Synchronization of a chaotic particle with ϕ6 potential

Physics Letters A 353 (2006) 179–184 www.elsevier.com/locate/pla

Synchronization of a chaotic particle with φ 6 potential Xiaoli Yang a,b , Wei Xu a,∗ , Zhongkui Sun a a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China b College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China

Received 24 September 2005; received in revised form 23 December 2005; accepted 23 December 2005 Available online 5 January 2006 Communicated by A.R. Bishop

Abstract The phenomenon whether common noise can induce complete synchronization in identical chaotic systems has been a topic of great relevance and longstanding controversy in the past decade. The present Letter extends to explore the injection of a common signal, either bounded noise or another chaotic driving, to the trajectories of chaotic systems can successfully induce complete synchronization. We illustrate a particular example of a particle with φ 6 potential in the chaotic region, and give numerical evidence showing that the addition of a common signal to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization when the largest Lyapunov exponent becomes negative. © 2006 Elsevier B.V. All rights reserved. PACS: 05.45.Xt; 05.45.G; 05.40.Ca Keywords: Chaos synchronization; Bounded noise; Extended Duffing oscillator

1. Introduction As it is well known sensitivity to initial conditions is a generic feature of chaotic dynamical systems. Two chaotic orbits, starting from slightly different initial points in state space, separate exponentially with time, and become totally uncorrelated. Thus, it is indeed surprising to find two chaotic orbits which are initiated differently could be brought to synchronization with each other [1]. Considerable attention has been shifted to chaos synchronization for its potential applications in a wide range from optics, chemical and biological systems, neural networks to secure communications. In the mean time various methods, such as linear and nonlinear feedback control [2,3], adaptive control [4,5], sliding mode control method [6] and active control [7,8], have been proposed for the synchronization of chaotic systems. In the context of coupled chaotic elements [9], there are several different types of synchronization of coupled chaotic os* Corresponding author. Tel.: +86 29 88495453; fax: +86 29 88494174.

E-mail addresses: [email protected] (X. Yang), [email protected] (W. Xu). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.082

cillators which have been described theoretically and observed experimentally. These are the complete synchronization [1,10], phase synchronization [11,12], lag synchronization [13], generalized synchronization [14,15], intermittent lag synchronization [13,16], imperfect phase synchronization [17], and almost synchronization [18]. The complete synchronization, which implies gradual coincidences of two states of coupled oscillators in the course of time, was the first discovered and is the simplest form of synchronization in chaotic systems. This mechanism was first shown to occur when two identical chaotic systems are coupled unidirectional, provided that the conditional Lyapunov exponents of the subsystem to be synchronized are all negative [1]. The influence of common noise on complete synchronization of identical chaotic systems was investigated by Mritan and Banavar in 1994 [19], which spurred a longstanding dispute on the general conclusion that strong enough noise is able to synchronize chaotic systems. Some authors [20,21] found that synchronization of the logistic maps in Ref. [19] is an artifact due to finite precision in numerical simulations. Others claimed that it is the nonzero mean of the applied noise that plays a decisive role and an unbiased noise cannot lead to syn-

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chronization [22,23]. This claim, however, has been disproved by examples [24–26] in which unbiased noise indeed induces complete synchronization. It is the significant work of Zhou and Kurths in 2002 [27] that clarified the mechanism of noiseinduced synchronization implying that noise may change the balance between contraction or expansion, synchronization occurs when contraction is dominating and whether the noise is biased or unbiased is not the key point. Motivated by the above findings, the main interest in this Letter is to give further evidence that it is possible to synchronization two identical chaotic systems linked by a common signal (bounded noise or another chaotic driving). We carry out our investigation in a Duffing oscillator with φ 6 potential (extended Duffing oscillator) described by the following equation [28] x¨ + a x˙ + bx + cx 3 + dx 5 = f cos Ω1 t.

(1)

Depending on the set of the parameters b, c and d, the potential of system (1) can be considered at least three physically interesting situations where the potential is (i) single well, (ii) double well, (iii) triple well. The present authors [29] have investigated the effect of Gaussian white noise on the chaotic motion of system (1). The present interest, however, is shifted here to chaos synchronization of two identical extended Duffing oscillators with three wells of the form of Eq. (1) driven by bounded noise and another chaotic signal, respectively. 2. Synchronization driven by bounded noise Firstly, we consider the influence of bounded noise on the extended Duffing oscillator, and then system (1) with addition of the noise can be written as x¨ + a x˙ + bx + cx 3 + dx 5 = f cos Ω1 t + Dξ(t).

