Bifurcations in two coupled Rössler systems

MATHEMATICS AND COMPUTERS IN SIMULATION ELSEVIER

Mathematics and Computers in Simulation 40 (1996) 247 270

Bifurcations in two coupled R6ssler systems J. Rasmussen a, E. Mosekilde a'*, C. Reick b, a

Physics Department, Center for Chaos and Turbulence Studies, The Technical University of Denmark, DK-2800 Lyngby, Denmark b I. lnstitutfu'r Theoretische Physik, UniversiNit Hambur 9, D-20355 Hamburg, Germany

Abstract

This paper presents a bifurcation analysis of two symmetrically coupled R6ssler systems. The assumed symmetry does not allow any one direction to become preferred, and the coupled system is therefore an example of a higher-dimensional dissipative system which does not become effectively one-dimensional. The results are presented in terms of one- and two-parameter bifurcation diagrams. A particularly interesting finding is the replacement of some of the period-doubling bifurcations by torus bifurcations with the result that instead of the Feigenbaum transition to chaos a quasiperiodic scenario with frequency locking occurs. Calculation of the largest Lyapunov exponents reveals that the system is hyperchaotic in a significant fraction of parameter space.

Keywords: Quasiperiodic transition; Symmetry; Hyperchaos

1. Introduction During the last decade we have come to understand most of the nonlinear dynamical phenomena that can arise in low-dimensional systems. The interest is rapidly shifting towards higher-dimensional phenomena as exemplified through studies of coupled nonlinear oscillators [1-5], map lattices [6-8] and spatially extended systems [9-11]. In particular, Van Buskirk and Jeffries [2] have studied the bifurcation structure of coupled electronic oscillators, Kapitaniak and Steeb [12] have studied the transition to hyperchaos in coupled generalized van der Pol equations, Klein et al. [13] have examined a two-compartment reaction-diffusion system with exchange of one of the species between the compartments, and Kli~ [14] has analyzed the period-doubling bifurcations in a two-compartment Brusselator model. Besides their immediate application to various electronic and chemical systems these types of studies reveal a variety of universal features. Hence, they are extremely relevant, for instance, for biological systems which generally involve a large number of interacting nonlinear oscillators with different periodicities and amplitudes. Within the individual cell more than 100 such self-sustained * Corresponding author. 0378-4754/96/$15.00 (-9 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 7 5 4 ( 9 5 ) 0 0 0 3 6 - 4

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

248

oscillators have been identified. It is also known that many cells exhibit complicated patterns of fast and slow bursts in their membrane potential [-15]. The cells communicate with one another through pulsatile signals [16], and many of the hormones which control the overall function of the body are released in pulses with 1 3 h intervals. This is the case, for instance, for insulin [17], growth hormone [ 18], and the male and female sex hormones [ 19]. The tubular pressure in the nephrons of the kidney have been found to exhibit self-sustained oscillations with a period of 20-30 s [20], and in specific cases period-doubling transitions have been observed [21]. The interaction of all of these self-oscillatory systems holds the potential of an extremely complicated behavior which, generally speaking, is still beyond the reach of modern nonlinear dynamics. In some cases one can expect a collapse of the many-dimensional system into a collective, low-dimensional behavior. In other cases, however, this will not occur, and a truly high-dimensional dynamics will be displayed. Hence, it is of interest to gradually build up our understanding of coupled nonlinear oscillators, and particularly of oscillators which display additional bifurcations. In this paper we present a two-parameter bifurcation analysis of two coupled R6ssler systems. The R6ssler system [22] is the prototype of a simple dynamical system which can develop continuous chaos through a cascade of period-doubling bifurcations. However, the R6ssler system also has a foundation in chemical reaction kinetics [23], and hence one can study the diffusive coupling of two such systems. A particular interesting phenomenon which arises in the coupled system is the replacement of some of the period-doubling bifurcations of the uncoupled systems with torus bifurcations leading to quasiperiodicity and frequency-locking. The emergence of these torus bifurcations is a generic feature of symmetrically coupled, identical period-doubling systems, and the phenomenon is stable against small nonsymmetric perturbations [24]. A similar phenomenon was observed by Van Buskirk and Jeffries [2] in a study of driven passive resonators consisting of an inductance in series with a pn-junction acting as a nonlinear capacitance. When varying the drive voltage, the single resonator showed a period-doubling transition to chaos, whereas two resistively coupled oscillators displayed a quasiperiodic transition. The replacement of period-doubling bifurcations by torus bifurcations was also reported by Anishchenko [25] who investigated a system of two identically coupled electronic frequency generators. Both Van Buskirk and Jeffries [2] and Anischenko E25] noted that the replacement of a perioddoubling transition to chaos by a quasiperiodic transition already occurs in a system of two symmetrically coupled logistic maps. This type of system has been investigated, for instance, by Frqlyland [26] and by Hogg and Huberman [1] who derived relatively detailed phase diagrams illustrating the bifurcation structure in a two-parameter plane. Paulus et al. [27] also found the change from period-doubling to quasiperiodic transition jn a pair of multiplicatively coupled maps. So far, however, detailed analytical and numerical studies of these phenomena for continuous-time systems appear not to have been performed. 2. The model

