Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 762807, 18 pages
http://dx.doi.org/10.1155/2012/762807
Research Article

Inverse Projective Synchronization between Two Different Hyperchaotic Systems with Fractional Order

1State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400030, China
2School of Automation, Chongqing University, Chongqing 400030, China
3School of Mathematics, Anhui University, Hefei 230039, China

Received 14 July 2011; Accepted 5 October 2011

Academic Editor: Yuri Sotskov

Copyright © 2012 Yi Chai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper mainly investigates a novel inverse projective synchronization between two different fractional-order hyperchaotic systems, that is, the fractional-order hyperchaotic Lorenz system and the fractional-order hyperchaotic Chen system. By using the stability theory of fractional-order differential equations and Lyapunov equations for fractional-order systems, two kinds of suitable controllers for achieving inverse projective synchronization are designed, in which the generalized synchronization, antisynchronization, and projective synchronization of fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system are also successfully achieved, respectively. Finally, simulations are presented to demonstrate the validity and feasibility of the proposed method.

1. Introduction

The history of fractional calculus is more than three centuries old. It has been found that the behavior of many physical systems can be properly described by fractional-order systems, for example, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, viscoelastic systems, quantitative finance, and diffusion waves [13]. Compared with the classical integer-order models, fractional-order derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. Nowadays, fractional-order systems have attracted more and more people’s attention. Many authors began to investigate the chaotic dynamics of fractional-order nonlinear systems with the help of fractional calculus. For examples, Podlubny proposed a generalization of the PID controller for the control of the fractional-order systems [4], by using fractional calculus theory. Reference [5] investigated the chaos revealed by the pseudo inverse-based trajectory planning algorithms. With the introduction of fractional derivatives, it was proved that many fractional-order differential systems behave chaotically, such as, Chua’s circuit of order as low as 2.7 can produce a chaotic attractor [6], chaos could exist in the fractional-order Rössler equation with order as low as 2.4 [7]. In [8], the chaotic behavior and its control in the fractional-order Chen system is investigated. Sheu et al. studied the dynamics of the Newton-Leipnik system with fractional-order and found chaos existing in the fractional-order system with order less than 3 [9]. Lü numerically investigate the chaotic behaviors of the fractional-order Lü system, in which the lowest order to have chaos is 0.3 [10]. It was shown that fractional-order modified Duffing system of order less than 2.2 can still behave in a chaotic manner [11]. A hyperchaotic attractor is characterized as a chaotic attractor with more than one positive Lyapunov exponent which can increase the randomness and higher unpredictability of the corresponding system. It is believed that chaotic systems with higher-dimensional attractors have much wider applications. Recently, some authors studied the fractional-order hyperchaotic system by adding nonlinear terms in fractional-order systems. Wu and Liu found that hyperchaotic behavior does exist in the fractional-order hyperchaotic Chen system of order as low as 3.72 through numerical simulations [12]. Reference [13] discussed numerically the lowest orders for hyperchaos in fractional-order hyperchaotic Lorenz system which is 3.884. Hyperchaos can exist in the fractional-order Rössler hyperchaotic equation with order as low as 3.8 [14].

It has been well known that a chaotic system is a nonlinear deterministic system with unpredictable complexity. In the past decade, synchronization of chaotic systems has attracted considerable attention since the pioneering work of Pecora and Carroll [15]. The control and synchronization of chaotic systems have been developed and thoroughly studied over the past two decades for their applications in some engineering applications such as image processing, chemical and biological systems, information science, and secure communication [16, 17]. So far, many different types of synchronization have been introduced, such as complete synchronization [18], lag synchronization [19], generalized synchronization [20], impulsive synchronization [21], and phase synchronization [22]. In 2001, experimental investigations of antisynchronization of optically coupled semiconductor lasers with external cavities were presented by Wedekind and Parlitz [23]. Then Chil et al. [24] observed antisynchronization phenomena in coupled identical chaotic oscillators in 2003. Antisynchronization is a phenomenon that the state vectors of synchronized systems have the same amplitude but opposite signs as those of the driving system, which can be characterized by the vanishing of the sum of relevant variables. In the past ten years, various synchronization control laws are obtained by using different control methods, see [2527]. In 1996, projective synchronization phenomenon was first reported and discussed by González-Miranda [28]. In 1999, Mainieri and Rehacek first proposed the concept of projective synchronization which is characterized by the fact that the drive and response systems could be synchronized up to a scaling factor [29]. Then, Li [30] considered a new synchronization method called modified projective synchronization (MPS), where the phases were locked and the amplitudes of the two coupled systems synchronized up to a scaling factor. The scaling factor is a constant transformation between the synchronized variables of the master and slave systems. In recent years, projective synchronization of chaotic systems has attracted increasing attention; some sufficient conditions for projective synchronization have been obtained, see [3133].

