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Strange nonchaotic attractors in the externally modulated Rayleigh–Bénard system
Chaos, Solitons and Fractals 20 (2004) 1141–1148 www.elsevier.com/locate/chaos
Strange nonchaotic attractors in the externally modulated Rayleigh–Benard system E.J. Ngamga Ketchamen a, L. Nana
b,c
, T.C. Kofane
a,c,*
a
c
Laboratoire de M ecanique, D epartement de Physique, Facult e des Sciences, Universit e de Yaound e I, B.P. 812, Yaound e, Cameroon b Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon 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–Benard system of binary mixtures near the codimension-two point is considered. Strange nonchaotic dynamics and chaotic behaviour are investigated numerically. The creation of strange nonchaotic attractors as well as the onset of chaos are studied through an analysis of Poincare 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]. Ó 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–Benard 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] €x ¼ ½a þ e1 cosðxtÞ_x þ ½b þ e2 cosðxt þ UÞx þ f1 x3 þ f2 x2 x_
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
*
Corresponding author. E-mail addresses: [email protected] (E.J.N. Ketchamen), [email protected] (L. Nana), [email protected] (T.C. Kofane). 0960-0779/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/j.chaos.2003.09.040
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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 characterized 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 exponents 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, Poincare surfaces 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 advantage 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 d€ þ ½a f2 x20 e1 cos xtd_ þ ½b 3f1 x20 e2 cos xt 2f2 x0 x_ 0 d ¼ 0
ð3Þ
It is interesting to note that we may eliminate the friction coefficient from Eq. (3) by letting D ¼ degðtÞ
ð4Þ
1 g_ ðtÞ ¼ ½a þ f2 x20 þ e1 cos xt 2
ð5Þ
where
Inserting Eq. (4) into Eq. (3) yields € þ b 3f1 x2 e2 cos xt f2 x0 x_ 0 1 e1 x sin xt 1 ða þ f2 x2 þ e1 cos xtÞ2 D ¼ 0 D 0 0 2 4
ð6Þ
This linearized equation (6) describes the dynamics of a Hamiltonian system with one degree of freedom given by 1 1 1 1 b 3f1 x20 e2 cos xt f2 x0 x_ 0 e1 x sin xt ða þ f2 x20 þ e1 cos xtÞ2 D2 ð7Þ H ¼ v2 þ 2 2 2 4 where v¼
dD : dt
ð8Þ
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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] dM ¼ JSM dt
ð9Þ
The evolution of the product MM þ is governed by the equation d ðMM þ Þ ¼ JSMM þ MM þ SJ dt
ð10Þ
where M þ denotes the matrix transpose of M, while S denotes the symmetric matrix given by H¼
2m 1 X ¼ Sij Di vj 2 i;j¼1
ð11Þ
and where J¼
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 S¼
S11 0
0 S22
ð13Þ
where 1 1 S11 ¼ b 3f1 x20 e2 cos xt f2 x0 x_ 0 e1 x sin xt ða þ f2 x20 þ e1 cos xtÞ2 2 4 S22 ¼ 1
ð14Þ
The general two-dimensional symplectic matrix can be written in the form M ¼ eJSa eJSc ¼ elðB2 cos aþB3 sin aÞ ebB1
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 0 1 0 1 1 0 ; B2 ¼ ; B3 ¼ ð16Þ B1 ¼ 1 0 1 0 0 1 It follows that MM þ ¼ e2JSa ¼ e2lðB2 cos aþB3 sin aÞ 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 sin a sinh 2l þ cosh 2l cos a sinh 2l MM þ ¼ cos a sinh 2l cosh 2l sin a sinh 2l Next, some manipulations of equations (18) and (10) yield to the following system ( dl ¼ 12 ðs22 S11 Þ cos a dt da dt
¼ S11 þ S22 ðS22 S11 Þ sin a coth 2l
ð17Þ
ð18Þ
ð19Þ
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These differential equations form the basis of symplectic method for calculating the Lyapunov exponents of our system. The Lyapunov exponents k are given by k ¼ lim
t!1
1 ½gðtÞ lðtÞ t
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 Poincare 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. Poincare surfaces of section are useful to determine in particular the period of the system’s response. We define the Poincare map in our case as the T -stroboscopic map, where T ¼ 2p=x. Surfaces of section are here obtained by re-
0.4 (a)
0.2
x
0
-0.2
-0.4 -1
-0.8
-0.6
-0.4
-0.2
0
f2 0.07
(b) 0.06 0.05 0.04
λ 0.03 0.02 0.01 0
-0.01 -0.02 -1
-0.8
-0.6
f2
-0.4
-0.2
0
Fig. 1. (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 0:05, b ¼ 0:045, x ¼ 0:395, e1 ¼ 0:25 and e2 ¼ 0:14.