As previously noted, the R6ssler system )(= -Y--Z, Y=X+AY, Z, = B + Z ( X -- C),

(1)

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

249

is the prototype of a simple dynamical system that develops continuous chaos through a cascade of period-doubling bifurcations. Fig. 1 shows the bifurcation diagram for this system as the parameter A is increased from 0.00 to 0.55. The parameters B and C are kept constant at B = 2 and C -- 4, respectively. With these values the model exhibits two equilibrium points with X o = 2 - 2x/1 - A/2 and X o = 2 + 2 x / i - - A/2, respectively. The latter of these is unstable, and trajectories started in the neighborhood of this point diverge to infinity. The equilibrium point at X 0 = 2 - 2 x / i - - A/2, is stable until A ~ 0.1228 where a supercritical H o p f bifurcation takes place. This leads to an unstable equilibrium point and a stable limit cycle. The limit cycle again loses its stability for A ~ 0.3349 in the first period-doubling bifurcation, and the system hereafter continues to chaos through a Feigenbaum cascade of period-doubling bifurcations, accumulating approximately at A = 0.3857. In the chaotic region, the system displays the usual periodic windows. Much of this behavior is described by the one-dimensional logistic map z,+ a = f ( z , ) = 2z,(1 - z,), 8.0

(2)

-

7.06.0 5.0 X 4.0

3.0

/

2.0 -

/

,/

1.0

0.00

O. 15

0.30

0.45

0.60

A Fig. 1. One-parameter bifurcation diagram for the single R6ssler system. This diagram also applies to the symmetrical solution of the coupled R6ssler model, except that in this case the solution may be unstable in certain intervals.

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

250

although there is an interesting interaction between the unstable equilibrium point and the chaotic attractor which does not arise in the logistic map. Let us hereafter consider a system of two identical, symmetrically coupled R6ssler systems, X= -Y--Z, Y=X+AY, 2 = B + Z ( X - C) + D ( W - Z), =

_

(3)

V-W,

i'=U+AV, I~= B +

W(U - C ) + D(Z

W),

where D is the coupling parameter. We denote the first, second and third state variable by X, Y, Z and U, V, W, respectively, for the two subsystems. The linear and symmetric coupling is realized by adding a term proportional to the difference between W and Z to the right-hand side of the equation for Z. A similar term but with opposite sign is added to the eqaution for IYV.For a chemical system this type of coupling could represent the selective diffusion of the species Z (respectively W) through a semipermeable membrane that separates two continuously stirred tank reactors (CSTRs). Experiments of this type have been performed at several laboratories [28]. Without coupling (D = 0) each of the previously observed bifurcations for the single R6ssler system now occurs doubly degenerate in the total system. When coupling is introduced (D :~ 0), this degeneracy is generally lifted, and each bifurcation immediately splits into two bifurcations occurring at slightly different values of A. Generically, the split between two corresponding bifurcation points will be proportional to D as long as the coupling is relatively weak. It is a characteristic feature of the coupling we have chosen here that a symmetric solution will always exist in which X = U, Y = V and Z = W with X, Y and Z as given by the equations of motion for the single R6ssler system (1). Hence, one set of bifurcation points will occur at precisely the same A-values for the coupled system as for the uncoupled system, and these bifurcations will involve transitions between symmetric solutions of the coupled system. It is important to notice, however, that the stability of these symmetric solutions may be different from the stability of the corresponding solutions of the uncoupled system. The other bifurcation points may shift upwards or downwards depending on the sign of the coupling parameter D and the details of the considered model. They do not necessarily all shift to the same side. In general, a degenerate bifurcation point of the uncoupled system (D = 0) will split into a bifurcation that produces a symmetric solution (and hence remains unshifled) and a bifurcation that produces an antisymmetric solution in which the two subsystems oscillate 180 ° out of phase. As we shall see below, D > 0 implies that the Hopf bifurcation leading to an antisymmetric limit cycle will occur for smaller values of A in our coupled R6ssler system than will the Hopf bifurcation leading to a symmetric limit cycle. Moreover, while the antisymmetric limit cycle is born as a stable solution from the stable equilibrium point, the symmetric limit cycle is born as an unstable solution. 3. Linear stability analysis Let us first perform a linear stability analysis for the coupled R6ssler system. We keep B, C and D constant at B = 2, C = 4 and D = 0.25 and consider the stability of the symmetrical equilibrium

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

251

points X o = U o = ½ C + ~ f l ¼ C 2 - AB = 2 + 2x/l_

--~A,1 (4)

Yo = Vo = X o / A Z o = W o = Xo/A ,

as A is increased from 0. The system may also have asymmetric equilibrium points. These will not be considered in the present analysis, however, and usually such equilibrium points are also unstable. For 0 ~
(5)

where J i s the Jacobian matrix in the equilibrium point, a n d / i s the unit matrix. By evaluating the determinant we find (1 - 2 ( A - - 2 ) ) ( X

-

C -- 2) + (A - 2)Z

(1 - 2 ( A - 2 ) ) ( X

-

C -

(6)

= 0

and 2D - 2) + (A -- 2)Z

= 0.