Recently, synchronization of chaotic fractional differential systems becomes a challenging and interesting problem and starts to attract increasing attention due to its potential applications to secure communication and control processing. Most of the methods and results of chaos synchronization in the integer-order differential systems cannot be simply extended to the fractional-order systems because fractional differential dynamical systems are very complex. Some scholars did some research about projective synchronization of fractional systems. For instance, Shao et al. proposes a method to achieve projective synchronization of the fractional-order chaotic Rössler system [34]. Based on the idea of a nonlinear observer, Peng and Jiang proposed a new method and applied it to generalized projective synchronization for a class of fractional-order chaotic systems [35]; Wu and Wang discussed a new fractional-order system and investigated its projective synchronization by designing the suitable nonlinear controller [36]. The hybrid projective synchronization of different dimensional fractional-order chaotic systems with different dimensions is investigated in [37]. In the meantime, some scholars have obtained some results about antisynchronization of fractional-order chaotic system; for example, based on the idea of active control, a novel active pinning control strategy is presented to achieve the antisynchronization of new uncertain fractional-order unified chaotic systems in paper [38]. In fact, adopting higher-dimensional chaotic systems has been proposed for secure communication, and the presence of more than one Lyapunov exponent clearly improves the security of communication schemes by generating more complex dynamics. Interest in synchronization of the fractional-order hyperchaotic systems is again motivated by secure communication applications. References [12, 13, 39] all discussed the synchronization of the fractional-order hyperchaotic systems. The antisynchronization of two identical and two nonidentical hyperchaotic fractional-order systems is investigated via sliding mode controller in [40], but the results about the projective synchronization and inverse synchronization are relatively few.

However, to our best knowledge, there are few results about inverse projective synchronization of fractional-order hyperchaotic system. Wu and Lu investigated the generalized projective synchronization of fractional-order Chen hyperchaotic systems [12]. The issue of full state hybrid projective synchronization of fractional Chen-Lee hyperchaotic system is investigated in [41]. They all concentrated on projective synchronization of fractional hyperchaotic systems, but without considering inverse synchronization. References [38, 40] only discussed inverse synchronization without considering projective synchronization. Motivated by the above discussions, in this letter, we study the inverse projective synchronization between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system. Two novel control laws are derived for synchronizing fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system.

The remainder of this paper is organized as follows. In Section 2, fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system are described, and preliminary results are presented. In Section 3, two control schemes of inverse projective synchronization for fractional-order hyperchaotic systems of different structures are presented. In Section 4, numerical simulations are given to illustrate the effectiveness of the main results. Finally, conclusions are drawn in Section 5.

2. System Description and Preliminaries

Fractional calculus is a generalization of integration and differentiation to a noninteger-order integrodifferential operator 𝐷𝛼𝑡 defined by𝐷𝛼𝑡=𝑑𝛼𝑑𝑡𝛼,𝑅(𝛼)>0,1,𝑅(𝛼)=0,𝑡𝑎(𝑑𝜏)𝛼,𝑅(𝛼)<0.(2.1)

There are some definitions for fractional derivatives. The commonly used definitions are Grunwald-Letnikov, Riemann-Liouville, and Caputo definitions. The Riemann-Liouvill fractional derivatives, defined by𝑑𝛼𝑓(𝑡)𝑑𝑡𝛼=𝐽𝑛𝛼𝑑𝑛𝑓(𝑡)𝑑𝑡𝑛,(2.2) where 𝑛=[𝛼], that is, 𝑛 is the first integer which is not less than 𝛼, 𝐽𝛽 is the 𝛽-order Riemann-Liuvile integral operator which is described as follows:𝐽𝛽=1Γ(𝛽)𝑡0𝜑(𝜏)(𝑡𝜏)1𝛽𝑑𝜏,(2.3) where Γ() is the gamma function, 0<𝛽1.

In this paper, the following definition is used:𝐷𝛼𝑥(𝑡)=𝐽𝑛𝛼𝑥(𝑛)(𝑡),(2.4) where 𝑛=[𝛼], it is a common practice to call the operator 𝐷𝛼 the Caputo differential operator of order 𝛼 because it has apparently first been used for the solution of practical problems by Caputo.

To obtain our results, the following lemma is presented.

For a given autonomous linear system of fractional order𝐷𝛼𝑥=𝐴𝑥,(2.5) with 𝑥(0)=𝑥0, where 𝑥𝑅𝑛 is the state vector, one has the following.

Lemma 2.1. The system (2.5) is (i)asymptotically stable if and only if |arg(𝜆𝑖(𝐴))|>𝛼𝜋/2, 𝑖=1,2,3,, where arg(𝜆𝑖(𝐴)) denotes the argument of the eigenvalue 𝜆𝑖 of A. In this case, the component of the state decay towards 0 like 𝑡𝛼;(ii)stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy |arg(𝜆𝑖(𝐴))|=𝛼𝜋/2 have geometric multiplicity one.

