Adaptive Projective Synchronization between Two Different Fractional-Order Chaotic Systems with Fully Unknown Parameters

Abstract

Projective synchronization between two different fractional-order chaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the stability theory of fractional-order differential equations, a suitable and effective adaptive control law and a parameter update rule for unknown parameters are designed, such that projective synchronization between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown parameters is achieved. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed method.

1. Introduction

In the past three decades, the control and synchronization of chaotic systems have gained a lot of attention for their potential applications in some engineering application, such as image processing, chemical and biological systems and information science, and in particular in secure communication [1–5]. Since Pecora and Carroll introduced a method to realize complete synchronize between two identical chaotic with different initial conditions, several types of synchronization have been proposed, such as complete synchronization [6], lag synchronization [7], phase synchronization [8], antisynchronization [9], partial synchronization [10], generalization synchronization [11], impulsive synchronization [12], and so on. Among all kinds of chaos synchronization, projective synchronization phenomenon was first reported and discussed by González-Miranda [13]. In 1999, Mainieri and Rehacek first proposed the concept of projective synchronization in which drive and response systems could be synchronized up to a scaling factor [14]. It has been proven that projective synchronization is not in the category of generalized synchronization because the response system of projective synchronization is not asymptotically stable [15]. Recently, some scholars extended the concept of projective synchronization and proposed modified projective synchronization [16], function projective synchronization [17], and modified function projective synchronization [18]. Because of the proportionality between its synchronized dynamical states, the feature can be used to M-nary digital communication for achieving fast communication [19]. Projective synchronization has attracted increasing attention during recent years, and some necessary conditions for projective synchronization have been obtained, see references [20–23].

In the investigation of dynamical systems, many physical models could be properly described by the fractional order systems, see, for example, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, viscoelastic systems, quantitative finance and diffusion wave [24–26]. 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. With the introduction of fractional derivatives, it was proved that many fractional-order differential systems behave chaotically with order less than 3, such as fractional-order Chua’s circuit system [27], fractional-order Rössler system [28], fractional-order Chen system [29], fractional-order Lü system [30] and fractional-order modified Duffing system [31]. In recent years, the control and synchronization of fractional-order chaotic systems have caught much attention, see references [32–34]. However, there exists a substantial difference between fractional-order differential systems and integer-order differential ones. Most of the properties, conclusions, and methods for dealing with the integer-order systems cannot be simply extended to the case of the fractional-order ones, such as Lyapunovs direct method. Therefore, results bout the synchronization of fractional-order chaotic systems are much less rich than those of integer-order systems. It is still a challenging problem.

On the other hand, at present, most of the theoretical results concerning synchronization of the fractional-order systems mainly focus on systems whose models are identical or similar, and parameters are exactly known in advance. But in many practical situation, the parameters of many systems cannot be known entirely, the synchronization will be greatly affected by these uncertainties. How to effectively synchronize two fractional-order chaotic systems with unknown parameters is an important problem for the theoretical and practical applications. As we know, we usually take adaptive control methods to deal with such problems. However, to our best knowledge, much of the literature on projective synchronization of fractional-order chaotic systems all focused on the case of parameter determination, for instances [35–38]. There are few results about synchronization of different fractional-order chaotic systems with fully unknown parameters [39], not to mention projective synchronization. Motivated by the above discussions, in this paper, we study projective synchronization between fractional-order chaotic Chen system and fractional-order chaotic Lü system with fully unknown parameters in the derive system and the response one, which is more general and practical than the one discussed in paper [39], where only master systems are assumed to be uncertainty and a novel parameter identification and an adaptive control law are derived based on the stability theory of fractional-order differential equations. Corresponding theoretical analysis and numerical simulations are presented to verify the validity and feasibility of the proposed method.

The remainder of this paper is organized as follows. In Section 2, preliminary results are presented and fractional-order chaotic Chen system and fractional-order chaotic Lü system are described. In Section 3, adaptive synchronization scheme with a parameter update law is presented. In Section 4, numerical simulations are given to illustrate the effectiveness of the main results. Finally, conclusions are drawn in Section 5.

2. Preliminaries and System Description

The fractional-order integrodifferential operator is the extending concept of integer-order integrodifferential operator, which can be denoted by a general fundamental operator as follows:_𝑎𝐷^𝑞_𝑡=𝑑^𝑞𝑑𝑡^𝑞𝑞𝑞,𝑅>0,1,𝑅=0,_𝑡𝑎(𝑑𝜏)^−𝑞,𝑅𝑞<0, (2.1) where 𝑞 is the fractional-order, which can be a complex number and 𝑎 and 𝑡 are the limits of the operation. There are some definitions for fractional derivatives. The commonly used definitions are Grunwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo (C) definitions.

The Grunwald-Letnikov (GL) derivative with fractional-order 𝑞 is given by_𝑎^GL𝐷^𝑞_𝑡𝑓(𝑡)=lim_ℎ→0𝑓_ℎ^(𝑞)(𝑡)=lim_ℎ→0ℎ^{−𝑞[𝑡−𝑞/ℎ]}_𝑖=0(−1)^𝑖𝑞_𝑖𝑓(𝑡−𝑖ℎ), (2.2) where [⋅] means the integer part.

The Riemann-Liouvill (RL) fractional derivatives are defined by𝑎^𝑅𝐷^𝑞_𝑡𝑓(𝑡)=𝑑^𝑛𝑑𝑡^𝑛1Γ𝑛−𝑞^𝑡_𝑎𝑓(𝜏)(𝑡−𝜏)^{(𝑞−𝑛+1)}𝑑𝜏,𝑛−1<𝑞<𝑛, (2.3) where Γ(⋅) is the gamma function and Γ(𝜏)=∫^∞₀𝑡^𝜏−1𝑒^−𝑡𝑑𝑡.