(7)

It is easy to see that the set of eigenvalues determined by (6) is identical to the eigenvalues for the single R6ssler system. The other set of eigenvalues is found by replacing C by C + 2D. This implies that no matter what the value of D is, three of the eigenvalues will always be identical to the eigenvalues of the single R6ssler system. This corresponds to the symmetrical case where the two systems move in-phase, and the coupling vanishes. For D = 0.25 the eigenvalues are calculated to be 2 1 ~ - 3.8791 + 0.5161 A,

22,3~ - 0.0604 + 0.4920A _+i(1.0137- 0.0301A),

2 4 2 -- 4.3918 + 0.5126A,

)~5,6 ~ -- 0.0541 + 0.4937A _+ i(1.0108- 0.0270A).

Hence, we can expect a Hopf bifurcation of the stable equilibrium point for A _-__0.1096 when the complex conjugate pair ofeigenvalues 25, 6 pass the imaginary axis and another Hopfbifurcation for A ~ 0.1228 when the real value of 22,3 becomes positive. The second Hopf bifurcation involves the now unstable equilibrium point and produces, as already noted, an unstable symmetrical period- 1 orbit. 4. One-parameter

bifurcation

diagrams

We have chosen to give a preliminary illustration of the behavior of the coupled system by displaying a series of one-parameter bifurcation diagrams showing the maxima in the temporal variation of the state variable X as a function of.4. The diagrams were obtained by slowly scanning .4 from the largest value that produces a stable attractor and downwards while neglecting for each new value of .4 a sufficiently long transient to ensure that only steady state behavior is revealed.

252

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

Fig. 2(a) shows the bifurcation diagram obtained for D = 0.25 when following the stable solutions that develop from the first Hopf bifurcation B and C attain their usual values. Starting at A = 0.0 the system has a stable equilibrium point. For A ~ 0.1113 this equilibrium point undergoes a supercritical Hopf bifurcation and produces an antisymmetric period-1 cycle in which the two subsystems oscillate precisely 180 ° out of phase. This agrees well with the results of the linear stability analysis. By means of continuation techniques [29, 30] we can also follow the unstable branches of the bifurcation tree. Fig. 2(b) shows an example of such an analysis. Starting again at A = 0.00, we follow the stable equilibrium point until at A ~ 0.1113 the first Hopf bifurcation occurs. Hereafter the equilibrium point is unstable, and two of its eigenvalues have positive real parts. At A ~ 0.1249 the second Hopf bifurcation occurs, and the unstable, symmetric period-1 solution is born. As A is further increased, this solution undergoes a pitchfork bifurcation at point a. In this bifurcation two mutually symmetric period-1 cycles are born, each with two eigenvalues numerically larger than 1. If we follow these solutions, they proceed through normal period-doubling bifurcations, the first

8.0 :I 7.0

6.0

5.0

X

4.0

3.0

2.0

1.0

O•0

I

0.0 (a)

I

I

I

]

0.I

I

I

I

I

[

I

I

|

0.2

I

l

0.3

i

i

i

i

i

0,4

I

I

I

I

[

0.5

A

Fig. 2. Bifurcation diagram for the coupled R6ssler system with asymmetric initial conditions (a). By comparison with Fig. 1 we note how the first period-doubling bifurcation has been replaced by a torus bifurcation. Bifurcation diagram for the unstable solutions which develop from the symmetric period-1 orbit (b) (next page).

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247--270

253

7.01 6.0

5.0

4.0 X 3.0

2.0

1.0 b c 0.0 0.0 (b)

0.1

0.2

0.3

0.4

0.5

A Fig. 2. Continued.

occurring in points b and c. After point a, the symmetric period-1 solution only has a single eigenvalue which is numerically larger than 1. As we follow this solution it proceeds through a pair of period-doubling bifurcations producing, respectively, an unstable, symmetric period-2 solution (in point d) and a doubly unstable, antisymmetric period-2 solution (in point e). Point d occurs at A = -~ 0.3348 and coincides with the first period-doubling bifurcation of the uncoupled R6ssler system. The symmetric period-2 solution born in this bifurcation again proceeds through a pair of period-doubling bifurcations producing, respectively, an unstable, antisymmetric period-4 solution (in point f ) and a doubly unstable, symmetric period-4 solution (in point 9). Point 9 occurs at A-~ 0.375 and coincides with the second period-doubling bifurcation for the uncoupled R6ssler system. The antisymmetric period-2 solution born in point e proceeds through a pitchfork bifurcation (in point h), and each of the mutually symmetric period-2 solutions hereafter perform a sequence of period-doubling bifurcations. Fig. 2(b) clearly illustrates the complicated bifurcation structure which governs the number and character of the unstable solutions. This structure is usually not revealed in bifurcation diagrams although it is important to understand properly the bifurcations that arise for the stable solutions. Let us hereafter return to Fig. 2(a).

