Chaos, Solitons and Fractals 20 (2004) 1141–1148

www.elsevier.com/locate/chaos

Strange nonchaotic attractors in the externally modulatedRayleigh–B�eenard system

E.J. Ngamga Ketchamen a, L. Nana b,c, T.C. Kofane a,c,*

a Laboratoire de M�eecanique, D�eepartement de Physique, Facult�ee des Sciences, Universit�ee de Yaound�ee I, B.P. 812,

Yaound�ee, Cameroonb Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon

c The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586 Strada Costiera, 11, I-34014 Trieste, Italy

Accepted 19 September 2003

Abstract

The amplitude equation associated with an externally modulated Rayleigh–B�eenard system of binary mixtures near

the codimension-two point is considered. Strange nonchaotic dynamics and chaotic behaviour are investigated nu-

merically. The creation of strange nonchaotic attractors as well as the onset of chaos are studied through an analysis of

Poincar�ee surfaces, a construction of the bifurcation diagram and a new method for computing Lyapunov exponents

that exploits the underlying symplectic structure of Hamiltonian dynamics [Phys. Rev. Lett. 74 (1995) 70].

� 2003 Published by Elsevier Ltd.

1. Introduction

Nonlinear equations play a central role in modern science. In particular, ordinary differential equations of nonlinear

type are very often encountered in the theoretical description of a broad variety of phenomena and processes. Examples

are found in a wide variety of systems, including biological systems, weather models, mechanical devices, plasmas, and

fluids, to name a few.

For various systems subjected to external temperature gradients, one finds two kinds of instabilities: stationary and

oscillatory [1,2]. For certain values of the external parameters, the stationary and oscillatory bifurcation lines intersect

at a so-called codimension-two (CT) point [3–7]. The behaviour of these systems near such a point may often be exactly

described by amplitude equations which are two dimensional and are differential equations for the amplitudes of the

critical modes at the instability [8,9].

The model we use as our working example is a Rayleigh–B�eenard system of binary fluid in which the instantaneous

Rayleigh number is given by R ¼ R0 þ R1 cosðxtÞ, where R0 is the Rayleigh number in the absence of the modulation, R1

is the amplitude of the modulation and is considered as a small perturbation ðR1=R0 � 1Þ.The system close to the CT point is described by the following nonlinear amplitude equation [10]

* Co

E-m

Kofan

0960-0

doi:10.

€xx ¼ ½aþ e1 cosðxtÞ� _xxþ ½bþ e2 cosðxt þ UÞ�xþ f1x3 þ f2x2 _xx ð1Þ

where overdot means time derivative and x is related to the vertical component of the velocity. For example, e1 and e2

are proportional to R1. The parameters a and b are functions of the temperature gradient and the concentration. The

rresponding author.

ail addresses: eulaliejoelle@yahoo.com (E.J.N. Ketchamen), la_nanad@yahoo.fr (L. Nana), tkofane@uycdc.uninet.cm (T.C.

e).

779/$ - see front matter � 2003 Published by Elsevier Ltd.

1016/j.chaos.2003.09.040

1142 E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148

coefficients a, b, e1, e2, f1, f2 and U are directly related to the physical properties of the application taken from [10]. On

the basis of this assumption, a particularly rich structure has been studied by Zielinska et al. [10].

In this case it has been reported that the presence of modulation close the CT point results in rich variety of new

bifurcations and in particular in regions in parameter space of chaotic behaviour. These authors found that for a

physically realistic range of parameters for binary mixtures, the conductive phase loses its stability and becomes chaotic

via intermittency [10]. However, our understanding of its behaviour is still far from complete. The motivation of the

present paper is to show that in addition to the well-known conductive state and chaotic attractors, it is possible to

observe another type of behavior leading to strange nonchaotic attractors [11–27]. Strange nonchaotic attractors

(SNAs) were first described by Grebogi et al. [11]. SNAs exhibit some properties of regular as well as chaotic regimes.

Like regular attractors, they have only negative Lyapunov exponents; as for usual chaotic attractors they are char-

acterized by fractal structure but typical nearby trajectories on it do not diverge exponentially with time. It has been

found that the transition to chaos in quasiperiodically forced systems is generally mediated by SNAs. They have been

investigated in a number of numerical [11–27] and experimental [28,29] studies.

