Phase-locking chaos in long Josephson junctions

Volume 144, number 8,9

PHYSICS LETFERS A

19 March 1990

PHASE-LOCKING CHAOS IN LONG JOSEPHSON JUNCTIONS Mario SALERNO Dipartjmento di Fisica Teorica, Università di Salerno, 1-84100 Salerno, Italy Received 8 November 1989; accepted for publication 9 January 1990 Communicated by D.D. Hoim

The dynamical behaviour of a fluxon in a long Josephson junction in the presence ofan external microwave field is studied by means ofa two dimensional map. The map shows that fluxon oscillations may become chaotic through a sequence of period doubling bifurcations. This chaos appears in the central portion of a subharmonic rf-induced step in the current—voltage characteristic and may be observable in a real junction both as a deformation of the step and as a broadening ofthe linewidth of the emitted radiation, as the bias point is moved from the edges toward the centre of the step.

1. Introduction

One ofthe most exciting developments in the field of nonlinear dynamics is the possibility of coexistence of solitons (spatial order) with chaos (temporal disorder). Temporal chaos with coherent spatial structures was indeed shown to exist in the perturbed sine-Gordon equation by direct numerical integration of the partial differential equation (p.d.e.) [1—3].Analytical descriptions of this phenomenon were given in terms of suitable reductions ofthe original p.d.e. with infinite degrees of freedom to systems with few freedoms [4,51. This approach was recently used to describe the phase locking phenomena observed in long Josephson junctions, where the sine-Gordon equation with damping and external periodic forcing was reduced to a two dimensional map for the time of flight and the fluxon’s energy inside the junction [6,7]. The reduction was obtained under two fundamental assumptions: (i) the influence of the external microwave field on the junction is felt through boundary conditions, (ii) only a single fluxon is present in the system, whose dynamics is adequately described by soliton perturbation theory. In spite of the drastic simplification involved in this reduction, the map was shown to capture all the known experimental features of phase locking such as: existence of subharmonics, super-

harmonics, zero crossing steps, hysteresis, frequency pulling etc. [8,9]. The aim of the present paper is to use the map model to show that fluxon oscillations in long Josephson junctions may become chaotic through a Sequence of period doubling bifurcations. In particular this chaos appears in the central portion of subharmonic rf-induced steps and disappears as the bias point is moved toward the step’s edges. This phenomenon may be observable in real long Josephson junctions both in the current—voltage (I—V) characteristic, as a deformation of the step in its central region and in the power spectrum of the emitted radiation, as a broadening of the linewidth (noise rise) as the bias point of the junction moves from the edges toward the centre of the step.

2. The map model We start by briefly sketching the derivation of the map model for phase locking in long Josephson junctions in the specific case ofthe in-line geometry (the results are easily extended to the overlap geometry as well). More details will appear in ref. [7]. In this case the dynamics of a fluxon in the junction is governed by the damped sine-Gordon equation [10] ~

—

—

sin ~ = a~.

(1) 453

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PHYSICS LETTERS A

Here x denotes the spatial coordinate normalized to the Josephson penetration length, t is time normalized to the inverse of the Josephson angular plasma frequency and a represents shunt dissipation due to quasi-particle tunnelling. The influence ofthe rf-field on the fluxon is modelled through the boundary conditions

19 March 1990

t= t,~÷ on the fluxon’s motion is computed by observing that during a reflection the change in the fluxon’s energy (neglecting dissipation) is given by

(7) which can be rewritten in terms of a~1and ak as cosh(ak+l)=cosh(ak—aL)

1

k+1

(8) (2) where Xdc represents the normalized bias current and ~j=K 1-1-sin(w1.,t+8) (3) is the external rf-field. We distinguish between electric and magnetic coupling according to whether K,.f has the same signs or opposite ones at the two ends of the junction [11]. In the following we consider only electric coupling but the results are easily generalized to the magnetic case as well. The back and forth motion of the fluxon inside thejunction can be described as unidirectional motion on a semi-infinile line, by periodically extending the junction of length L along the x-axis. In this description the boundary conditions (2) act on the fluxon as kicks at discrete times Ik: X( 1k) = kL, keN, with X( 1k) being the position of the fluxon’s center of mass. The dynamics of the fluxon between consecutive kicks is obtained from soliton perturbation theory on an infinite line as described by McLaughlin and Scott [12]. Following these authors we obtain for the fluxon momentum dP —=—aP, (4) dt which can be integrated twice to give [13, 14] X(t)=Xk+a~(sinh~ (Zk) —

sinh

{Zk

exp [

—

a

(1— 1k) ] })