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

254

For A ~ 0.3127, a supercritical torus bifurcation occurs instead of the first period-doubling bifurcation. This is a generic phenomenon for symmetrically coupled, nearly identical perioddoubling systems [-24]. Hereafter, the dynamics takes place on a two-torus and is quasiperiodic, interrupted by narrow regions (Arnol'd tongues) where frequency locking occurs. The quasiperiodic solution exists until A _-__0.3353 where it is replaced by two mutually symmetric, stable period-2 cycles. As illustrated more clearly in Fig. 3(a) the stable period-2 cycles are born together with two mutually symmetric, unstable period-2 cycles in a couple of simultaneous saddle-node bifurcations occurring at A ~ 0.335272. As A is further increased, the stable period-2 cycles undergo a perioddoubling bifurcation at A _-__0.34615, and the unstable period-2 cycles undergo a similar bifurcation at A _-__0.3509. Fig. 3(b) shows a Poincar6 section of the torus which exists immediately before the formation of the two period-2 solutions (i.e. for A = 0.335268). More precisely the figure shows the points of intersection between the stationary solution and the plane X = 0, projected onto the UV-plane. We clearly observe the near resonant behavior with most of the intersection points 5.25 -

// j./"

/

\ \\

5.15

5.05 >< 4.95

4.85

4.75 (a)

--r" 0.330

\

0.335

0.340

0.345

0.350

0.355

A

Fig. 3. The two simultaneous saddle-node bifurcations in which pairs of mutually symmetric stable and unstable period-2 solutions are born. At higher values both types of solution go through a period-doubling bifurcation. Also shown is a Poincar6 section of the torus that exists for A = 0.335.

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

-3.5

255

iI

\ \ \

-4.0

/

-4.5

/ /

,/

-5.0

i

/

-

--5.5

I

-0.5 (b)

i

/

I

I

I

I

;

;

;

0.0

I

I

0.5

;

I

I

I

I

1.0

U

Fig. 3. Continued.

concentrating in four regions of the cross section. These are the regions from which the two period-2 orbits emerge. Each of the two period-2 orbits hereafter appear to develop through a normal period-doubling cascade to chaos which is reached approximately at A = 0.356. This is illustrated in the close-ups of this part of the bifurcation diagram reproduced in Fig. 4. For A < 0:3675 the system has two coexisting, mutually symmetric chaotic attractors. For larger values of A, however, these attractors merge into a single, chaotic attractor. It is interesting to observe that the chaotic band is visited more uniformly than for the single RSssler system, and that some of the structure associated with iterations of the critical point [31] has disappeared. With the resolution of Fig. 2 it is possible to observe only a single periodic window. The solution in this window is a stable, antisymmetric period-3 limit cycle. If one considers the bifurcations on this cycle, one also finds that they start with a torus bifurcation instead of the first period-doubling bifurcation. This is illustrated in Fig. 5 which shows a magnification of one of the branches of the period-3 solution. For slightly higher values of A (A ~ 0.4065), the quasiperiodic region ends as the system develops two mutually symmetric, stable period-6 orbits. Each of these solutions again passes

256

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

an apparently normal period-doubling sequence to chaos. In the range up to A ~ 0.40705 the system displays two mutually symmetric chaotic attractors developed from the period-6 solutions, but hereafter the attractors merge into a single chaotic attractor. The single branch of the period-3 solution in Fig. 5 clearly shows a period-7 window in the chaotic regime. This corresponds to a period-21 window for the system as such. As illustrated in Fig. 6, the bifurcations on the period-21 orbit again start with a torus bifurcation for A ~ 0.407253. The system hereafter exhibits quasiperiodic behavior with frequency-locking until A _-__0.407265 where two mutually symmetric period-42 orbits arise. For A ~ 0.407269, the system turns chaotic in an intermittency transition, and it remains chaotic until A = 0.407272 where two mutually symmetric period-84 orbits arise. These orbits hereafter proceed to chaos via period-doubling and/or quasiperiodicity. In the beginning the chaotic attractor falls in very narrow bands to suddenly increase dramatically in a crisis [32].

7.0

--

6.0

--

5.0

4.0 X

• ......#.

3.0 . . . . . ~!'~!~.:~,..'., , ,, t ~~ ~-,,,:..~ ~:~~.,

2.0

1.0

0.0

I

0.29 (a)

;

I

I

I

0.,31

I

I

I

i

I

i

I

i

0.,33

I

I

0.35

I

i

I

i

I

i

0.,37

I

i

t

I

0.39

A

Fig. 4. Close-ups of part of the bifurcation diagram in Fig. 2 to illustrate the two coexisting, mutually symmetric period-2 orbits.