Figure 1 illustrates the stability region of the fractional-order system.

762807.fig.001
Figure 1: Stability region of the fractional-order system.

The hyperchaotic Lorenz system with fractional order [13] is described as follows:𝐷𝑞𝑥1𝑥=𝑎2𝑥1+𝑥4,𝐷𝑞𝑥2=𝑐𝑥1𝑥2𝑥1𝑥3,𝐷𝑞𝑥3=𝑥1𝑥2𝑏𝑥3,𝐷𝑞𝑥4=𝑥2𝑥3+𝑑𝑥4,(2.6) where 𝑞 is the fractional order, 0<𝑞<1, 𝑥1, 𝑥2, 𝑥3, and 𝑥4 are state variables, 𝑎, 𝑏, 𝑐, and 𝑑 are parameters. When the parameters are chosen as 𝑎=10, 𝑏=8/3, 𝑐=28, 𝑑=1, and 𝑞=0.98, the values of Lyapunov exponents are 𝜆1=0.3362, 𝜆2=0.1568, 𝜆3=0, and 𝜆4=15.1724, the system (2.6) exhibits hyperchaotic behaviors as shown in Figure 2.

fig2
Figure 2: The projections of attractors of fractional-order hyperchaotic Lorenz system with order 𝑞=0.98.

In 2009, Wu and Lu investigated the hyperchaotic behaviors in fractional-order Chen hyperchaotic system, which is described by𝐷𝑞𝑦1𝑦=𝛼2𝑦1+𝑦4,𝐷𝑞𝑦2=𝛽𝑦1𝑦1𝑦3+𝜃𝑦2,𝐷𝑞𝑦3=𝑦1𝑦2𝛾𝑦3,𝐷𝑞𝑦4=𝑦2𝑦3+𝜂𝑦4.(2.7) System (2.7) admits a hyperchaotic attractor for 𝛼=35, 𝛽=7, 𝜃=12, 𝛾=3, and 𝜂=0.5. Figure 3 displays the hyperchaotic attractor of fractional-order Chen system with fractional-order 𝑞=0.98.

fig3
Figure 3: The projections of attractors of fractional-order hyperchaotic Chen system with order 𝑞=0.98.

3. Inverse Projective Synchronization

In the section, we will discuss inverse projective synchronization behavior between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system. We assume that fractional-order hyperchaotic Lorenz system is the drive system and the response system is fractional-order hyperchaotic Chen system, which are governed by systems (3.1) and (3.2), respectively,𝐷𝑞𝑥1𝑥=𝑎2𝑥1+𝑥4,𝐷𝑞𝑥2=𝑐𝑥1𝑥2𝑥1𝑥3,𝐷𝑞𝑥3=𝑥1𝑥2𝑏𝑥3,𝐷𝑞𝑥4=𝑥2𝑥3+𝑑𝑥4,𝐷(3.1)𝑞𝑦1𝑦=𝛼2𝑦1+𝑦4+𝑢1,𝐷𝑞𝑦2=𝛽𝑦1𝑦1𝑦3+𝜃𝑦2+𝑢2,𝐷𝑞𝑦3=𝑦1𝑦2𝛾𝑦3+𝑢3,𝐷𝑞𝑦4=𝑦2𝑦3+𝜂𝑦4+𝑢4,(3.2) where 𝑥𝑖 and 𝑦𝑖 stand for state variables of the master system and the slave one, respectively. 𝑎, 𝑏, 𝑐, 𝑑, 𝛼, 𝛽, 𝜃, 𝛾, and 𝜂 are parameters. 𝑢1, 𝑢2, 𝑢3, and 𝑢4 are four nonlinear controllers will be designed later.

