Elsevier

Optik

Volume 126, Issue 23, December 2015, Pages 3854-3858
Optik

Partial switched modified function projective synchronization of unknown complex nonlinear systems

https://doi.org/10.1016/j.ijleo.2015.07.075Get rights and content

Abstract

Many studies on the switched modified function projective synchronization of nonlinear real dynamic systems have been carried out, whereas the switched modified function projective synchronization of chaotic complex systems has not been studied extensively. In this paper, the concept of the partial switched modified function projective synchronization of complex nonlinear systems with fully unknown parameters is proposed. Partial switched synchronization of chaotic systems means that the state variables of the drive system synchronize with partial different state variables of the response system. Based on the parameter modulation and the adaptive control technique, the adaptive control laws and the parameter update laws are derived to make the states of two different complex nonlinear systems asymptotically synchronized up to a desired scaling function matrix. The given scaling function can be an equilibrium point, a periodic orbit, or even a chaotic attractor in the phase space. Finally, numerical simulations are performed to verify the feasibility and effectiveness of the proposed controllers and identifiers.

Introduction

In the past decades, chaos synchronization has become a hot subject in the field of nonlinear science due to its great potential applications in physical systems, biological networks, information sciences and secure communications, etc. Up to now, a variety of synchronization approaches have been proposed which include complete synchronization [1], phase synchronization [2], generalized synchronization [3], lag synchronization [4], projective synchronization [5], modified projective synchronization [6], [7], function projective synchronization [8], [9], [10] and modified function projective synchronization [11], [12], [13], [14], [15], [16]. Modified function projective synchronization is the more general type of synchronization, which means that the master and slave systems could be synchronized up to a scaling function matrix. The complete synchronization, projective synchronization, modified projective synchronization and function projective synchronization are all its special case.

Because the unpredictability of the scaling function factors in modified function projective synchronization can additionally enhance the security of secure communications, this new synchronization phenomenon has been developed and studied extensively recently. Refs. [12], [13], [14] investigated the adaptive modified function projective synchronization problem of chaotic and hyperchaotic systems with fully unknown or partially unknown parameters. Ref. [15] considered the modified function projective synchronization of unidirectional coupled multiple time-delayed Rössler chaotic systems by adaptive controls. Based on chaos mask and parameter modulate, two different hyperchaotic secure communication schemes by using generalized function projective synchronization are presented in [16]. More recently, a new synchronization, called switched modified function projective synchronization, was proposed in [17]. Compared with the projective synchronization and its variants, switched synchronization of chaotic systems in which a state variable of the drive system synchronize with a different state variable of the response system is a promising type of synchronization as it provides greater security in secure communications [17], [18]. Therefore, it would be very instructive and significant to study switched modified function projective synchronization in such systems with unknown parameters.

Most of the research efforts mentioned above have concentrated on studying a presetting scaling function with numerical examples. In fact, the given scaling function can be an equilibrium point, a periodic orbit or even a chaotic attractor in the phase space [19]. In addition, all the above-mentioned studies only involve with the chaotic systems with real variables. For describing the real world better, many complex dynamical systems have been proposed and studied [20], [21], [22], [23], [24], [25], [26], [27], [28]. Fowler et al. [20] firstly introduced the complex Lorenz equations. The complex Chen and Lü systems are introduced and the global synchronization of coupled identical systems are well investigated in [21] and their controls and modified projective synchronization [22] are also discussed. By adding a state feedback controller and using complex periodic forcing, a new hyperchaotic complex Lü system [23] is constructed. An active control scheme is applied to achieve phase and antiphase synchronization of two identical hyperchaotic complex Lorenz systems [24]. Anti-synchronization is examined in different types of chaotic complex systems with certain and uncertain parameters in [25], [26], respectively. The adaptive control is used to achieve synchronization of two identical n-dimensional chaotic complex nonlinear systems with uncertain parameters [27], and the nonlinear control technique is used to realize lag synchronization of n-dimensional hyperchaotic complex nonlinear systems [28].

Inspired by the above discussions, many studies on the switched modified function projective synchronization of nonlinear real dynamic systems have been carried out, whereas the switched modified function projective synchronization of chaotic complex systems has not been studied extensively. In this paper, we intend to apply the adaptive control technique to investigate the partial switched modified function projective synchronization problems of a class of chaotic (hyperchaotic) complex systems with unknown parameters. Based on the adaptive feedback method, sufficient conditions for two different chaotic (hyperchaotic) complex systems have been obtained by constructing a suitable drive chaotic complex system and a controlled slave chaotic complex system. The adaptive laws of the unknown parameters are given as well. They guarantee that the unknown parameters are identified when the partial switched modified function projective synchronization occurs.

The rest of this paper is organized as follows. Section 2 gives some preliminaries. In Section 3, the adaptive laws and synchronization criteria are derived given. One example is provided to illustrate the effectiveness of the obtained scheme in Section 4. Finally, some conclusions are given in Section 5.

Section snippets

Problem description

Consider a class of n-dimensional complex dynamical system, which is described by the following form of the differential equation:x˙=H(x,α)where x = (x1, x2, …, xn)T  Cn is a n-dimensional state complex vector with xl=xlr+jxli, l = 1, 2, …, n and j=1; α is an n × 1 real (or complex) vector of system unknown parameters and superscripts r and i stand for the real and imaginary parts of the state complex vector x, respectively. Further, it can be rewritten asx˙=f(x)+F(x)αwhere f(x) : Cn  Cn is the

Main results

In this section, we discuss the partial switched modified function projective synchronization of two different chaotic complex systems with uncertain parameters via the adaptive control method. The purpose of synchronization is to design a controller U, which is able to drive the slave system synchronize to the master system according to desired scaling function matrix.

Substituting (2) and (3) into (4), the error dynamical system ise˙=y˙iM(t)x˙jM˙(t)xj=g(yi)+G(yi)β+UM(t)[f(xj)+F(xj)α]M˙(t

Numerical simulation

In the foregoing section, Theorem 1 and Corollaries essentially provide the criteria for synchronization and unknown system parameters identification between controlled uncertain complex systems. Here, our scheme is illustrated by applying it for two different chaotic complex Chen and Lorenz systems.

The complex chaotic Chen system is described byx˙1=a(x2x1)x˙2=(ba)x1+bx2x1x3x˙3=x1¯x2+x1x2¯2cx3in which a, b, c, are constant parameters. When a = 27, b = 23, c = 1, the system (16) shows chaotic

Conclusion

This paper is concerned with the partial switched modified function projective synchronization problem of a class of chaotic complex systems with unknown parameters. Specifically, some uncertain factors are taken into account in systems, such as some unknown system parameters. Based on adaptive control and Lyapunov stability theory, by constructing another suitable slave complex chaotic system, several novel adaptive laws and synchronization criteria are derived. These criteria are very useful

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (Grant Nos. 11402226 and 11302086), the Society Science Foundation from Ministry of Education of China (Grant No. 12YJAZH002), the Natural Science Foundation of Zhejiang Province (Grant No. LY15G010005) and the Foundation of Zhejiang Provincial Education Department (Grant No. Y201328316).

References (28)

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