257

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

7.0

6.0

5.0

4.0 X

3.0

2.0

1.0 m

0.0

I

0.29 {b)

i

i

i

I

0.3]

i

i

i

i

I

I

i

I

0.33

I

I

0.35

I

I

I

I

I

0.37

i

i

i

i

1

0.39

A Fig. 4. Continued.

For higher values of the coupling parameter, the overall bifurcation diagram shows that the quasiperiodic range developing after the torus bifurcation of the antisymmetric period-1 orbit expands at the expense of the range with two mutually symmetric period-2 orbits. This is illustrated in Fig. 7 for D = 0.5. Here, in fact, the quasiperiodic region is terminated by the generation of two mutually symmetric period-4 orbits, and the whole range with period-2 orbits has disappeared. Moreover, before the quasiperiodic region ends, the system has become chaotic. This is clearly revealed by a positive value for the largest Lyapunov exponent for 0.327 ~
258

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

-

2.0

~,'? ~

"-1

X

'


1.0-b--r0.4058

0.4062

0.4066

0.4070

0.4074

0.4078

A Fig. 5. Magnificationofoneofthebranchesoftheantisymmetricperiod-3orbit againshowshow the first period-doubling bifurcation is replacedby a torus bifurcation.

and that the system hereafter proceeds through torus destabilization. This cannot be proved through numerical studies, however, since we are dealing with details of the asymptotic behavior that cannot easily be resolved by computer simulation. For the symmetric period-3 solution which exists in the window from A = 0.4053 to A -- 0.4161, we observe a somewhat different scenario. Here, the system first passes through a symmetry breaking pitchfork bifurcation to produce two mutually symmetric period-3 solutions. Each of these then proceed to chaos through a supercritical torus bifurcation followed by torus destabilization.

5. Emergence of quasiperiodicity in coupled logistic maps

One of the most interesting observations in the preceding section was the replacement of the first period-doubling bifurcation of the antisymmetric solution by a torus bifurcation. This is a generic

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

259

1.58

1.s7

X

-1--91

1.56

-

1.55

-

1.54

-

! ........q 1.55

-

1.52

,

,

0.40725

I

I

I

I

I

I

0.40726

I

I

I

0.40727

I

i

i

I

0.40728

A Fig. 6. Magnification of 3 branches of the period-21 solution. Besides the torus bifurcation we here see two intermittency transitions to chaos.

phenomenon for symmetrically coupled nearly identical period-doubling systems which has been dealt with in considerable detail in another publication [24]. To understand the essentials of the phenomenon let us for a m o m e n t consider the system of two coupled logistic maps Xk +1 ~"f (Xk) "{- C(Yk -- Xk)'

Yk+ 1 =f(Yk) + e ( x k -- Yk),

(8)

withf(z,) as given by (2). The coupling parameter is now denoted by e, but the coupling itself is very similar to the one we have used for the R6ssler systems. As for the coupled R6ssler systems all symmetric solutions to (8) follow directly from the solutions to the logistic map. In particular, the symmetric fixed point is given by s

s

Xo = Y0 = (2 - 1)/2

(9)

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

260

8.0-

7.0

6.0

5.0

X

4.0

5.0

2.0

1.0 t

0.0 0.0

, c-i--i--; ,-,-,-; 0.1

(,-, 0.2

i ,

I, 0.3

,,,

":7~

I , , , , t 0.4 0.5

A Fig. 7. B i f u r c a t i o n d i a g r a m f o r t h e c o u p l e d R 6 s s l e r s y s t e m s w i t h a c o u p l i n g p a r a m e t e r D= 0.5. The quasiperiodic solution obtained after the torus bifurcation at A ~_ 0.294 now develops into chaos through torus destabilization.

and the symmetric period-2 orbit by xS

s

1,2 = Yl,2 - -

2_~21

~---

X/(2--1)2--4

(10)

-4-~

On the other hand, the coupled system may also display an antisymmetric period-2 m o t i o n where the two subsystems are precisely one iteration out of phase. This orbit is determined by the solutions to (8) with X1,2 a ~ Y2,1, , i.e., a

a

X1,2 = Y2,1 --

2 + 1 -- 2e

2Z

/(2

+ X/

--

1) 2 --

4(1 -

,~)2

422

(1 1)

The period-2 orbits first arise when the arguments of the square roots turn positive, i.e., for 2] = 3

(12)

261

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

and 2] = 3 - 2~,

(13)

respectively. It follows from this that for ~ > 0 the antisymmetric period-2 orbit arises for lower values of 2 than does the symmetric period-2 orbit, and again we observe that the splitting between the two bifurcations is proportional to the coupling parameter. It is i m p o r t a n t to notice that although the symmetric orbits of the coupled system follow directly from the orbits of the logistic map, these orbits do not necessarily have the same stability properties. If (x, + 2, Y, + 2) = (x,, y,) denotes a period-2 orbit of the coupled system, its stability is determined by the eigenvalues of the matrix

j = n=l~I1

n) -- e

E

=

(14)

if(y,)--

7.0

e

"

--

6.0

5.0

4.0 X 5,0

2.0 :i~': :::'::

1.0

0.0

I

0.27 (a)

I

I

I

I

i

0.29

i

i

i

I

i

0.,.'31

i

i

i

I

i

0,,.33

'

I

i

I

i

0.35

'

i

I

[

i

0.,37

I

i

|

[

0.39

A

Fig. 8. Magnifications of part of Fig. 7 to show the development of two mutually symmetric period-4 solutions.