(5)

Here the subscript k denotes evaluation at time tk and z represents the normalized momentum z= P/8. The time of flight tk~I~ tk between consecutive kicks is obtained from eq. (4) with t = tk+ 1, ~—

tk+ j

—

tk =

~ log( sinh(ak_aL))’ sinh ak

—

where a,= sinh

1Z~. The

(6)

effect of the kick at time

~kc~’~’11(

~tJ+6)J.

/

~

Eq. (8) together with eq. (6) constitute explicit tk• In this an form howtwo dimensional map for a,. and ever, the map has memory. To eliminate this fact, we define the variable T~ ~ t~,

(9)

j1

which appears in the map only in the argument of the sine function and may therefore be defined modulo 2x/w~i.e. on a cylindrical phase space. By introducing the normalised fluxon energy at the nth boundary,

U~ cosh(an) the in-line map can be finally written as ,

2

T~+

)

(10)

1 log( (U~—~” 1=T~+ a C( U~ 1)1/2 SU~ —

__________________ —

mod(2it/w,4,

—

2

U~~1=CU~—S(U~—l)” +~7t[xdc+Krfsin(WrfTn÷l+&)],

(lla) (llb)

with S and C denoting respectively sinh(aL) and cosh(aL). eq. (11) is easily seen to be a smooth invertible map (diffeomorphism) of the half cylinder S’ xo~into itself. The determinant of the Jacobian J of the mapping is given by IdetJI=exp[—a(T~÷i—T~)] (12) which is not a constant but depends on the fluxon’s time of flight. Note that for a ~E~ the expression in (12)is always <1, i.e. the map is area contracting, ,

while for a = 0 the log term in eq. (11 a) becomes LU~(U~—l)”2and the map is area preserving. In the following we concentrate only on the dissipative case. Phase locking of a fluxon oscillation inside the

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junction corresponds to periodic orbits of the map, i.e. to fixed points of (11). The equal time of flight fixed points, i.e. those for which at the (n + 1) st reflection the energy and the time of flight of the fluxon are the same as at the nth but m rf field oscillations later, are easily found as

the equal time of flight fixed point is at the center of the step. Moreover, violation of inequalities (17), (19) occurs when the eigenvalue of the Jacobian of the mapping leaves the unit circle at 1 in a flip bifurcation with the birth of a stable period-two orbit.

1*

(13a)

3. Numerical experiment

U31~

Experience with maps of the plane leads us to expect that the period-two orbit obtained in the previous section loses stability by a sequence of period doubling bifurcations, or by a reverse bifurcation

arcsin[(K—xdC)/K~-f-]

8

—

Wrf

C—E [(C—E)2—S2]”2’ with E exp (

—

mi~a/w~)

(14)

and K —

2(C— 1) (1 +E) 2EC+ E2)”2’

~t (1

(15)

—

with m in eq. (14) denoting the order of the subharmonic. From eqs. (11 b) and (1 3a) it is apparent that K is just the value of the bias current Xdc at which the rf-induced step intersects the unperturbed zerofield step. The range ofbias current in which the fixed point exists is derived from eq. (1 3b) as K—K~
(17)

with 2 A~irw~(l—2EC+E2)312 4aE(l+E)S

back to period-one orbit. This is indeed what we find by numerical iterations of (11). More precisely we find a period doubling route to chaos when Xdc increases from the bottom (xdC = K— K~)to the middle (XdC=K) of the step, and reverse bifurcations from chaos back to period-one orbit when Xdc is increased from the middle to the top (xdC=K+K~)of the step. As consequence ofthe symmetry of the stability range (19) of the equal time of flight fixed point around Xdc = K, we have that the chaotic states are confined in the central part ofthe step and disappear when the bias point is moved toward the edges. By fixing all parameters in (11) except Xdc and the amplitude of the rf field, it is however possible to find values of K 1.~-for which the sequence of period doubling bifurcations in the lower half of the step terminate before reaching chaos. These results are easily verified by direct iterations of the map. To this end we have constructed in fig. 1 a portion of the I—V curve by plotting age defined the bias as current Xdc versus the average volt-