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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 kþ ðkþ > 0Þ for a chaotic state and kþ 6 0 for nonchaotic states. For a modulated Rayleigh–Benard 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
(a) 0.1
x
0
-0.1
-0.2 -1
-0.8
-0.6
-0.4
-0.2
0
-0.4
-0.2
0
f2 0.06
(b)
0.04 0.02
λ
0 -0.02 -0.04 -0.06 -1
-0.8
-0.6
f2 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.
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0.15 (a)
0.1 0.05
x
0 -0.05 -0.1 -0.15 -1
-0.8
-0.6
-0.4
-0.2
0
f2 0.06
(b) 0.04 0.02
λ
0 -0.02 -0.04 -0.06 -1
f
-0.8
-0.6
-0.4
-0.2
0
f2 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.
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 ½1; 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, illustrate 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. Remarkably, 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 exponent 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
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1147
Fig. 4. Poincare 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.
)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 ½0:491; 0 in Fig. 2(b) and f2 2 ½0:433; 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 Poincare maps 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 ½0:433; 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–Benard 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.
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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.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Steinberg V. J Appl Math Mech USSR 1971;35:335. Hurle DTJ, Jakeman F. J Fluid Mech 1971;47:667. Brand H, Hohenberg PC, Steinberg V. Phys Rev A 1983;27:591. Brand H, Hohenberg PC, Steinberg V. Phys Rev A 1984;30:2548. Coullet PH, Spiegel EA. SIAM J Appl Math 1983;43:776. Knobloch E, Proctor MRB. J Fluid Mech 1981;108:291. Knobloch E, Guckenheimer J. Phys Rev A 1983;27:408. Newell AC, Whitehead JA. J Fluid Mech 1969;38:279. Segel LA. J Fluid Mech 1969;38:203. Zielinska BJA, Mukamel D, Steinberg V, Fishman S. Phys Rev A 1986;34:4171. Grebogi C, Ott E, Pelikan S, Yorke JA. Physica D 1984;13:261. Bondeson A, Ott E, Antonsen TM. Phys Rev Lett 1985;55:2103. Romeiras FJ, Bondensen A, Ott E, Antonsen T, Grebogi C. Physica D 1987;26:277. Romeiras FJ, Ott E. Phys Rev A 1987;35:4404. Ding M, Grebogi C, Ott E. Phys Rev A 1989;39:2593. Ding M, Grebogi C, Ott E. Phys Lett A 1989;137:167. Brindley J, Kapitaniak T. Chaos, Solitons & Fractals 1991;1:323. Kapitaniak T, Ponce E, Wojewoda J. J Phys A 1990;23:L383. Ditto W, Spano M, Savage H, Rauseo S, Heagy J, Ott E. Phys Rev Lett 1990;65:533. Heagy JF, Ditto W. J Nonlinear Sci 1991;1:423. Zhou T, Moss F, Bulsara A. Phys Rev A 1992;45:5394. Kapitaniak T. Phys Rev E 1993;47:1408. Heagy JF, Hammel SM. Physica D 1994;70:140. Pikovsky A, Feudel U. J Phys A 1994;27:5209. Ding M, Scott Kelso J. Int J Bifurcat Chaos 1994;4:553. Lai YC. Phys Rev E 1996;53:57. Nishikawa T, Kaneko K. Phys Rev E 1996;54:6114. Kuznetsov S, Feudel U, Pikovsky A. Phys Rev E 1998;57:1585. Habib S, Ryne RD. Phys Rev Lett 1995;74:70.