262

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

7.0-

6.0

5.0

4.0 X 3.0

2.0

1.0

0.0

i

0.27

i

,

,

I

i

0.29

i

,

i

I

0.31

'

i

i

I

I

i

0.33

i

i

i

I

'

'

0.35

i

,

I

i

0.,37

i

i

i

I

0.39

A

(b)

Fig. 8. Continued.

For a symmetric period-2 orbit, the stability matrix has the eigenvalues 1/] = 5 - ( 2 -

1)2

(15)

and r/2 = 5 - - (~. - - 1) 2 -+- 4 e ( e + 1).

(16)

The first of these two eigenvalues is equal to the stability coefficient for the period-2 orbit of the logistic map. This orbit is stable for 3 ~<2 ~< 1 +~/-6 corresponding to the interval in which Ir/1 [ <~ 1. However, for the symmetric period-2 to be stable we must also have Itt2[ ~< 1. To the first order in e this requires 3 + e ~< 2 ~< 1 + . , f 6 + ~ 3 .

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

263

Hence, it follows that for finite coupling the symmetric period-2 orbit of the coupled map system is born as an unstable solution for 2 = 3 and is stabilized in a secondary bifurcation for 2 = 3 + e. For the antisymmetric period-2 orbit (x]. 2 = Yz,x) the stability matrix has the eigenvalues ~/1,2 =

5(1 - 8)2 --(X

--

1)2 q-/32 q- 2e X/5(1 -- a) 2 --(2

- - 1) 2,

(17)

which assume the values 1 and 1 - 2a for 2 = 3 - 2e. Hence, the antisymmetric period-2 solution is born to be stable. For 2 > 1 + x/-5(1 - a), the two eigenvalues are complex and they leave the unit circle as a pair of complex conjugate eigenvalues for 2 = 2n = 1 + x/Z(1 - e)(3 - 2e)

(18)

In this way, the second period-doubling bifurcation in the coupled map system is replaced by a Hopf bifurcation. The first period-doubling bifurcation s e r v e s - in the same way as the H o p f bifurcation for the time-continuous system - to generate an antisymmetric solution. It is this antisymmetry that gives rise to the imaginary c o m p o n e n t in the eigenvalues To clarify this statement it may be of interest to expand the eigenvalues ql,2 for the antisymmetric period-2 orbit in terms of the stability coefficient for the period-2 orbit of the logistic map. This expansion gives [24] ,1,2 = 7 - 2e(S -t- x/7) + O(e2) •

(19)

Without coupling (e = 0) this reduces to q],2 = 7. The period-2 solution of the logistic map undergoes a period-doubling bifurcation when 7 = - 1. For g # 0 and negative values of 7 there is an additional imaginary contribution to r/i, 2 arising from x~" Close to the period-2/period-4 bifurcation for the logistic map, the antisymmetric solution of the coupled system has two complex conjugated eigenvalues. Moreover, since it is possible to change 2 so that 7 passes - 1, for sufficiently small 8, it will also be possible to change 2 so that q1,2 cross the unit circle with imaginary components. With the transformation of the period-doubling bifurcation into a torus bifurcation a whole new set of resonance p h e n o m e n a arise [33], and an infinity of Arnol'd tongues with their associated saddle-node bifurcations now emerge from this bifurcation point. Even the slightest coupling thus changes the character of this point completely. As we have shown elsewhere [24], this transformation is generic and can be extended to symmetrically coupled, nearly identical time continuous period-doubling systems such as, for instance, the coupled R6ssler systems discussed in this paper.

6. Two-parameterphase diagrams By means of the continuation routines implemented in Simpack [29] and P A T H [30], we have determined the basic structure of a two-parameter phase diagram for the coupled R6ssler system. The parameters considered are the bifurcation parameter A and the coupling parameter D. Fig. 9 provides an overview of the obtained results. Here, curve 1 represents the H o p f bifurcation of the stable equilibrium point in which the antisymmetric period-1 orbit is born. Curve 2 shows the H o p f bifurcation which gives rise to the unstable, symmetric period-1 orbit. It is evident that the position of this bifurcation is independent of the coupling parameter. For D = 0 both H o p f bifurcation curves meet in the Hopfbifurcation point of the single R6ssler system. Curve 3 represents