Definition 3.1. System (3.1) and (3.2) are inverse projective synchronization, if there exist a scaling matrix 𝜎 such that lim𝑡𝑒(𝑡)=lim𝑡𝑦+𝜎𝑥=0,(3.3) where is the Euclidean norm and 𝜎=diag(𝜎1,𝜎2,𝜎3,𝜎4).
It follows from (3.1)–(3.3) that we have the following error dynamical system: 𝐷𝑞𝑒1=𝜎𝑦2𝜎𝑦1+𝑦4+𝑎𝜎1𝑥2𝑎𝜎1𝑥1+𝜎1𝑥4+𝑢1,𝐷𝑞𝑒2=𝛽𝑦1𝑦1𝑦3+𝜃𝑦2+𝑐𝜎2𝑥1𝜎2𝑥2𝜎2𝑥1𝑥3+𝑢2,𝐷𝑞𝑒3=𝑦1𝑦2𝛾𝑦3+𝜎3𝑥1𝑥2𝑏𝜎3𝑥3+𝑢3,𝐷𝑞𝑒4=𝑦2𝑦3+𝜂𝑦4𝜎4𝑥2𝑥3+𝑑𝜎4𝑥4+𝑢4,(3.4) where 𝑒1=𝑦1+𝜎1𝑥1, 𝑒2=𝑦2+𝜎2𝑥2, 𝑒3=𝑦3+𝜎3𝑥3, 𝑒4=𝑦4+𝜎4𝑥4.
Our aim is to find suitable control laws 𝑢𝑖(𝑖=1,2,3,4) for stabilizing the error variables of the system at the origin. For this goal, we propose the following control laws for the error dynamical system 𝑢1=(𝑎𝛼)𝜎1𝑥1+𝛼𝜎2𝑎𝜎1𝑥2+𝜎4𝜎1𝑥4𝛽+𝜎3𝑥3𝑒2+𝜎2𝑥2𝑒3,𝑢2=𝛽𝜎1𝑐𝜎2𝑥1+(𝜃+1)𝜎2𝑥2+𝜎1𝜎3+𝜎2𝑥1𝑥3+𝑒3𝑒𝛼12𝜃𝑒2+𝜎3𝑥3𝑒4,𝑢3𝜎=1𝜎2+𝜎3𝑥1𝑥2(𝛾𝑏)𝜎3𝑥3𝑒1𝑒2+𝜎2𝑥2𝑒4,𝑢4=𝜎4𝜎2𝜎3𝑥2𝑥3+(𝜂𝑑)𝜎4𝑥4𝑒1𝑒2𝑒32𝜂𝑒4.(3.5)

Theorem 3.2. For a given constant scaling matrix 𝜎=diag(𝜎1,𝜎2,𝜎3,𝜎4) and any initial conditions, if parameters 𝛼, 𝜃, 𝛾, and 𝜂>0, then the inverse projective synchronization between fractional-order hyperchaotic Lorenz system (3.1) and Chen system (3.2) will occur under control law (3.5).

Proof. Substituting (3.5) into (3.4) leads to the following error system: 𝐷𝑞𝑒1=𝛼𝑒1+𝛼𝛽𝜎3𝑥3𝑒2+𝜎2𝑥2𝑒3+𝑒4,𝐷𝑞𝑒2=𝛽+𝜎3𝑥3𝑒𝛼1𝜃𝑒2+𝜎1𝑥1𝑒3+𝜎3𝑥3𝑒4,𝐷𝑞𝑒3=𝜎2𝑥2𝑒1𝜎1𝑥1𝑒2𝛾𝑒3+𝜎2𝑥2𝑒4,𝐷𝑞𝑒4=𝑒1𝜎3𝑥3𝑒2𝜎2𝑥2𝑒3𝜂𝑒4.(3.6)
Error system (3.6) can be rewritten in the following matrix form: 𝐷𝑞𝑒1,𝐷𝑞𝑒2,𝐷𝑞𝑒3,𝐷𝑞𝑒4𝑇𝑒=𝐴1,𝑒2,𝑒3,𝑒4𝑇,(3.7) where 𝐴=𝛼𝛼𝛽𝜎3𝑥3𝜎2𝑥21𝛽+𝜎3𝑥3𝛼𝜃𝜎1𝑥1𝜎3𝑥3𝜎2𝑥2𝜎1𝑥1𝛾𝜎2𝑥21𝜎3𝑥3𝜎2𝑥2𝜂.(3.8)
Suppose 𝜆 is one of the eigenvalues of matrix 𝐴 and the corresponding eigenvector is 𝜀=(𝜀1,𝜀2,𝜀3,𝜀4)𝑇, that is, 𝐴𝜀=𝜆𝜀.(3.9) Taking conjugate transpose (𝐻) on both sides of (3.9), one obtains (𝐴𝜀)𝑇=𝜆𝜀𝐻.(3.10) Taking (3.9) multiplied left by (1/2)𝜀𝐻 plus (3.10) multiplied right by (1/2)𝜀, we derive that 𝜀𝐻121𝐴+2𝐴𝐻1𝜀=2𝜆+𝜆𝜀𝐻𝜀.(3.11) From (3.11), we have 12𝜆+𝜆=𝜀𝐻(1/2)𝐴+(1/2)𝐴𝐻𝜀𝜀𝐻𝜀.(3.12) By substituting 𝐴 into (3.12), we can obtain 12𝜆+𝜆=1𝜀𝐻𝜀𝜀𝐻𝛼0000𝜃0000𝛾0000𝜂𝜀.(3.13) Since parameters 𝛼, 𝜃, 𝛾, and 𝜂>0,𝜆+𝜆<0, that is, any eigenvalue of matrix 𝐴 satisfies ||||𝜋arg(𝜆)2>𝑞𝜋2(𝑞<1).(3.14) According to the stability theory of fractional-order systems (Lemma 2.1), error system (3.6) is asymptotically stable, which implies that the inverse projective synchronization between systems (3.1) and (3.2) is achieved.