(18) 2,t

from which we see that the stability decreases rap-

V=
idly with increasing subharmonic order. Furthermore, for A> K~the range of stability is the entire range of existence, while for A ~ K~,we may cast the stability condition into a more useful form by substituting eq. (13a) in (17), this giving 2) 1/2 (1 9a) K— K~
—

Tk> AVG

(20)

with the parameters L, a and W,-t~in eqs. (11) fixed, here and in the following, respectively to 12, 0.05, 1.5. The average in eq. (20) is computed over a time span of l0~map iterations. The smooth curve in fig. 1 denoteswith thethe zero fieldprocedure step i.e. the curve constructed above butI—V in the absence of the microwave field (K,f-=0) in eq. (llb), while the discontinuous curve represents a subharmonic step of order m = 4 (V= w~/4)induced by an rf field of amplitude K~=0.18. Note that the step is symmetric around the value xdC=K= 0.4211 in agree455

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0.8

____________

00~•3

~

0.5

VØLTAGE Fig. I. Current—voltage characteristic of in-line junction. The smooth curve is with no field while the discontinuous one denotes the m = 4 subharmonic induced by an rf field of amplitude Kff= 0.18 and frequency o~=1.5. The remaining parameters were fixed as a=0.05, L=12.

0.8

,,,,

the 0.4

0.0

Li

•1

2.0

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the above analytical values up to three significant digits. It is of interest to note that the step in fig. 1 slightly deviates from its phase locked value in the range of bias current over which the period-two solution exists. This becomes more evident at high values of K~as we show in the following. The range of existence of the period-two solution rapidly decreases with increasing values of K~,this quickly leading to a succession of period doubling bifurcations. In fig. 3 we report the bifurcation diagram [1* versus Xdc for an rf field amplitude of 0.21, from which we see the fully developed period doubling cascade to chaos in the central region of the bias current interval. The ratio between successive bias current ranges, corresponding to successive period doubling states, was seen to approach Feigenbaum’s universal constant ö=4.6692 within three significant digits. The breakdown of phase locking in the central part of the step is evidenced in the I— Vcurve constructed in fig. 4 for the same parameter values as fig. 3. From this figure it is clear that the step is perfectly vertical at the edges where periodic solutions exist, while it appears as a ragged function of XdC in the central part ofthe step where chaotic states exist. We remark that the average voltage continued to be a ragged function of the bias current also when time span over which the average was computed was increased from 1 0~to 1 0~map iterations. The chaotic nature of this phenomenon is clearly seen in terms of the power spectrum analysis of the emitted radiation from the junction. By representing the sig-

3.0

0.6

U Fig. 2. Bifurcation diagram for the fixed point of the in-linemap as the bias current is varied along the step of fig. 1. The parameterswerefixedasinfig. 1.

/ t( ~

0.4

ment with eq. (15) and its extension in current coincides with the range ofexistence (16) ofthe equal time offlight fixed point. According to eqs. (19) this fixed point loses stability in a flip bifurcation respectively at K1=0.3481 and K2=0.4941. In fig. 2 we report the bifurcation diagram U~versus Xdc as the bifurcation parameter Xdc is varied along the step of fig. 1. The numerical values at which the bifurcations occur are found to be in good agreement with 456

1.0

2.0

3.0

U Fig. 3. Same as fig. 2 but with an rffield amplitude of 0.21.