264

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270 0,8

-

31

0.6

E3 0.4

0.2

0.0

I l l ,

0.0

I

0.1

1 i

i

i

i

i

i

i

0.2

i

I

0.3

i

i

i

i

0.4

i

i

I

0.5

A Fig. 9. Overview of the bifurcation structure. To the left of curve 1 the equilibrium point is stable. Between curves 1 and 3 the antisymmetric period-1 orbit is the only stable solution, and between curves 3 and 4 we have quasiperiodicity and frequency-locking. Curve 2 represents the supercritical Hopf bifurcation in which the symmetric period-1 orbit is born. This orbit is unstable.

the supercritical torus bifurcation in which the antisymmetric period-1 orbit becomes quasiperiodic, and curve 4 shows the saddle-node bifurcation where the quasiperiodic region ends with the emergence of two stable, mutually symmetric period-2 orbits. Between these two curves we have an infinity of Arnol'd tongues spreading out as a fan from the bifurcation point of the uncoupled system. The next bifurcation curves can more clearly be seen in Fig. 10 where we have stretched part of the phase diagram along the A-axis. In this figure curves 5, 7 and 9 represent period-doubling bifurcations in which the two mutually symmetric period-2 solutions lose their stability to period-4, period-8, and period-16 solutions. For D = 0 all of these curves start at the corresponding bifurcation points for the single R6ssler system. Curves 4 and 5 meet at the point a (A,D) (0.33603386, 0.75864322), curves 6 and 7 meet at the point b (A, D) --- (0.3330824, 1.0556738), and curves 8 and 9 meet at the point c (A, D ) ~ (0.33289116, 1.12703295). These are all codimension-2 points. To the right of curve 9 the system is mostly chaotic.

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

265

0.8-

0.6

,o L\,l

0.4

0.2 -~

/ / /

5"~

0.0 . - i 0.32

0.54

7\'~ / /

0.,.56

0.58

f

J

/

0.40

0.42

A Fig. 10. Detail of phase diagram. Between curves 4 and 5 the system has two stable, mutually symmetric period-2 solutions. Between curves 5, 6 and 7 two stable period-4 solutions exist, and between curves 7, 8 and 9 two stable period-8 solutions are found. Curves 10 and 11 are associated with the period-7 window in the chaotic regime. Between curves 9 and 10 the system is generally chaotic.

In point a curve 5 turns around as an unstable period-doubling curve and bends downwards to the right to pass through the point in which the simultaneous period-doublings of the unstable period-2 orbits in Fig. 2(a) occur. The period-2 saddle-node curve continues upwards at least for a while, now representing an unstable saddle-node bifurcation. Tilted slightly to the left, a new stable saddle-node bifurcation curve arises. This curve (curve 6) represents the bifurcations in which the two mutually symmetric period-4 solutions are born. Around points b and c the picture is more or less the same. The period-4/period-8 and period-8/period-16 bifurcation curves turn around and bend downwards to the right, now representing unstable period-doubling bifurcations. The saddle-node bifurcation curves continue upwards, also representing saddle-node bifurcations of unstable solutions, and new curves for stable period-8 and period-16 saddle-node bifurcations demarcate the end of the quasiperiodic region. For small values of D the whole area between curves 3 and 4 is quasiperiodic or

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247-270

266

periodic. For D > 0.379, however, one can observe the quasiperiodic solutions destabilize into chaotic solutions before the period-4 saddle-node bifurcation curves are reached. Curve 10 represents the saddle-node bifurcation in which the two stable, mutually symmetric period-7 cycles are born, and curve 11 shows the supercritical torus bifurcation in which these solutions become quasiperiodic. For D < 0.375 it is not possible to observe the period-7 window in the one-parameter bifurcation diagrams. This is because the period-7 solutions coexist with a chaotic solution which has a much larger basin of attraction. Hence, it is difficult to find the period-7 solutions, and the associated bifurcation curves cannot be followed accurately to low values of D. Finally, Fig. 11 shows the main bifurcation curves connected with the period-3 window. Here, curve 12 represents the saddle-node bifurcation in which two stable period-3 cycles are born, curves 13 and 15 are supercritical torus bifurcation curves in which the period-3 solutions become

0.8

--

0.6

0.2

,

0.0 0.40

,

i

,~~I 0.41

,

,

,

,

I

0.4"2

A Fig. 11. The main bifurcation curves connected with the period-3 window. Between curves 12 and 13 we have two mutually symmetric period-3 solutions, and between curves 16 and 17 we have two mutually symmetric period-6 solutions. Between curves 13 and 16 the solution is quasiperiodic. Around the window the solution is mostly chaotic.

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

267

quasiperiodic, curve 14 represents a pitchfork bifurcation in which a symmetrical period-3 solution is replaced by two mutually symmetric period-3 solutions, curve 16 is a saddle-node bifurcation in which two stable period-6 solutions are born, and curve 17 is a period-doubling curve in which these period-6 solutions lose their stability to period-12 solutions. The codimension-2 bifurcation points d and e are located at (A, D) = (0.40585728, 0.51205142) and (0.4063659, 0.27676693), respectively.

