TEMPORAL CHAOS VIA PERIOD-DOUBLING ROUTE IN SINE-GORDON SYSTEM

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TEMPORAL CHAOS VIA PERIOD-DOUBLING ROUTE IN SINE-GORDON SYSTEM

M. Taki, K. Spatschek

To cite this version:

M. Taki, K. Spatschek. TEMPORAL CHAOS VIA PERIOD-DOUBLING ROUTE IN SINE-GORDON SYSTEM. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-77-C3-84.

�10.1051/jphyscol:1989311�. �jpa-00229452�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3, Tome 50, mars 1989 C3-77

TEMPORAL CHAOS VIA PERIOD-DOUBLING ROUTE IN SINE-GORDON SYSTEM (x >

M. TAKI and K.H. SPATSCHEK

Instltut fur Theoretlsche Physlk der Universitat DiisselcLorf, D-4000 DUsseldorf, F.R.G.

Résumé: On étudie l'équation de sine-Gordon perturbée par un terme de disssipation et un forçage périodique dans la limite Schrôdinger nonlinéair. On montre que l'évolution temporelle d'une condition initiale type breather devient chaotique via les cascades de doublement de périodes.

Abstract: The damped, driven sine-Gordon equation is investigated in the nonlinear Schrodinger.limit. It is shown that the temporal evolution of a breather as initial data becomes chaotic via period-doubling cascades.

I Introduction

Although in the past significant progress was made in the study of the dynamics of nonlinear partial differential equations (PDE's), the occurrence and the nature of chaos in such systems are very often not quite clear.

The complexity due to the spatio-temporal description renders them more difficult, particularly when they loose their integrability under external perturbations. Nevertheless, some PDE's present a variety of a rich and clear phenomena for weak perturbations. Among the near-integrable systems, the (1 + 1)—dimensional sine-Gordon and the NLS equations are paradigms which have many applications, in plasma physics / l , 2/, solid state physics / 3 , 4/, electric circuits / 5 / , Josephson transmission lines / 6 / , and others. Different analytical and numerical approaches have been used to study the evolution of weakly perturbed nonlinear modes / 7 / , transitions (chaotic or not) between stable (or metastable) regimes / 8 / , and the characterization of attractors in these systems / 9 / .

Recently the nature of temporal chaos with coherent stable spatial structures in a driven sine-Gordon equation (the nondissipative case) was studied / 7 / by using the Melnikov theory and a collective-coordinate description.

The occurrence of horseshoe chaos was shown for the system which is associated with the zero-frequency breather (homoclinic orbit). The resonant breathers are spatially preserved, but their chaotic temporal dynamic reveals itself in the presence of islands in an appropriate Poincare section. In this paper we consider the dissipative case (i.e., the damped, driven sine-Gordon equation). In contrast to Ref. 7, where the procedure is valid for any driving frequency entering the perturbative term Fsin ust, here we restrict ourselves to the NLS limit. We show that in a typical parameter regime the temporal evolution of a breather as inital data in the damped, driven sine-Gordon equation becomes chaotic via period-doubling cascades. First, the stable spatial coherent structure (perturbed breather) is locked to the driver. Subsequently, this state looses its stability, giving rise to the period-doubling cascades before resulting in a temporal chaotic regime.

Within the NLS approximation, we present a reduced model of four ordinary differential equations (ODE's), which are derived with the variational method (Lagrangian method). We perform numerical simulations on the original (PDE) as well as the reduced (ODE) systems which turn out to be in good agreement, both qualitatively and quantitatively. However, by increasing the.control parameter (the amplitude of the driver) one reaches the limit of the validty of our method in two respects. Firstly, the amplitude of the nonlinear mode (perturbed breather) becomes large and the NLS approximation fails. Secondly, the complexity of the stable spatial coherent structure renders the collective-coordinate picture (based on a one-breather solution) no longer reliable. The study of the dynamics in this region of the control parameter is left for future work.

The rest of this paper is organized as follows. The next section contains the general setting: the description

}Work performed together with G. Reinische and J . C Fernandez

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989311

C3-78 JOURNAL DE PHYSIQUE

and the results of the numerical experiments. The reduced model, the comparison and the discussions of the results together with our conclusion constitute the last section.