Strange nonchaotic attractors in the externally modulated Rayleigh–Benard system E.J. Ngamga Ketchamen a, L. Nana
b,c
, T.C. Kofane
a,c,*
a
c
Laboratoire de M ecanique, D epartement de Physique, Facult e des Sciences, Universit e de Yaound e I, B.P. 812, Yaound e, Cameroon b Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon 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–Benard system of binary mixtures near the codimension-two point is considered. Strange nonchaotic dynamics and chaotic behaviour are investigated numerically. The creation of strange nonchaotic attractors as well as the onset of chaos are studied through an analysis of Poincare 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]. Ó 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–Benard 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] €x ¼ ½a þ e1 cosðxtÞ_x þ ½b þ e2 cosðxt þ UÞx þ f1 x3 þ f2 x2 x_
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
*
Corresponding author. E-mail addresses: [email protected] (E.J.N. Ketchamen), [email protected] (L. Nana), [email protected] (T.C. Kofane). 0960-0779/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/j.chaos.2003.09.040
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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 characterized 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 exponents 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, Poincare surfaces 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 advantage 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 d€ þ ½a f2 x20 e1 cos xtd_ þ ½b 3f1 x20 e2 cos xt 2f2 x0 x_ 0 d ¼ 0
ð3Þ
It is interesting to note that we may eliminate the friction coefficient from Eq. (3) by letting D ¼ degðtÞ
ð4Þ
1 g_ ðtÞ ¼ ½a þ f2 x20 þ e1 cos xt 2
ð5Þ
where
Inserting Eq. (4) into Eq. (3) yields € þ b 3f1 x2 e2 cos xt f2 x0 x_ 0 1 e1 x sin xt 1 ða þ f2 x2 þ e1 cos xtÞ2 D ¼ 0 D 0 0 2 4
ð6Þ
This linearized equation (6) describes the dynamics of a Hamiltonian system with one degree of freedom given by 1 1 1 1 b 3f1 x20 e2 cos xt f2 x0 x_ 0 e1 x sin xt ða þ f2 x20 þ e1 cos xtÞ2 D2 ð7Þ H ¼ v2 þ 2 2 2 4 where v¼
dD : dt
ð8Þ
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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] dM ¼ JSM dt
ð9Þ
The evolution of the product MM þ is governed by the equation d ðMM þ Þ ¼ JSMM þ MM þ SJ dt
ð10Þ
where M þ denotes the matrix transpose of M, while S denotes the symmetric matrix given by H¼
2m 1 X ¼ Sij Di vj 2 i;j¼1
ð11Þ
and where J¼
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 S¼
S11 0
0 S22
ð13Þ
where 1 1 S11 ¼ b 3f1 x20 e2 cos xt f2 x0 x_ 0 e1 x sin xt ða þ f2 x20 þ e1 cos xtÞ2 2 4 S22 ¼ 1
ð14Þ
The general two-dimensional symplectic matrix can be written in the form M ¼ eJSa eJSc ¼ elðB2 cos aþB3 sin aÞ ebB1
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 0 1 0 1 1 0 ; B2 ¼ ; B3 ¼ ð16Þ B1 ¼ 1 0 1 0 0 1 It follows that MM þ ¼ e2JSa ¼ e2lðB2 cos aþB3 sin aÞ 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 sin a sinh 2l þ cosh 2l cos a sinh 2l MM þ ¼ cos a sinh 2l cosh 2l sin a sinh 2l Next, some manipulations of equations (18) and (10) yield to the following system ( dl ¼ 12 ðs22 S11 Þ cos a dt da dt
¼ S11 þ S22 ðS22 S11 Þ sin a coth 2l
ð17Þ
ð18Þ
ð19Þ
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These differential equations form the basis of symplectic method for calculating the Lyapunov exponents of our system. The Lyapunov exponents k are given by k ¼ lim
t!1
1 ½gðtÞ lðtÞ t
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 Poincare 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. Poincare surfaces of section are useful to determine in particular the period of the system’s response. We define the Poincare map in our case as the T -stroboscopic map, where T ¼ 2p=x. Surfaces of section are here obtained by re-
0.4 (a)
0.2
x
0
-0.2
-0.4 -1
-0.8
-0.6
-0.4
-0.2
0
f2 0.07
(b) 0.06 0.05 0.04
λ 0.03 0.02 0.01 0
-0.01 -0.02 -1
-0.8
-0.6
f2
-0.4
-0.2
0
Fig. 1. (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 0:05, b ¼ 0:045, x ¼ 0:395, e1 ¼ 0:25 and e2 ¼ 0:14.
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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 kþ ðkþ > 0Þ for a chaotic state and kþ 6 0 for nonchaotic states. For a modulated Rayleigh–Benard 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
(a) 0.1
x
0
-0.1
-0.2 -1
-0.8
-0.6
-0.4
-0.2
0
-0.4
-0.2
0
f2 0.06
(b)
0.04 0.02
λ
0 -0.02 -0.04 -0.06 -1
-0.8
-0.6
f2 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.
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0.15 (a)
0.1 0.05
x
0 -0.05 -0.1 -0.15 -1
-0.8
-0.6
-0.4
-0.2
0
f2 0.06
(b) 0.04 0.02
λ
0 -0.02 -0.04 -0.06 -1
f
-0.8
-0.6
-0.4
-0.2
0
f2 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.
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 ½1; 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, illustrate 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. Remarkably, 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 exponent 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
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Fig. 4. Poincare 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.
)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 ½0:491; 0 in Fig. 2(b) and f2 2 ½0:433; 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 Poincare maps 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 ½0:433; 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–Benard 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.
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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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