7. Lyapunov exponents and hyperchaos Fig. 12 shows the variation of the three largest Lyapunov exponents for the coupled Rrssler systems with D = 0.25. The calculations were performed using the method described by Wolf [34]. The variation of the Lyapunov exponents clearly reflects the existence of a stable periodic orbit up to A = 0.313 where two exponents become equal to zero, and a two-torus is produced. At A ~- 0.335, a stable limit cycle again appears. This orbit passes through a series of period-doubling bifurcations

0.20 1 0.10

-1

0.00 -0.10 -0.20 -0.30

-0.4-0 -0.50

I

-0.60

-

-0.70 0.0

0.1

0.2

0.3

0.4

0.5

A Fig. 12. Variation of the three largest Lyapunov exponents for the coupled R6ssler system with D = 0.25. For A > 0.3675 two Lyapunov exponents are positive, and the system is hyperchaotic.

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

268

leading the system to become chaotic at A ~ 0.356. In the chaotic regime we can observe a couple of periodic windows, before the system becomes hyperchaotic at A ~ 0.368. The hyperchaotic behavior, which is characterized by having two positive Lyapunov exponents, emerges in the crisis where the two chaotic attractors merge. The observed variation of the largest Lyapunov exponents confirms the results of our one- and two-parameter bifurcation analysis. The whole scenario appears to be quite characteristic for systems which develop hyperchaos [35, 36]. The route to hyperchaos typically starts with a torus bifurcation leading to quasiperiodic behavior with frequency locking. Then a saddle-node bifurcation follows producing a relatively simple periodic orbit which after an often incomplete perioddoubling cascade is destabilized into chaos that subsequently turns into hyperchaos. The initial quasiperiodic behavior reflects the fact that the system contains two more or less evenly strong oscillators, and because of this symmetry the dynamics does not effectively collapse into a onedimensional behavior. Fig. 13 shows a Poincar6 section of the hyperchaotic attractor for A = 0.42 and D = 0.25. The section was obtained by plotting Z versus V for 10 6 subsequent intersection points of the stationary 1.4"

--

1.2

..

%

1.0 I'4 0.8

0.6-

0.4

i t , l i i

--7.0

-5.0

'

I

I

i

i

-3.0

,

I

i

-I.0

i

i

i

I

1.0

'

i

i

i

I

3.0

V Fig. 13. Poincar6 section of the hyperchaotic solution existing in the coupled R6ssler system for D = 0.25 and A = 0.42.

J. Rasmussen et al./Mathematics and Computers in Simulation 40 (1996) 247 270

269

trajectory with the plane X = 0. The Poincar6 section clearly reveals the folded towel structure characteristic of many hyperchaotic solutions [35].

8. Conclusion The bifurcation structure observed in the present analysis is rather complex, and many details still remain to be revealed, particularly for higher values of A and D. We have identified regions with two coexisting periodic attractors as well as regions with coexisting chaotic attractors. By calculating the largest Lyapunov exponent we have shown that the system becomes hyperchaotic for large enough values of A. An interesting finding is the replacement of some of the period-doubling bifurcations with torus bifurcations leading the coupled system to follow a quasiperiodic transition to chaos. If instead of a positive coupling constant we had considered negative values for D, the order of the two initial Hopf bifurcations would have been interchanged, and a stable symmetric period-1 orbit would have appeared. In this case the first period-doubling bifurcation would have remained a bifurcation of this type. Nonetheless, the Feigenbaum route to chaos might still have been interrupted at higher bifurcations, because the same transformation we have seen for the first period-doubling bifurcation with D > 0 may arise at later stages in the cascade for D < 0. Thus, the replacement of the Feigenbaum route to chaos by a quasiperiodic transition is generic to coupled, nearly-identical period-doubling systems. Another interesting aspect of the coupling of nearly identical systems is that the chaotic motion may retain a relatively high dimensionality, and the dimension may continue to grow with the number of coupled subsystems [36]. If instead of two coupled period-doubling systems we had considered three such systems, we would have expected each bifurcation of the uncoupled systems to immediately split into three bifurcations. And generically for N coupled identical systems each bifurcation would split into N bifurcations, depending, of course, on the symmetries of the introduced coupling. To the extent that some of the period-doubling bifurcations are replaced by torus bifurcations, frequency-locking resonances may arise again. The formation of higher tori can also be expected, and these are known to be hard to stabilize. Hence, for large N we would observe bands of bifurcations. The occurrence of such bands is likely to produce new routes to chaos and hyperchaos.

Acknowledgements We have applied the software packages SimPack and PATH in our numerical calculations of oneand two-parameter bifurcation diagrams. We are thankful to Jesper Skovhus Thomsen, Rasmus Feldberg and Carsten Knudsen, who developed SimPack, for many helpful suggestions concerning numerical procedures and interpretations of the observed bifurcation structure.

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