It follows from the control law (3.5) that the control law concerns the derive state variables and error variables, but is not related to the response ones. Now, we will propose another control scheme, which is associated with the response state variables and error variables.

Theorem 3.3. For a given constant scaling matrix 𝜎=diag(𝜎1,𝜎2,𝜎3,𝜎4) and any initial conditions, if parameters 𝑎,𝑏>0 and 𝑑<0, then fractional-order hyperchaotic Lorenz system (3.1) and Chen system (3.2) are inverse projective synchronized by the following control law: 𝑢1=(𝛼𝑎)𝑦1𝜎𝛼𝑎1𝜎2𝑦2𝜎11𝜎4𝑦4𝑐𝜎2𝜎1+𝜎2𝜎1𝜎3𝑦3𝑒2+𝜎3𝜎1𝜎2𝑦2𝑒3,𝑢2=𝑐𝜎2𝜎1𝑦𝛽1(𝜃+1)𝑦2+𝜎1+2𝜎1𝜎3𝑦1𝑦3+𝜎2𝜎1𝜎3𝑒1𝑒3𝜎𝑎1𝜎2𝑒1+𝜎3𝜎1𝜎2𝑦1𝑒3𝜎4𝜎2𝜎3𝑦3𝑒4,𝑢3=(𝑟𝑏)𝑦3𝜎1+3𝜎1𝜎2𝑦1𝑦2𝜎3𝜎1𝜎2𝑒1𝑒2𝜎2𝜎1𝜎3𝑦1𝑒2𝜎4𝜎2𝜎3𝑦2𝑒4,𝑢4=(𝑑𝜂)𝑦4+𝜎4𝜎2𝜎3𝑦12𝑦3𝜎1𝜎4𝑒1+𝜎4𝜎2𝜎3𝑒2𝑒3.(3.15)

Proof. Combining (3.15) with (3.4), the error dynamical system can be described as 𝐷𝑞𝑒1=𝑎𝑒1+𝑎𝜎1𝜎2𝜎𝑐2𝜎1𝜎2𝜎1𝜎3𝑦3𝑒2+𝜎3𝜎1𝜎2𝑦2𝑒3+𝜎1𝜎4𝑒4,𝐷𝑞𝑒2=𝑐𝜎2𝜎1𝜎𝑎1𝜎2+𝜎2𝜎1𝜎3𝑦3𝑒1𝑒2+𝜎2𝜎1𝜎3+𝜎3𝜎1𝜎2𝑦1𝑒3𝜎4𝜎2𝜎3𝑦3𝑒4,𝐷𝑞𝑒3𝜎=3𝜎1𝜎2𝑦2𝑒1𝜎2𝜎1𝜎3+𝜎3𝜎1𝜎2𝑦1𝑒3𝑏𝑒3𝜎4𝜎2𝜎3𝑦2𝑒4,𝐷𝑞𝑒4𝜎=1𝜎4𝑒1+𝜎4𝜎2𝜎3𝑦3𝑒2+𝜎4𝜎2𝜎3𝑦2𝑒3+𝑑𝑒4.(3.16) System (3.16) can be rewritten in the following matrix form: 𝐷𝑞𝑒1,𝐷𝑞𝑒2,𝐷𝑞𝑒3,𝐷𝑞𝑒4𝑇𝑒=𝐴1,𝑒2,𝑒3,𝑒4𝑇,(3.17) where 𝜎𝐴=𝑎𝑎1𝜎2𝜎𝑐2𝜎1𝜎2𝜎1𝜎3𝑦3𝜎3𝜎1𝜎2𝑦2𝜎1𝜎4𝑐𝜎2𝜎1𝜎𝑎1𝜎2+𝜎2𝜎1𝜎3𝑦3𝜎12𝜎1𝜎3+𝜎3𝜎1𝜎2𝑦1𝜎4𝜎2𝜎3𝑦3𝜎3𝜎1𝜎2𝑦2𝜎2𝜎1𝜎3+𝜎3𝜎1𝜎2𝑦1𝜎𝑏4𝜎2𝜎3𝑦2𝜎1𝜎4𝜎4𝜎2𝜎3𝑦3𝜎4𝜎2𝜎3𝑦2𝑑.(3.18)
The following proof process is similar to that of Theorem 3.2, so is omitted here. Finally, we have 12𝜆+𝜆=1𝜀𝐻𝜀𝜀𝐻𝑎000010000𝑏0000𝑑𝜀.(3.19) Parameters 𝑎,𝑏>0 and 𝑑<0, therefore 𝜆+𝜆<0, which means any eigenvalue of matrix 𝐴 satisfies ||||𝜋arg(𝜆)2>𝑞𝜋2(𝑞<1).(3.20) Based on Lemma 2.1, error system (3.16) is asymptotically stable, which implies the inverse projective synchronization between the drive system (3.1) and the response system (3.2) is achieved under the nonlinear controller (3.15).
If we set the scaling matrix 𝜎=diag(𝜎,𝜎,𝜎,𝜎)(𝜎1=𝜎2=𝜎3=𝜎4=𝜎), then inverse projective synchronization will be degraded to inverse generalized synchronization. So we have the following Corollaries 3.4 and 3.5 to realize inverse generalized synchronization between the drive system (3.1) and the response system (3.2).

