Adaptive Switched Generalized Function Projective Synchronization between Two Hyperchaotic Systems with Unknown Parameters

Xiaobing Zhou <sup>1</sup>, \*, Lianglin Xiong <sup>2</sup> and Xiaomei Cai <sup>3</sup>

<sup>1</sup> School of Information Science and Engineering, Yunnan University, Kunming 650091, China


*Received: 17 October 2013; in revised form: 15 December 2013 / Accepted: 16 December 2013 / Published: 31 December 2013*

Abstract: In this paper, we investigate adaptive switched generalized function projective synchronization between two new different hyperchaotic systems with unknown parameters, which is an extension of the switched modified function projective synchronization scheme. Based on the Lyapunov stability theory, corresponding adaptive controllers with appropriate parameter update laws are constructed to achieve adaptive switched generalized function projective synchronization between two different hyperchaotic systems. A numerical simulation is conducted to illustrate the validity and feasibility of the proposed synchronization scheme.

Keywords: generalized function projective synchronization; switched state; hyperchaotic system; stability

### 1. Introduction

Hyperchaos, which was first introduced by Rössler [1], is usually characterized as a chaotic attractor with more than one positive Lyapunov exponent. The degree of chaos of a system can be measured by a generalization of the concept of entropy for state space dynamics [2,3]. It is a highly desired property to ensure security in a chaos encryption scheme that the larger the entropy, the larger the unpredictability of the system [4]. After the hyperchaotic Rössler system, many other hyperchaotic systems have been reported, including the hyperchaotic Lorenz system [5], hyperchaotic Chen system [6], hyperchaotic Lü system [7]. In [8], the positive topological entropy was calculated, which indicated that the system from two coupled Wien-bridge oscillators was hyperchaotic.

Since the concept of synchronizing two identical chaotic systems from different initial conditions was introduced by Pecora and Carroll in 1990 [9], synchronization in chaotic systems has been extensively investigated over the last two decades. Many synchronization schemes have been proposed, which include complete synchronization [10,11], lag synchronization [12], generalized synchronization [13], phase synchronization [14], anti-synchronization [15,16], partial synchronization [17,18], Q-S synchronization [19,20], projective synchronization [21–32], anticipating synchronization [33], inverse lag synchronization [34] and inverse π-lag synchronization [35,36].

Among the above-mentioned synchronization phenomena, projective synchronization has been investigated with increasing interest in recent years due to the fact that it can obtain faster communication with its proportional feature [23–26]. The concept of projective synchronization was first introduced by Mainieri and Rehacek in 1999 [27], in which the drive and response systems could be synchronized up to a constant scaling factor. Later on, Li [28] proposed a new synchronization scheme called modified projective synchronization (MPS), where the drive and response dynamical states synchronize up to a constant scaling matrix. Afterwards, Chen *et al*. [29] extended the modified projective synchronization and proposed function projective synchronization (FPS), where the drive and response dynamical states synchronize up to a scaling function matrix, but not a constant one. Recently, Du *et al*. [30] discussed a new type of synchronization phenomenon, modified function projective synchronization (MFPS), in which the drive and response systems could be synchronized up to a desired scaling function matrix. Many of these synchronization schemes have been applied to investigate chaotic or fractional chaotic systems [37–44]. More recently, Yu and Li [31] have proposed a new synchronization scheme by choosing a more generalized scaling function matrix, called generalized function projective synchronization (GFPS), which is an extension of all the aforementioned projective synchronization schemes. Lately, Sudheer and Sabir [32] reported switched modified function projective synchronization (SMFPS) in hyperchaotic Qi system using adaptive control method, in which a state variable of the drive system synchronize with a different state variable of the response system up to a desired scaling function matrix.

Inspired by the previous works, in this paper, we propose the switched generalized function projective synchronization (SGFPS) between two different hyperchaotic systems using adaptive control method by extending the GFPS and SMFPS schemes, in which a state variable of the drive system synchronizes with a different state variable of the response system up to a more generalized scaling function matrix. Due to the unpredictability of the switched states and scaling function matrix, this synchronization scheme can provide additional security in secure communication.

The rest of this paper is organized as follows. Section 2 gives a brief description of the SGFPS scheme and two new hyperchaotic systems. In Section 3, we propose appropriate adaptive controllers and parameter update laws for the adaptive switched generalized function projective synchronization of two different hyperchaotic systems. Section 4 presents a numerical example to illustrate the effectiveness of the proposed method. Finally, conclusions are given in Section 5.
