logistic maps Hogg and Huberman [22]

have shown that an increase in the coupling strength

can lead to permanent chaotic motion. Unlike that

we have calculated homoclinic bifurcations which

are accompanied by transient chaotic behaviour we

have not checked whether the lowering ^{of the}

threshold also exists in the case of permanent ^chaos.

For the calculation of the Melnikov functions the

separatrix ^solutions (2.7) ^{of both} subsystems ^are

used. There is also the possibility ^{that one} subsystem

has the trivial solution. The corresponding ^fixed point ^{of the} unperturbed system ^{has two} imaginary eigenvalues ^{which are} connected with a two dimen- sional centre manifold. Stable and unstable man-

ifolds have the dimension one. If the nonresonance

condition f2 2 :0 1 or f2 2 =/ = 5 is _given, ^the _perturbed

system ^{has a} hyperbolic ² 7T / n periodic ^solution

and the stable manifold is of dimension three. For

e

0 this stable manifold tends to the centre stable manifold [3].

Only ^{one Melnikov} function is obtained but there

are two cases depending ^on which of the subsystems

has the trivial solution.

case 1 : separatrix ^of system 1, trivial solution of system ²

case 2 : ^.’ trivial solution of system 1, separatrix ^of system ²

these functions do not depend upon ^the coupling

parameters, ^i.e. they ^are the Melnikov functions of the single pendulum [12-16]. ^{The onset of homoclinic}

bifurcations is at larger values of the currents than in the coupled ^{case. But now} only ^{one condition must}

be fulfilled.

Of course, there are still other possibilities ^to

demonstrate the existence of homoclinic orbits in the

perturbed system. E.g. by using ^{the homoclinic}

solution for one subsystem ^{and a} periodic ^solution

for the other system ^one will find conditions analog-

ous to equation (3.1). ^{However, the} analytical ^calcu-

lations become very complicated. ^{Without an} explicit

40

treatment the results on the homoclinic bifurcation

are between the borderline cases (2.11) ^and (3.2/3),

i.e. by increasing ^the alternating ^{currents the case} given by equation (2.11) ^{is the first to occur.}

Acknowledgments.

The authors would like to thank Prof. R. W. Leven for helpful discussions.

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