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0.3

0.4

0.5

19 March 1990

—

~3Q

VØLTIRGE

0.40

0.45

FREQUENCY

Fig. 4. Same as fig. I but with parameter fixed as in fig. 3. 0.0

0.35

Fig. 6. Same as fig. 5 but with Xdc= 0.36. 36, from which we see that in step in fig.to4 the at Xdc=O. addition discrete line at w 1.1./4, a noisy con-

_________________________

tinuous component, typical of chaos, appears. —5.0

4. Conclusion

-io.o

0.35

o.so

‘

o.is

FREQUENCY Fig. 5. Power spectrum of the radiation emitted 26S. from a junction biased on the step offig. 4 at the value Xdc=O.

nal at one end of the junction as a sequence of delta functions at the discrete times T 2,., k= 1, 2, obtamed from the map, we may construct the power spectrum of the emitted radiation simply as the Fourier transform of the above sequence. In fig. 5 such a spectrum is reported for a junction biased on the lower vertical edge of the step in fig. 4 at Xdc = 0.265. As expected, the spectrum looks regular with a discrete component at the rf drive frequency and its harmonics (in the figure only a window in the frequency around the m =4 subharmonic line is shown). In fig. 6 we show the power spectrum when the junction is biased in the ragged portion of the ...

Before closing the paper we feel compelled to point out the limitations of our approach and discuss the conditions of its applicability to real Josephson junctions. The map is clearly based on the assumption of single fluxon dynamics. Phenomena such as creation or annihilation of fluxons, which do occur in real junctions, are therefore not included. In order to avoid the occurrence of such phenomena, the amount of (from) energy the which system should is injected be kept (orsmall. extracted) This is posinto sible if we restrict the analysis to small rf field amplitudes, i.e. to subharmonics of high order. From eqs. (17), (18) it is indeed clear that on subharmonics of high order the equal time of flight fixed point, from which the bifurcation cascade originates, may becomesunstable at low values of K,.1.. This suggests that the above chaotic phenomenon should be experimentally searched on high subharmonic steps and should manifest itself both as a deformation of the step in its central portion and as a noise rise effect in the spectrum of the emitted radiation as the bias point is moved from the edges toward the centre ofthe step. We expect experiments on real Josephson junctions and numerical simulations on the partial 457

Volume 144, number 8,9

PHYSICS LETTERSA

differential equation to be soon performed.

Acknowledgement This work was supported by the G.N.S.M. (C.N.R.) and the M.P.I. (Italy).

References [11 J.C. Eilbeck, P.S. Lomdahl and A.C. Newell, Phys. Lett. A 87 (1981)1. [21 A.R. Bishop, K. Fesser, P.S. Limdahl, W.C. Kerr, M.B. Williams and S.E. Trullinger, Phys. Rev. Lett. 50 (1983) 1095. [3] E.A. Overman II, D.W. McLaughlin and A.R. Bishop, PhysicaD 19(1986)1.

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[4] A.R. Bishop, MG. Forest, D.W. McLaughlin and E.A. Overman II, PhysicaD 23 (1986) 293. [5] AR. Bishop, D.W. McLaughlin and M. Salerno, Phys. Rev. A (1989), tobe published. [6] M. Salerno, M.R. Samuelsen, G. Filatrella, S. Pagano and R.D. Parmentier, Phys. Lett. A 137 (1989) 75. [71M. Salerno, M.R. Samuelsen, G. Filatrella, S. Pagano and R.D. Parmentier, Phys. Rev. B, submitted. [8] M. Scheuermann, J.T. Chen and J.J. Chang, J. AppI. Phys. 54(1983) 3286. [9] M. Cirillo and F.L. Lloyd, J. App!. Phys. 61(1987) 2581. [101 A. Barone and G. Paternô, Physics and applications of the Josephson effect (Wiley, New York, 1982). [111 J.J. Chang, Phys. Rev. B 34 (1986) 6137. [12] D.W. McLaughlin and A.C. Scott, Phys. Rev. A 18 (1978) 1652. [131 P.L. Christiansen and O.H. Olsen, Wave Motion 2 (1980) 185. [141 O.A. Levring, N.F. Pedersen and M.R. Samuelsen, J. AppI. Phys. 54(1983)987.