Fig. 1. (a) Bifurcation diagram and (b) the largest Lyapunov exponent of system (1).

(2)

Obviously, the noisy extended Duffing system is the same as that in Eq. (1), except with a noise term Dξ(t) added to Eq. (1). The introduced bounded noise [30] is a harmonic function with constant amplitude and random frequency and phase which can be expressed by mathematical presentation as ξ(t) = cos(Ω2 t + σ B(t) + Γ ), where Ω2 and σ are positive constants, B(t) is a standard Wiener process, Γ is a random variable uniformly distributed in [0, 2π). The bounded noise ξ(t) is a stationary random process in wide sense with zero mean. It was first employed by Stratonovich [31] and has since been applied in certain engineering applications by many researchers [30,32,33] and chaotic motion by Yang et al. [34,35]. It should be note that studying the convergence or divergence of trajectories of Eq. (2) starting from different initial conditions under the same driving is equivalent to analyzing the converge or divergence of trajectories from two identical systems of the form Eq. (2) driven by the same excitation. Selecting the parameters as a = 0.678, b = 1.0, c = −0.5, d = 0.038 and Ω1 = 1.0 and remaining them fixed in this Letter, the bifurcation diagram and largest Lyapunov exponent as a function of the amplitude of harmonic excitation in the noiseless case is plotted respectively in Fig. 1. The typical windows in which the system behaves chaotically can be observed obviously in Fig. 1(a). The corresponding largest Lyapunov expo-

nent, vividly depicted in Fig. 1(b), is positive in these regions. For instance, for f = 2.0 (the case will be considered throughout the Letter) the largest Lyapunov exponent is 0.16274. To investigate the effect of the addition of noise on system (1), we have integrated numerically Eq. (2) using the stochastic Euler method [36] with the time step 0.001 when the parameters of bounded noise are selected as Ω2 = 1.0 and σ = 0.1. The largest Lyapunov exponent is computed using a simultaneous integration of the linearized equations of Eq. (2). In Fig. 2 we observe the largest Lyapunov exponent becomes negative when the noise level is more than a critical value Dc (Dc ≈ 0.39). For the deterministic case of the extended Duffing system, i.e., D = 0 in Eq. (2), trajectories starting with different initial conditions are completely uncorrelated during the evolution. This is also the situation for small values of noise level D (D < Dc ), for instance, see Fig. 3(a) which displays the evolution of the two trajectories x1 (t), x2 (t) starting from two completely different initial points of system (2) for D = 0.1 and Fig. 4(a) which displays their corresponding synchronization error e(t) (e(t) = x1 (t) − x2 (t)). However, when using a noise level more than Dc the noise is strong enough to induce perfect coincidence of the trajectories, for instance, see Fig. 3(b) which displays the two trajectories x1 (t), x2 (t) starting from two completely different initial points of system (2) for D = 1.0 and

X. Yang et al. / Physics Letters A 353 (2006) 179–184

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Fig. 2. The largest Lyapunov exponent vs. the level of bounded noise of system (2) for f = 2.0.

Fig. 4. Plot of the synchronization error of the two trajectories x1 , x2 starting from two completely different initial points of system (2) driven by bounded noise in the case (a) D = 0.1 and (b) D = 1.0.

Fig. 3. Plot of the evolution of two trajectories x1 , x2 starting from two completely different initial points of system (2) driven by bounded noise in the case (a) D = 0.1 and (b) D = 1.0.

Fig. 4(b) which displays their corresponding synchronization error e(t). From Figs. 3(b) and 4(b) we can see two different trajectories which have started in completely different initial conditions and behaved chaotically become identical after some transient time, i.e., they are perfectly synchronized. One can also observe that the basic structure of the attractor of system (2) remains fixed as shown in Fig. 5, although the attraction region has an expansion trend as the noise intensity increasing. The result is similar with that obtained in Ref. [27] which investigated the effect of the addition of Gaussian noise on the complete synchronization of two identical Lorenz systems. Therefore, all these results implies that complete synchronization by the addition of common noise in the chaotic Duffing system with φ 6 potential does occur when the noise intensity is more than the critical value Dc . In addition, when the frequency of Ω2 is fixed at 1.0, further numerical simulations have been carried out for various intensity of σ . And the results imply that noise-induced syn-

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Fig. 6. The largest Lyapunov exponent vs. the intensity of driving signal of system (2) for f = 2.0.

source of excitation is the classical Lorenz system working in the chaotic region with the parameters ρ = 10, r = 28, δ = 8/3, give by the following equations  u˙ 1 = −ρ(u1 − u2 ), (3) u˙ 2 = −u1 u3 + ru1 − u2 , u˙ 3 = u1 u2 − δu3 .