The paper is organized as follows. In Section 2, we use the symplectic calculation to evaluate the Lyapunov ex-

ponents for the system under consideration. As indicated by Habib and Ryne [29], this approach obviates analytically

the need for rescaling and reorthogonalization in the numerical computation of the exponents. In Section 3, we present

the results of our numerical simulations which evolve the computation of largest Lyapunov exponents, Poincar�ee sur-

faces and bifurcation diagrams. Section 4 concludes the paper.

2. Symplectic calculation of Lyapunov exponents

The Lyapunov exponents quantify the exponential divergence or convergence of initially nearby trajectories. Over

the past two decades several methods for calculating these exponents have been developed.

In what follows, we will make use of symplectic calculation to evaluate the Lyapunov exponents of our system [see

Eq. (1)].

Symplectic methods have been applied with success to classical dynamical problem [2,3]. These methods take ad-

vantage of the Hamiltonian structure of many systems and avoid the renormalization and reorthogonalization in the

numerical computation of the exponents [29].

We begin by defining the deviation variable d by

d ¼ x� x0 ð2Þ

where x0 denotes the fiducial trajectory.

In terms of the parameter just defined, Eq. (1) can be written, after the linearization, as

€dd þ ½�a� f2x20 � e1 cos xt� _dd þ ½�b� 3f1x2

0 � e2 cos xt � 2f2x0 _xx0�d ¼ 0 ð3Þ

It is interesting to note that we may eliminate the friction coefficient from Eq. (3) by letting

D ¼ de�gðtÞ ð4Þ

where

_ggðtÞ ¼ 1

2½aþ f2x2

0 þ e1 cosxt� ð5Þ

Inserting Eq. (4) into Eq. (3) yields

€DD þ�� b� 3f1x2

0 � e2 cos xt � f2x0 _xx0 �1

2e1x sinxt � 1

4ðaþ f2x2

0 þ e1 cos xtÞ2�D ¼ 0 ð6Þ

This linearized equation (6) describes the dynamics of a Hamiltonian system with one degree of freedom given by

H ¼ 1

2v2 þ 1

2

�� b� 3f1x2

0 � e2 cosxt � f2x0 _xx0 �1

2e1x sin xt � 1

4ðaþ f2x2

0 þ e1 cos xtÞ2�D2 ð7Þ

where

v ¼ dDdt

: ð8Þ

E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148 1143

The functions D and v may be considered as the generalized coordinate and momentum, respectively.

As it is well-known, the dynamics of classical Hamiltonian systems has an underlying symplectic structure [4]. Those

systems such as (6) are governed by a symplectic matrix M that maps the initial variables into time-evolved variables.

It has been shown that the symplectic matrix M satifies the equation of motion [5]

dMdt

¼ JSM ð9Þ

The evolution of the product MMþ is governed by the equation

d

dtðMMþÞ ¼ JSMMþ �MMþSJ ð10Þ

where Mþ denotes the matrix transpose of M , while S denotes the symmetric matrix given by

H ¼ 1

2¼

X2mi;j¼1

SijDivj ð11Þ

and where

J ¼ 0 1

�1 0

� �ð12Þ

in which 1 represents the m m identity matrix, and m is the site index.

Comparing Eq. (7) with (11), one finds that the matrix S is of the form

S ¼ S11 0

0 S22

� �ð13Þ

where

S11 ¼ �b� 3f1x20 � e2 cos xt � f2x0 _xx0 �

1

2e1x sinxt � 1

4ðaþ f2x2

0 þ e1 cos xtÞ2

S22 ¼ 1

ð14Þ

The general two-dimensional symplectic matrix can be written in the form

M ¼ eJSaeJSc ¼ elðB2 cos aþB3 sin aÞebB1 ð15Þ

where Sa is a symmetric matrix that anticommutes with J and Sc is an another symmetric matrix that commutes with J .

The parameters a, b, and l are real coefficients and where B1, B2 and B3 are basis elements of the Lie algebra given by

B1 ¼0 1�1 0

� �; B2 ¼

0 11 0

� �; B3 ¼

1 00 �1

� �ð16Þ

It follows that

MMþ ¼ e2JSa ¼ e2lðB2 cos aþB3 sin aÞ ð17Þ

since the second matrix on the right-hand side of Eq. (15) is unitary.