Corollary 3.4. For a given constant scaling matrix 𝜎=diag(𝜎,𝜎,𝜎,𝜎) and any initial conditions, if parameters 𝛼, 𝜃, 𝛾 and 𝜂>0, then the inverse generalized synchronization between fractional-order hyperchaotic Lorenz system (3.1) and Chen system (3.2) will occur under the following control law: 𝑢1=(𝑎𝛼)𝜎𝑥1+(𝛼𝑎)𝜎𝑥2𝛽+𝜎𝑥3𝑒2+𝜎𝑥2𝑒3,𝑢2=(𝛽𝑐)𝜎𝑥1+(𝜃+1)𝜎𝑥2+𝜎2𝑥+𝜎1𝑥3+𝑒3𝑒𝛼12𝜃𝑒2𝜎𝑥3𝑒4,𝑢3𝜎=2𝑥+𝜎1𝑥2(𝛾𝑏)𝜎𝑥3𝑒1𝑒2+𝜎𝑥2𝑒4,𝑢4=𝜎𝜎2𝑥2𝑥3+(𝜂𝑑)𝜎𝑥4𝑒1𝑒2𝑒32𝜂𝑒4.(3.21)

Corollary 3.5. For a given constant scaling matrix 𝜎=diag(𝜎,𝜎,𝜎,𝜎) and any initial conditions, if parameters 𝑎,𝑏>0 and 𝑑<0, then inverse generalized synchronization between fractional-order hyperchaotic Lorenz system (3.1) and Chen system (3.2) are achieved under the following control law: 𝑢1=(𝛼𝑎)𝑦1(𝛼𝑎)𝑦21𝑐+𝜎𝑦3𝑒2+1𝜎𝑦2𝑒3,𝑢2=(𝑐𝛽)𝑦1(𝜃+1)𝑦2+11+𝜎𝑦1𝑦3+1𝜎𝑒1𝑒3𝑎𝑒1+1𝜎𝑦1𝑒31𝜎𝑦3𝑒4,𝑢3=(𝑟𝑏)𝑦311+𝜎𝑦1𝑦21𝜎𝑒1𝑒21𝜎𝑦1𝑒21𝜎𝑦2𝑒4,𝑢4=(𝑑𝜂)𝑦4+1𝜎𝑦12𝑦3𝑒1+1𝜎𝑒2𝑒3.(3.22)

Further, we take constant scaling matrix 𝜎 as 𝜎=diag(1,1,1,1)(𝜎1=𝜎2=𝜎3=𝜎4=1), antisynchronization between the derive system (3.1) and the response system (3.2) will occur by the following Corollaries 3.6 and 3.7.

Corollary 3.6. For any initial conditions, if parameters 𝛼, 𝜃, 𝛾, and 𝜂>0, then antisynchronization between fractional-order hyperchaotic Lorenz system (3.1) and Chen system (3.2) will occur under the following control law: 𝑢1=(𝑎𝛼)𝑥1+(𝛼𝑎)𝑥2𝛽+𝑥3𝑒2+𝑥2𝑒3,𝑢2=(𝛽𝑐)𝑥1+(𝜃+1)𝑥2+2𝑥1𝑥3+𝑒3𝑒𝛼12𝜃𝑒2+𝑥3𝑒4,𝑢3=2𝑥1𝑥2(𝛾𝑏)𝑥3𝑒1𝑒2+𝑥2𝑒4,𝑢4=(𝜂𝑑)𝑥4𝑒1𝑒2𝑒32𝜂𝑒4.(3.23)

Corollary 3.7. For any initial conditions, if the parameters 𝑎,𝑏>0 and 𝑑<0, then fractional-order hyperchaotic Lorenz system (3.1) and Chen system (3.2) are inverse generalized synchronized under the following control law: 𝑢1=(𝛼𝑎)𝑦1(𝛼𝑎)𝑦2𝑐+𝑦3𝑒2+𝑦2𝑒3,𝑢2=(𝑐𝛽)𝑦1(𝜃+1)𝑦2+2𝑦1𝑦3+𝑒1𝑒3𝑎𝑒1+𝑦1𝑒3𝑦3𝑒4,𝑢3=(𝑟𝑏)𝑦32𝑦1𝑦2𝑒1𝑒2𝑦1𝑒2𝑦2𝑒4,𝑢4=(𝑑𝜂)𝑦4𝑒1+𝑒2𝑒3.(3.24)

Similarly, scaling factor matrices are selected as 𝜎=diag(𝜎,𝜎,𝜎,𝜎) and 𝜎=diag(1,1,1,1); it is easy to obtain the corresponding corollaries to achieve generalized synchronization and complete synchronization, respectively. We omit here.