Fig. 5. Attractors (Poincaré map) of system (2) in the (x, y (y = dx/dt )) plane in the case (a) of no noise, (b) D = 0.1 and (c) D = 1.0.

chronization also occurs for some values of noise level D as that of the case discussed above.

In this case the system under consideration is excited using a signal given by a variable u3 from the Lorenz system, i.e., in Eq. (2). As the discussion of Section 2, the largest Lyapunov exponent of Eq. (2) with the addition of the chaotic driving is computed numerically and displayed in Fig. 6. From Fig. 6 we observe that the largest Lyapunov exponent becomes negative when the level of excitation is more than a critical value Dc (Dc ≈ 0.01). For the deterministic case of the extended Duffing system, i.e., D = 0 in Eq. (2), the two trajectories x1 (t), x2 (t) starting with different initial conditions, displayed in Fig. 5(a), are completely uncorrelated during the evolution, and their corresponding synchronization error e(t), displayed in Fig. 6(a), does not trend towards zero in the course of time. However, when the driving intensity is more than Dc , the presence of the driving term forces the largest Lyapunov exponent to become negative, i.e., the driving is strong enough to induce synchronization of the trajectories. For instance, see Fig. 7(b) which displays the two trajectories x1 (t), x2 (t) starting from two completely different initial points of system (2) for D = 0.2 and Fig. 8(b) which displays their corresponding synchronization error e(t). From Figs. 7(b) and 8(b) one can see two different trajectories which have started in completely different initial conditions and behaved chaotically become identical after some transient time, i.e., they are perfectly synchronized.

3. Synchronization driven by another chaotic signal 4. Concluding remarks Secondly, we show complete synchronization of the chaotic extended Duffing oscillator occurs when another chaotic signal, rather than noise, is used as the common external driving. The

Synchronization of nonlinear systems, which are not coupled or only weakly coupled but subjected to a common fluctu-

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Fig. 7. Plot of the evolution of two trajectories x1 , x2 starting from two completely different initial points of system (2) driven by chaotic signal in the case (a) of no driving D = 0.0 and (b) D = 0.2.

ating driving signal, has drawn considerable attention in recent years. The common fluctuating driving signal, assumed to be noise in many contexts, is also of great relevance to engineering structure, biological systems [37–39], etc. In the literature, the common driving signal generally assumed to be Gaussian white noise or wideband random process. However, quite often, the random loading of structures is a narrowband random process, for example the excitation caused by the deck and/or towers in vortex shedding and buffeting, the Dryden and von Karman of the air on-flow [40]. Hence, bounded noise, as a rather new model of narrowband random excitation, is a suitable model for common random fluctuating. In this Letter, complete synchronization of two identical systems subjected to a common driving signal, modeled by bounded noise or another chaotic signal, is considered in detail. The numerical results have shown that the injection of an additive common signal to different trajectories of the extended Duffing system, which start from different initial con-

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Fig. 8. Plot of the synchronization error of the two trajectories x1 , x2 starting from two completely different initial points of system (2) driven by chaotic signal in the case (a) of no driving D = 0.0 and (b) D = 0.2.

ditions, leads eventually to their perfect synchronization when the largest Lyapunov exponent becomes negative. Since, on the one hand, the mode of a particle with φ 6 potential is a universal nonlinear differential equation, and many nonlinear oscillators in physical, engineering and biological problems can be really descried by the model or analogous ones. On the other hand, the dynamical systems are inevitably exposed to the common fluctuating driving signal in practice and bounded noise is a suitable model for common random fluctuating. So, the synchronization procedure proposed in this Letter may have practical application in the future. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10472091 and 10332030) and NSF of Shaanxi Province (Grant No. 2003A03) and NSF of Guangdong Province (Grant No. 04011640) and the Doctorate

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