Eq. (17) leads, when taking into account Eq. (16) and properties of matrix’s exponential calculation, to

MMþ ¼ sin a sinh 2l þ cosh 2l cos a sinh 2lcos a sinh 2l cosh 2l � sin a sinh 2l

� �ð18Þ

Next, some manipulations of equations (18) and (10) yield to the following system

dldt ¼ 1

2ðs22 � S11Þ cos a

dadt ¼ S11 þ S22 � ðS22 � S11Þ sin a coth 2l

(ð19Þ

1144 E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148

These differential equations form the basis of symplectic method for calculating the Lyapunov exponents of our

system. The Lyapunov exponents k are given by

Fig. 1

of the

and e2

k ¼ limt!1

1

t½gðtÞ lðtÞ� ð20Þ

where l follows from solving numerically (19) and g from (5).

3. Numerical results

In this section, we present some results obtained by the direct numerical investigation of Eq. (1). Our numerical

routines are based on the classical fourth-order Runge–Kutta algorithm with the integration time step, which we chose

to be Dt ¼ 2p=100x.

We have used three standard indicators including Poincar�ee surfaces, bifurcation diagrams and computation of

largest Lyapunov exponents to characterize the long time dynamics of our model. These indicators reinforce each other

in the following way.

Poincar�ee surfaces of section are useful to determine in particular the period of the system’s response. We define the

Poincar�ee map in our case as the T -stroboscopic map, where T ¼ 2p=x. Surfaces of section are here obtained by re-

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-1 -0.8 -0.6 -0.4 -0.2 0

f2

λ

(b)

-0.4

-0.2

0

0.2

0.4

-1 -0.8 -0.6 -0.4 -0.2 0

(a)

x

f2

. (a) Bifurcation diagram with respect to the component x, by varying the value of f2 and (b) control parameter f2 dependence

largest Lyapunov exponent. The other physical parameters are fixed as: f1 ¼ �1, a ¼ �0:05, b ¼ �0:045, x ¼ 0:395, e1 ¼ 0:25

¼ 0:14.

E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148 1145

cording the positions of the oscillator every integer multiple of T , starting at some large enough time to ensure that

transients have died away. If one such surface of section consists of a finite number k of distinct points, the response of

the system is a subharmonic of order k, that is, it performs a period-k motion. When the number of points that form the

surface of section is infinite, with all the points lying on a smooth closed curve, the motion is quasiperiodic. Strange

attractors correspond to surfaces of section made of an infinite number of points that occupy a bounded domain of the

cross-section without forming a smooth closed curve. They may be chaotic or not. The bifurcation diagram is another

indicator of the presence or not of irregular movements. It permits the identification of order-chaos-order transitions in

the dynamics. The Lyapunov exponents now constitute a standard tool for the numerical identification of chaotic

states. They measure the average exponential rates of divergence or convergence of nearby orbits in phase space. The

chaotic behavior is characterized by the largest Lyapunov exponents kþðkþ > 0Þ for a chaotic state and kþ 6 0 for

nonchaotic states.

For a modulated Rayleigh–B�eenard system of binary mixtures, the parameters f1 and f2 are such that f2 < 0 and

f1 > 0 [10]. For these chosen values, the model described by Eq. (1) is not stable and therefore, there is a need to add a

fifth-order term to Eq. (1) to ensure stability. We restrict ourselves to Eq. (1) and take f1 < 0 which corresponds to some

-0.2

-0.1

0

0.1

0.2

-1 -0.8 -0.6 -0.4 -0.2 0

(a)

x

f 2

(b)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-1 -0.8 -0.6 -0.4 -0.2 0f2

λ

Fig. 2. (a) Bifurcation diagram with respect to the component x, by varying the value of f2 and (b) control parameter f2 dependence of

the largest Lyapunov exponent. The other physical parameters are fixed as: f1 ¼ �1, a ¼ �0:148, b ¼ �0:039, x ¼ 0:273, e1 ¼ 0:3 and

e2 ¼ 0:348.

(b)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-1 -0.8 -0.6 -0.4 -0.2 0f

2

(a)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-1 -0.8 -0.6 -0.4 -0.2 0f

f

2

x

λ

Fig. 3. (a) Bifurcation diagram with respect to the component x, by varying the value of f2 and (b) control parameter f2 dependence

of the largest Lyapunov exponent. The other physical parameters are fixed as: f1 ¼ �1, a ¼ �0:14, b ¼ �0:039, x ¼ 0:273, e1 ¼ 0:15

and e2 ¼ 0:35.

