Physica B 165&166 (1990) 1659-1660North-Holland

CHAOS BEYOND-THE ONSET IN AC-DRIVEN PHASE SLIP CENTERS

Rafael Rangel and Luis E. Guerrero

Facultad de Ciencias, Universidad Central de Venezuela, Apartado 21201, Caracas 1020 A,Venezuela and Centro de Fisica, IVIC, Apartado 21827, Caracas 1020 A, Venezuela

Chaos beyond the onset for the generalized time-dependent Ginzgurg-Landau equationsexhibits generic behavior. We present qualitative and quantitative evidence of amultifractal phase transition and of a symmetry-increasing bifurcation. We also discussthe underlying mechanisms of these phenomena.

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U(I + i /'vJ2) [a t+ ill + t ia tl'JI/2] 'V =

a;'V + (1- /'vJ2) 'V (l )

where 'Y(x,t) is the complex order parameter,j is the current density, Il(x,t) is thechemical potential, y(T) is a measure of thedamping of the system, ~(T) is the coherencedepth and u is a dimensionless parameterwhich arises from the microscopic theory.Variables are renormalized in the usualfashion (2). In this paper parameter valuesare u=5.79, ro=0.2 and y= 10. We employ

-2.8-3.2 -3.0

109,0 I

.............__ ~~5

/ ..~~; /......~;;:::~~~~

-/'.......-....

-3.4

-1.2

-2.0

a.

.; -1.6o

FIGURE Iperiodic boundary condi tions.

The ac-driven phase slip centers presentthe period doubl1ng route to chaos as theamplitude of the drIving force is decreased(2).

As will be reported elsewhere, at theonset of chaos (jac=0.474892) the system isruled by an strange attractor which exhibitsan universal multifractal spectrum t(ex) andas jac is decrea'sed higher dimensionaleffects appear: an internal crisis sequenceremoves a supercritical behavior like the onecharacteristIc of the logistic map, givingrise to an attractor more likely to the Henonattra~tor (j~c=0.47455).

FIgure I and the corresponding Table Ipresents the local scaling of the probabilitymeasure for a representative group of pointsof the c:attractor at jac=0.47455. ThiscalculatIOn allows us a quantitativeunderstanding of the scalin9 structure(p::ol-a) of the attractor, in partIcular allowsus to detect points wIth abnormally highlocal denSIty (curves 3 and 5 at Figure I).

(3)j =j cos(rot)ac

The different routes to chaos received agreat deal of attention in recent years; incontrast much less is known about d)'namlcalsystems beyond the onset of chaos. There isan increasmg theoretical effort in thisdirection and recent experiments confirmedsome universal features beyond the onset ofchaos ( I ).

In this paper we explore the supercriticalregime in ac-dri ven phase sl ip centers (2).We show that this system exhibits a coupleof interesting phenomena characteristic ofdiscrete maps: multifractal phasetransitions (4) and a symmetry restoringbifurcation (6).

The ac-driven phase slip centers aresolutions of the generalized tIme-dependentGinzburg-Landau equations which describes athin and long current carrying filament. Fordirty superconductors in the localequilibrium regime, these equations aregiven by (2),

0921-4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)

1660 R. Rangel, L.E. Guerrero

FIGURE 2

fL ( X= 0 I t + 8 t)0.98

0.98

-0.98l.-- -.J

-0.98

ACKNOWLEDGEMENTSThis work has been partially supported

by IBM of Venezuela Scientific Center.

REFERENCES(I) G. H. Gunaratne, P. S. Linsay and M. J.Vinson, Phys. Rev. Let t. 63 ( 1989) 1.(2) R. Rangel cmd L. Kramer, J. Low Temp.Phys. 74 (1989) 163. .(3) E. Ott, C. Grebogi and J. A. Yorke Phys.Lett. A 135 (1989) 343. '(4) M. H. Jensen, Phys. Rev. Lett. 60 (1988)1680.(5) D. Katzen and I. Procaccia Phys. Rev.Lett. 58 (1987) 1169. '(6) P. Chossat and P. Golubitsky Physica D32 (1988) 423. '(7) Y. Yamaguchi, Phys. Lett. A 135 (1989)259.

\. ~-'"

\~1\',~

-oIIx

:i.\,:~~;,:~,; '"

"<~..~'\_i~\

":,:~~,~ ~:~~\\·'0,.". \

Curve Slopes

Bottom Top

1 1.468961 1.070476

2 1.796748 0.831836

3 0.322568 0.464425

4 0.986235

5 0.569997 0.808245

TABLE IThese points are generated by tangenciesbetween the stable and the unstablemlmifolds of unstable periodic orbits (3).

Curves 1,2 and 4 shows local values of abigger or equal than one which arise fromhyperbolic parts of the attractor (3,4).

The simultaneous presence of regions ofthe attractor with two different localscalings (hyperbolic and non-hyperbolic)causes a first-order-multifractal phasetransition (3,5).

. Figure 2 shows a strobed plot /l(x=O,t) vs./l(x=O,t+~t) for jc:lc=0.4715 This plot revealsan attractor wnich has in average thefollowing symmetry of the equations (1 ), (2)and (3),

here T=21t/ro is the period. This symmetry ofthe solution was lost slightly before thebegginning of the period doubling cascade(symmetry breaking bifurcation) (2).

The attractors related through (4) arecalled conjugated attractors. Whichattractor settles in depends on the initialconditions. At a value of amplitude of thedriving force jac"'0.47358 a symmetryincreasing bifurcation occurs (6). Thesignature is the presence of only oddmultiples of ro in the power spectrum of/l(x=O,t). The symmetry increasingbifurcation occurs when the stable and theunstable manifold of the periodic solutionwith symmetry (4) touch in tangency (7).This fast solution exists before thesymmetry breaking bifurcation.

We have shown that a rather complexsystem exhibits a quite generic behaviorbeyond the onset of chaos, previouslyreported for low dimensional maps.