4. Numerical Simulations

In this section, to verify and demonstrate the effectiveness of the proposed controller laws, we discuss the simulation results of inverse projective synchronization between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system. The simulation is performed using Matlab software.

For this numerical simulation, the parameters are chosen to be 𝛼=35, 𝛽=7, 𝜃=12, 𝛾=3, 𝜂=0.5, 𝑎=10, 𝑏=8/3, 𝑐=28, and 𝑑=1, which all satisfy the conditions 𝛼,𝜃,𝛾,𝜂>0 and 𝑎,𝑏>0, 𝑑<0 in Theorems 3.2 and 3.3, respectively. The initial values of the drive and response systems are (𝑥1(0),𝑥2(0),𝑥3(0),𝑥4(0))=(0.8,0.2,0.3,0.1) and (𝑦1(0),𝑦2(0),𝑦3(0),𝑦4(0))=(1.2,1.2,0.70.3), respectively. Without loss of generality, the scaling matrix 𝜎 is chosen as diag(2,3,4,6) and 𝑞=0.98. Based on Theorems 3.2 and 3.3, under the control laws (3.5) and (3.15), the state variables during the synchronizing process between the fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system are shown in Figures 4 and 5, respectively. Figures 6 and 7 show the signals after synchronization under the control laws (3.5) and (3.15), respectively. It is obvious that the state vectors of the systems are synchronized in the same direction all the time from the simulations. Define 𝑒(𝑡)=𝑒1(𝑡)2+𝑒2(𝑡)2+𝑒3(𝑡)2+𝑒4(𝑡)2. Figures 8(a) and 8(b) display the synchronization errors between systems (3.1) and (3.2) under the control laws (3.5) and (3.15), respectively.

fig4
Figure 4: The state synchronization trajectories of the master system (3.1) and slave system (3.2) under control law (3.5).
fig5
Figure 5: The state synchronization trajectories of the master system (3.1) and slave system (3.2) under control law (3.15).
fig6
Figure 6: Signals after synchronization under control law (3.5).
fig7
Figure 7: Signals after synchronization under control law (3.15).
fig8
Figure 8: Error states of the drive system (3.1) and the response system (control law (3.5) : (a); control law (3.15) : (b)).

5. Conclusions

In this paper, based on the stability theory of fractional-order differential equations and Lyapunov equations for fractional-order systems, two criteria are provided to ensure the inverse projective synchronization between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system. There are few papers about the inverse projective synchronization of fractional-order hyperchaotic system, which often exists in real systems. The feasibility and effectiveness of proposed schemes have been validated by computer simulation. It should be noted that complete synchronization, antisynchronization, and general projective synchronization are the special cases of inverse projective synchronization. Therefore, the results of this paper are more applicable and representative. Meanwhile, results could be extended to other fractional-order hyperchaotic system, such as fractional-order hyperchaotic Lü system, and fractional-order hyperchaotic Rössler system.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 60974090), the Fundamental Research Funds for the Central Universities (no. CDJXS11172237; no. CDJRC10170005; no. CDJZR 11170005), the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20093401120001; no. 102063720090013), the Natural Science Foundation of Anhui Province (no. 11040606M12), and the Natural Science Foundation of Anhui Education Bureau (no. KJ2010A035).