1146 E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148

cases of magnetoconvection. The numerical values of the other parameters are taken close to those used by Zielinska

et al. [10]. The control parameter is f2.

Figs. 1(a), 2(a) and 3(a) display the usual bifurcation diagram as a function of f2, with f1 ¼ �1. The range of

variation of the control parameter f2 is ½�1; 0�. It is noted that as f2 increases, bistability goes through two period

doubling sequences leading to a chaotic attractor as shown in Fig. 1(a) where the other parameters are fixed to

a ¼ �0:05, b ¼ �0:045, x ¼ 0:395, e1 ¼ 0:25 and e2 ¼ 0:14. Fig. 2(a), which we obtained for a ¼ �0:148, b ¼ �0:039,

x ¼ 0:273, e1 ¼ 0:3 and e2 ¼ 0:348, and Fig. 3(a) for a ¼ �0:14, b ¼ �0:039, x ¼ 0:273, e1 ¼ 0:15 and e2 ¼ 0:35, il-

lustrate the qualitatively same behaviour. In both cases, a large band of chaos gives rise to bistability. We now turn to

the calculation of the largest Lyapunov exponents as a function of f2, related to those bifurcation diagrams. Re-

markably, one observes another type of behaviour emerging into SNAs. SNAs display geometric properties unlike

either limit cycles or quasiperiodic attractors. While chaotic behaviour are characterized by the largest Lyapunov ex-

ponent which is positive, SNAs and period-k attractors have their one which is negative. These variations in the sign of

the largest Lyapunov exponent according to a specific behaviour is clearly observed in Fig. 1(b). If f2 is small, k is

negative, so Eq. (1) does not show sensitive dependence on the initial conditions. When f2 is increasing up to about

Fig. 4. Poincar�ee surfaces with a ¼ �0:14, b ¼ �0:039, x ¼ 0:273, e1 ¼ 0:15, e2 ¼ 0:35 and f1 ¼ �1: (a) f2 ¼ �0:431; (b) f2 ¼ �0:851;

(c) a ¼ �0:05, b ¼ �0:045, x ¼ 0:395, e1 ¼ 0:25, e2 ¼ 0:14 and f2 ¼ �0:3 and (d) f2 ¼ �0:82.

E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148 1147

)0.491 in Fig. 2(b) and )0.433 in Fig. 3(b), k changes suddenly from positive to negative values and the behaviour of

Eq. (1) is nonchaotic. In the intervals f2 2 ½�0:491; 0� in Fig. 2(b) and f2 2 ½�0:433; 0� in Fig. 3(b), we observe long

transient aperiodic trajectories without sensitive dependence on the initial conditions. Those slight oscillations of the

non trivial Lyapunov exponent although this one is negative are strange [22], the word strange referring to the geometry

of the attractor.

We focus our attention on the critical value of the control parameter f2. This crucial value is the one where the

largest Lyapunov exponents change sign from positive to negative. This transition occurs at f2 � �0:4. Some Poincar�eemaps for the system around this particular value are shown in Fig. 4. On Fig. 4(a), one sees a quasiperiodic motion.

This motion is also observed for almost all values of f2 2 ½�0:433; 0�. At the left of the critical value of f2, we notice a

chaotic attractor as shown on Fig. 4(b). Another chaotic attractor is observed on Fig. 4(c) obtained for f2 ¼ �0:3, the

other parameters are fixed as in Fig. 1(a). Fig. 4(d) displays a strange nonchaotic attractor with kþ ¼ �0:002.

4. Conclusion

In this paper, we have considered the dynamics of a Rayleigh–B�eenard system of binary fluid. Our particular point of

interest was to show the existence of SNAs in that dynamics. This has been done by measuring the largest Lyapunov

exponent. We have found that the SNAs exist in a finite interval starting at the critical value of the control parameter

and going up to zero. We noticed that the transition from chaos to regular motion is mediated by SNAs. In our further

studies, we will consider a suitable model for binary mixtures described by an amplitude equation in which a fifth-order

term is required.

1148 E.J.N. Ketchamen et al. / Chaos, Solitons and Fractals 20 (2004) 1141–1148

Acknowledgements

The authors would like to thank the Abdus Salam International Centre for Theoretical Physics (AS-ICTP) and the

Swedish International Development Agency (SIDA) for sponsoring the visit of Dr. NANA Laurent as Junior Associate

at the AS-ICTP.

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