I1 Results of the numerical experiments

In this paper we consider the ac-driven, damped sine-Gordon equation in dimensionless units,

utt - u,,

+

^{sin u = -aut}

+ r

^{sin Rt (11.1)}

with reflective boundary conditions u,(-L,t) = u,(L,t) = 0 and u(x,t = 0) = ui,(x). For the following discussion, the dissipation parameter a and the driver frequency R are fixed at the values a = 0.004 and R = 0.98. The amplitude I'(0

5

I? << 1) of the driver plays the role 0f.a control parameter. As we shall see later, the choice of R = 0.98 places us in the NLS approximation regime. For all experiments we report here, the initial data, u,,(x), is a breather with the following characteristics. The temporal frequency and phase are W B = 0.98 and c p ~ = 2.154, respectively. However, we should mention that different breather initial data (0 << w&1) give the same qualitative sequences of bifurcations as described below. The only 2ffect of these different initial data is to shift slightly the boundaries between the different bifurcation sequences in the control parameter interval.

In fact, for very small values of the driver amplitude

r,

all initial breathers are damped out asymptotically.

Obviously, then the dissipation dominates the external driver. The single attractor is a flat (x-independent state) locked to the driver. The period of this flat attractor is exactly the period of the external driver (forced pendulum). When one increases the control parameter, the situation evolves as follows. Two attractors may co-exist (for the same external parameters); the previous one which is still stable, and a x-dependent attractor (perturbed breather) which is also locked to the driver. At this stage, the phase c p ~ of the breather initial data becomes crucial. Therefore, depending of its phase, a breather initial data may be attracted by either the flat or the x-dependent attractor. Once and for all we fix c p ~ = 2.154. The bassin of attraction of the phase locked breather is centered around this phase value. Numerical integrations of the system (1) was achieved with the characteristic method which is appropriate for hyperbolic partial differential equation. Applied t o the perturbed sine-Gordon equation (1) i t implies Ax = At (x is the space variable), which is in fact a restriction. However, the characteristic method is strongly stable. The description of the method is out of the scope of this paper. In most of the runs At = P/40, whpre P is the period of the external driver, (i.e., we have 40 steps of integration in each cycle). The optimum number of the discretized space points is 501 (i.e., we have 2L = 501Ax N 80). The initial breather is placed in the middle of the chain. Since the driving term is periodic and we are interested in the temporal evolution, one may use a Poincark mapping constructed by sampling the coordinates X ( t ) E u(x. = 0, t ) and Y(t) ^E ut (x = 0, t ) each time the driving force completes one period.

The following results of the numerical simulations show a very clear period doubling of bifurcations. For

I'

= 0.0032 an asymptotically fixed point appears in the X,Y-coordinates; see Fig.1. In fact, the fixed point corresponds to the case of the phase-locked breather. We note that even though there is a clear tendency of the "trajectory" to wind the fixed point, it is difficult to reach it numerically. (In Fig. 1, the maximum time of the integration is t = 15.000 units). Nevertheless, the fixed point is confirmed and predicted by the reduced model in the last section. It turns out that the space configuration of the solution is really a perturbed breather for all times. The time characteristics of this phase-locked state (i.e. the maximum of the amplitude and the appearing radiation) are predicted also by the reduced model. We note that the analytically predicted values are in a good agreement with the numerical ones (see the next section).

When increasing the control parameter the fixed point becomes unstable and at the value I? = 0.0038 it gives rise to a stable limit cycle in X,Y-coordinates. Fig. 2 shows this limit cycle. If one further increases the control parameter, first the limit cycle remains stable but its maximum amplitude becomes large, but finally at the value I? = 0.00432 it looses its stability and a "2-limit cycle" appears which is stable. Fig. 3 shows what we call the 2-limit cycle. This orbit in a turn becomes unstable at I? = 0.00434 and is replaced by a new stable orbit: a 4-limit-cycle (Fig.4). The process continues until the parameter value I? = 0.00456 a t which the system exhibits chaotic behavior (Fig.5). Although the dolutions of the original system reside in an infinite dimensional phase space they may approach attractors which are low dimensional /9,10,11/.

Therefore, the collective-coordinate decription may be useful in studying the temporal evolution as far as the spatial coherence remains. This is actually the case since we consider weak perturbations. In the X,Y- coordinates we have chosen, these bifurcation sequences reveal strongly the period doubling route t o chaos.

In fact, as one can see by Fourier analyzing these data that the bifurcation sequences are nothing but cascades

of period-doubling bifurcations. The period of the 2-limit cycle is twice the period of the limit-cycle and the period of the 2-limit-cycle is doubled giving rise to the $-limit cycle and so on. The temporal power spectra S P D ( w ) for period-1 limit cycle, period-4 4-limit-cycle, are shown in figures 6 and 7. It is now clear, a t least numerically, that the system exhibits temporal chaotic behavior via period-doubling cascades with spatial coherent structures. This is confirmed both qualitatively and quantitatively by a reduced model which we present in the next section.