References

  1. R. C. Koeller, “Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics,” Acta Mechanica, vol. 58, no. 3-4, pp. 251–264, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. O. Heaviside, Electromagnetic theory, Chelsea, New York, NY, USA, 1971.
  4. I. Podlubny, “Fractional-order systems and PIλDμ-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999. View at Publisher · View at Google Scholar
  5. F. B. M. Duarte and J. A. T. Machado, “Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 315–342, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,” Physica A, vol. 360, no. 2, pp. 171–185, 2006. View at Publisher · View at Google Scholar
  7. C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons & Fractals, vol. 22, no. 2, pp. 443–450, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. L.-J. Sheu, H.-K. Chen, J.-H. Chen et al., “Chaos in the Newton-Leipnik system with fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 98–103, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1–4, pp. 61–72, 2005. View at Publisher · View at Google Scholar · View at Scopus
  11. X. Gao and J. B. Yu, “Chaos in the fractional order periodically forced complex Duffing's oscillators,” Chaos, Solitons & Fractals, vol. 24, no. 4, pp. 1097–1104, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. X. Wu and Y. Lu, “Generalized projective synchronization of the fractional-order Chen hyperchaotic system,” Nonlinear Dynamics, vol. 57, no. 1-2, pp. 25–35, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. T. Wang and X. Wang, “Generalized synchronization of fractional order hyperchaotic lorenz system,” Modern Physics Letters B, vol. 23, no. 17, pp. 2167–2178, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. Y. G. Yu and H.-X. Li, “The synchronization of fractional-order Rössler hyperchaotic systems,” Physica A, vol. 387, no. 5-6, pp. 1393–1403, 2008. View at Publisher · View at Google Scholar
  15. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. G. R. Chen and X. Dong, From Chaos to Order—Perspectives, Methodologies and Applications, World Scientific, Singapore, 1998.
  17. S. K. Dana, P. K. Roy, and J. Kurths, Complex Dynamics in Physiological Systems, from Heart to Brain, Springer, New York, NY, USA, 2009.
  18. C. Li and X. F. Liao, “Complete and lag synchronization of hyperchaotic systems using small impulses,” Chaos, Solitons & Fractals, vol. 22, no. 4, pp. 857–867, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. H. Li, “Projective lag synchronization in chaotic systems,” Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2630–2634, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. X. D. Zhang, P. D. Zhao, and A. H. Li, “Construction of a new fractional chaotic system and generalized synchronization,” Communications in Theoretical Physics, vol. 53, no. 6, pp. 1105–1110, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. H. Jiang and Q. Bi, “Impulsive synchronization of networked nonlinear dynamical systems,” Physics Letters A, vol. 374, no. 27, pp. 2723–2729, 2010. View at Publisher · View at Google Scholar
  22. X. F. Lang, Q. S. Lu, and K. Jurge, “Phase synchronization in noise-driven bursting neurons,” Physical Review E, vol. 82, no. 2, pp. 1–6, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. I. Wedekind and U. Parlitz, “Experimental observation of synchronization and anti-synchronization of chaotic low-frequency-fluctuations in external cavity semiconductor lasers,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 11, no. 4, pp. 1141–1147, 2001. View at Publisher · View at Google Scholar · View at Scopus
  24. M. K. Chil, S. H. Rim, W. H. Kye, J. W. Ryu, and Y. J. Park, “Anti-synchronization of chaotic oscillators,” Physics Letters A, vol. 320, no. 1, pp. 39–46, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. Hu, S. Chen, and L. Chen, “Adaptive control for anti-synchronization of Chua's chaotic system,” Physics Letters A, vol. 339, no. 6, pp. 455–460, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. G. H. Li and S. P. Zhou, “Anti-synchronization in different chaotic systems,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 516–520, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. L. Pan, W. Zhou, J. Fang, and D. Li, “A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 417–425, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. J. M. González-Miranda, “Synchronization of symmetric chaotic systems,” Physical Review E, vol. 53, no. 6, pp. 5656–5669, 1996. View at Google Scholar · View at Scopus
  29. R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042–3045, 1999. View at Publisher · View at Google Scholar · View at Scopus
  30. G.-H. Li, “Modified projective synchronization of chaotic system,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1786–1790, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. Q. Jia, “Projective synchronization of a new hyperchaotic Lorenz system,” Physics Letters A, vol. 370, no. 1, pp. 40–45, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. M. F. Hu and Z. Y. Xu, “Adaptive feedback controller for projective synchronization,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 1253–1260, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. D. Ghosh, “Projective synchronization in multiple modulated time-delayed systems with adaptive scaling factor,” Nonlinear Dynamics, vol. 62, no. 4, pp. 751–759, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. S. Shao, “Controlling general projective synchronization of fractional order Rossler systems,” Chaos, Solitons & Fractals, vol. 39, no. 4, pp. 1572–1577, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. G. Peng and Y. Jiang, “Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal,” Physics Letters A, vol. 372, no. 22, pp. 3963–3970, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. X. Wu and H. Wang, “A new chaotic system with fractional order and its projective synchronization,” Nonlinear Dynamics, vol. 61, no. 3, pp. 407–417, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. S. Wang, Y. Yu, and M. Diao, “Hybrid projective synchronization of chaotic fractional order systems with different dimensions,” Physica A, vol. 389, no. 21, pp. 4981–4988, 2010. View at Publisher · View at Google Scholar · View at Scopus
  38. L. Pan, W. Zhou, J. Fang, and D. Li, “Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3754–3760, 2010. View at Publisher · View at Google Scholar
  39. H. M. Deng, T. H. Li, Q. Wang, and H. Li, “A fractional-order hyperchaotic system and its synchronization,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 962–969, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  40. N. Sun, H. G. Zhang, Z. L. Wang, and M. Dong, “Anti-synchronization of fractional-order hyperchaotic systems via sliding mode controller,” in Proceedings of the 29th Chinese Control Conference (CCC '10), pp. 755–759, July 2010. View at Scopus
  41. H. W. Songa, X. R. Lin, H. Yang, R. Zhen, Y. Chie, and X. Feng, “Full state hybrid projective synchronization in fractional Chen-Lee hyperchaotic system,” in Proceedings of the International Conference on Communications, Circuits and Systems (ICCCAS '10), pp. 777–780, July 2010. View at Publisher · View at Google Scholar · View at Scopus