I11 Reduced Model System and Conclusion

In this section we present a simple model consisting of four ordinary differential equations (ODE's) of first order. This reduced system is obtained from the analytical approach which is sometimes known as collective coordinate description. The main idea of this method consists in an ansatz for the dominating solution with slowly time-varying parameters. Inserting this ansatz into the Lagrange density, we integrate over the space coordinate to obtain a Lagrangian depending on coordinates (and velocities) which are the time-dependent parameters just mentioned. Because of this integration (averaging) over space the whole procedure is not too sensitive to the explicit x-dependence of the ansatz. To obtain a reduced system of ODE's we vary the space averaged Lagrangian (variation of action method) for the time-dependent Euler equations. When investigating the breather solution of the sine-Gordon equation by this approach, obviously two time-scales are involved: one originating from the fast oscillation of the breather and the other one due t o the slow time-variation of the breather parameters. This fact and the intention t o derive ODE's for the breather parameters suggests an additional averaging over the fast time-dependence. And indeed such a procedure was suggested by Whitham already for some other but similar problems of nonlinear propagation.

Let us consider t h e perturbed sine-Gordon equation (1) for which we can write the Lagrange density L in the form

-0.56 -0.58 -0.59 -0.61 ,-0.62 -0.64 -0.66

* _-0.67

-0.69 -0.70 -0.72

-

-

^{. . . .}

. . . .

-

. . . .

. . .

.

^.

-

_. . . . . . : . . . . .

. .

. . . . . .

-

^{. .} . . . . . _.. . .

. . . . . . .:

. . . . . . .. . . , : : . .,<;.;; . ... :,;:;:.;::;.,;;..,::.;.;;'.. 1

:

^{. . .}

-

. . . . . _. ._. ._...:..:.:::C. ...(< _. ....* j.4 .i?,*. ... ,:;: . . : : . . . . . . _{. . .} .:r~~,,J$$;,,j.!:z::,. ; . . . . . .

-

...:... : . . . . . . _{. . .} ^{,,!,.:::, ::.::.:} _: . . . _{. . . .} ,,., ' , : . . : . . . _{/' >}. . . . _{.. : .:} . . . _{. . . .} . . . . . . .

- .

.. .

.

.. : . . . : .

.

. . . ,; : . .^: ... ..

-

. . . . .

. . . .

-

_{. . .}

- 0 . 6 2 - 0 . 5 0 -0.55 - 0 . 5 1 - 0 . 4 0 - 0 . 4 7

X

Fig. 1 - Fixed point in the Poincar6 section defined as

X

= u(x,= O,T), Y = ut(x = 0 , T ) . T is the period of the driver. u and ut are oblained by the in~egration of (1) : R = 0.98, ar = 0.004 and I? = 0.0032.

0 . 0 0

- 0 . 2 0 -

Fig. 2

-

Stable limit-cycle for I' = 0.0038

-0.40

-

^-0.60

- 0 . 8 0 -

- 1 . 0 0

- 1 . 2 0

- ,,/-.,

,/

^-.

/ -.

-

1 \ \

1

^{\ \}

. .

\ . .

-

. .

\ . . . .

^\

\

^\ ₁

-\

I ^I

- -"-, /

I..... ... _{-. ,..}

.-'

/

- .,- I

- 1 . 0 0 -0.80 - 0 . 6 0 -0.40 - 0 . 2 0 0 . 0 0 0.2C

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-1.40

- 1 . 2 0 - 0 . 0 4 -0.48 - 0 . 1 2 0.24 0 . 6 0

X

Fig. 3 - Stable 2-limit-cycle for

I?

= 0.00432

X

Fig. 4 - Stable 4-limit-cycle for

r

^{= 0.00434}

- 1 . 6 0 1 ^I

- 1 . 2 0 - 1 . 0 0 -0.00 -0.60 - 0 . 4 0 -0.20 0 . 0 0 0 . 2 0 0 . 4 0 0.6C X

Fig. 5

-

The onset of chaos at

r

⁼ 0.00456

OHEOA

Fig. 6 - Power spectra for the time serie X,, in fig. 2, the frequency of the limit-cycle is wl .v 0.016

.

This frequency can be analytically obtained fxam the reduced model.

Via the variational equation 6JJ L dx dt = 0, and the corresponding Euler equation, we obviously obtain equation (1) (with

I?

= 2

1

^A

I

and arg(A) = -$).

When investigating equation (1) in the nonlinear Schrodinger liml't, we apply the Whitham approach, i.e. we write

=

C

00 e i ( ~ ~ - l ) f i t , z ~ - 2 F(zv-I) _{(X, T )}

+

^c.c,

w=l

where e is a smallness parameter and

fi

^N O(1). The stretched coordinates X = ex,T = ie2t have been introduced. Next we insert this ansatz into the (time-) averaged Lagrangian

-

6

^zn/b

L - J L a t 2n o

where the integration is only over the fast variable t (and not T). We have [when eat is not affected and

h

O(1)]

where: A N O(e3), 0 -

fi

^N 0(e2) and a ^N 0(e2). In

Z

we have also used the expression /12/

cos z = 1

+

azxz

+

^a4x4

+ S

(where

I S

15 9 x l W 4 for a2 = -0.4967 and a4 = 0.03705), which is valid for 0

5

x

5 q.

In order to get a consistent picture we set ft2 -2az (fi E 0.9967). Then we obtain (to simplify the notation we introduce

fil)

= 2 q * / m )

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where ia =

$5-

^{and y =}

6.

It is obvious that the Euler equation is now the NLSE in t h e standard form

Thus the breather amplitude is described by a driven and damped NLS equation with the relation ua ( u , t ) = EF(')(x, T)eint

+

^c.c.

+

^O(e3).

OHEQA

Fig. 7 ^- Power spectra for t h e time serie

X,

in fig. 4 corresponding to the period 4-limit-cycle.

- 1 . 6 0 I

- 1 . 4 0 - 1 . 2 0 - 1 . 0 0 - 0 . 0 0 - 0 . 6 0 -0.40 -0.20 0 . 0 0 0 . 2 0 0.4C X

Fig. 8 - Stable 4-limit-cycle in the Poincar6 section. We integrate the ODE'S and transform the results t o X,Y-coordinates.

OMEGA

Fig. 9 - Power spectra for the time serie

X,,

in fig. 8. Figures 8 and 9 have to be conipared with the figures 4 and 7 respectively.

When introducing collective coordinates, we use the ansatz /13,14/

where c(T) = qeiWT =/ c

I

^{ei~c+iWT, r ( X , T ) = -i} 27 ~ech(2qX)~-2"

+

^{rl(X, T),}

rl(X, T ) = -

f

{pe-4" sech2(2vX)

+

p* tanh2(2qx)) ;

and 71 = q(T), a = a ( T ) , p = p(T)

=I

^p(T)

I

e"(=) are the new slowly time-varying collective coordinates.

After lengthy calculations we get

f .

^{16 3}

L(q, a,

1

^p I,cp,ir,+) = eZyTB -8qu

+

^2yf

I

p

I

cos(2a - cp) - -cp _a

I

^cp

1'

^--q ₃

after which we obta.in a set of four ordinary differential equations for q, u and p.

Comparing with the driven damped sine-Gordon-equation (1) we have

Since we are limited in place here, we give only a typical period 4-limit-cycle which is obtained by integrating the set of the four ODE'S in q , o and p. Figs.8 and 9 show respectively this period 4-limit-cycle in the coordiantes X and Y and the corresponding power spectra. These figures have t o be compared with the figures 4 and 7.

Since the reduced model is based on the soliton picture, we expect that for large values of the control parameter

r

one may reach the limit of the validity of the method. This is actually what we observed. However, the reasons for this are (i) the amplitude of the nonlinear mode becomes large and the NLS approximation fails, (ii) the spatial structure is still stable and coherent but more complex than a perturbed one-breather solution.

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A new model based on a two-soliton state may be used in this region of the control parameters, and one may attack directly the sine-Gordon equation. This problem will be investigated in the future.

Acknowledgements:

This work has been performed within the PROCEOPE collaboration Financial support by the Ministerium fiir Wissenschaft und Forschung NRW and the DFG through SFB 237 as also acknowledged.

References

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/ 2 / Ikezi, H., Phys. Fluids

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(1973) 1668.

/3/ Scott, A.C., Chu, F.Y.F., and Reible, S.A., J. Appl. Phys. a(1976) 3272.

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/ 6 / Pedersen, N.F., Solitons, edited by Trullinger, S.E., Zakharov, V.E., and Pokrovsky, V.L. (Elsevier, Amsterdam, !.386), Chap. 9, p. 469.

/7/ Taki, M., Fernandez, J

C.,

and Reinisch, G., Phys. Rev. A z(1988) 3086, and references therein.

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B.

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(Private Communication).

/lo/

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1111 Bishop, A.R., Forest, M.G., McLaughlin, D.W., and Overrnan 11, E.A., P h y s i c a L D (1986) 293.

/12/ Abramowitz, M., and Stegun, I. A., Hand Book of Mathematical Functions, Dover Publications, INC., N.Y. (1965) p. 76.

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