The Caputo (C) fractional derivative is defined as follows:1Γ𝑛−𝑞^𝑡_𝑎(𝑡−𝜏)^{(𝑛−𝑞−1)}1𝑑𝜏⟶Γ𝑛−𝑞^𝑡_𝑎(𝑡−𝜏)^{(𝑛−𝑞−1)}𝑓^(𝑛)(𝜏)𝑑𝜏. (2.4)

It should be noted that the advantage of Caputo approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer-order differential, which have well understood physical meanings. Comparing these two formulas, one easily arrives at a fact that Caputo derivative of a constant is equal to zero, which is not the case for the Riemann-Liouville derivative. Therefore, in the rest of this paper, the notation 𝐷^𝑞_∗ is chosen as the Caputo fractional derivative operator ^𝐶_𝑎𝐷. For more about knowledge of fractional definitions and fractional calculus, see references [40, 41].

In the following section, an improved predictor-corrector algorithm [42] for fractional-order differential equations is presented in brief, which will be used in numerical simulation section. In comparison with the classical one-step Adams-Bashforth-Moulton algorithm, the numerical approximation of the improved algorithm is more accurate and the computational cost is more lower. Predictor-corrector algorithm for fractional differential equations is the generalization of Adams-Bashforth-Moultono one; the similarities, differences, merits, and demerits between them are shown in [42, 43].

Consider the following differential equation:𝐷^𝑞_∗𝑥(𝑡)()𝑥=𝑓𝑡,𝑥,0≤𝑡≤𝑇,_(𝑘)(0)=𝑥₀^(𝑘),𝑘=0,1,2,…,𝑞−1, (2.5) equivalent to the Volterra integral equation [44]𝑥(𝑡)=^[𝑞]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘+1𝑘!Γ𝑞^𝑡₀𝑓(𝜏,𝑥)(𝑡−𝜏)^1−𝑞𝑑𝜏. (2.6) Set ℎ=𝑇/𝑁,𝑡_𝑛=𝑛ℎ(𝑛=0,1,2,…,𝑁). Then (2.6) can be discretized as follows:𝑥_ℎ𝑡_𝑛+1=^[𝑞]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘_𝑛+1+ℎ𝑘!Γ^𝑞𝑓𝑡𝑞+2^𝑛+1,𝑥^𝜌_ℎ𝑡_𝑛+1+ℎ^𝑞Γ𝑞+2^𝑛_𝑗=0𝑎_𝑗,𝑛+1𝑓𝑡_𝑗,𝑥_ℎ𝑡_𝑗, (2.7) where𝑎_𝑗,𝑛+1=𝑛^𝑞+1−𝑛−𝑞(𝑛+1)^𝑞,𝑗=0,𝑛−𝑗+2^𝑞+1+𝑛−𝑗^𝑞+1−2𝑛−𝑗+1^𝑞+1𝑥,1≤𝑗≤𝑛,1,𝑗=𝑛+1,_𝜌ℎ𝑡_𝑛+1=^[𝑞]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘_𝑛+1+1𝑘!Γ𝑞^𝑛_𝑗=0𝑏_𝑗,𝑛+1𝑓𝑡_𝑗,𝑥_ℎ𝑡_𝑗, (2.8) in which 𝑏_𝑗,𝑛+1=(ℎ^𝑞/𝑞)((𝑛−𝑗+1)^𝑞−(𝑛−𝑗)^𝑞).

The error estimate 𝑒 is Max|𝑥(𝑡_𝑗)−𝑥_ℎ(𝑡_𝑗)|=𝑂(ℎ^𝜌)(𝑗=0,1,…,𝑁), where 𝜌=Min(2,1+𝑞).

In 2004, Li and Peng [29] and Li and Chen [45] studied the chaotic behaviors of chaotic Chen system with fractional-order, which is described as follows:𝐷^𝑞_∗𝑥₁(𝑡)=𝑎𝑥₂−𝑥₁,𝐷^𝑞_∗𝑥₂(𝑡)=(𝑐−𝑎)𝑥₁−𝑥₁𝑥₃+𝑐𝑥₂,𝐷^𝑞_∗𝑥₃(𝑡)=𝑥₁𝑥₂−𝑏𝑥₃, (2.9) where 𝑞 is the fractional order, 0<𝑞<1,𝑥₁,𝑥₂ and 𝑥₃ are state variables, 𝑎,𝑏 and 𝑐 are unknown parameters to be identified. When 𝑎=35,𝑏=3,𝑐=28 and 𝑞=0.9, system (2.9) exhibits chaotic behaviors as shown in Figure 1. References [46, 47] considered the synchronization of fractional-order chaotic Chen systems with determined parameters.

(a)

(b)

(c)

(d)

Figure 1 

Chaotic attractors of fractional-order chaotic Chen system with order 𝑞=0.9.

Meanwhile, Deng and Li [30] and Lu [48] numerically investigated the chaotic behaviors of the fractional-order Lü system, respectively. Fractional-order chaotic Lü system is given by𝐷^𝑞_∗𝑦₁(𝑡)=𝛼𝑦₂−𝑦₁,𝐷^𝑞_∗𝑦₂(𝑡)=−𝑦₁𝑦₃+𝛽𝑦₂,𝐷^𝑞_∗𝑦₃(𝑡)=𝑦₁𝑦₂−𝜃𝑦₃, (2.10) where 𝑞 is the fractional order; 0<𝑞<1; 𝑦₁,𝑦₂, and 𝑦₃ are state variables; 𝛼,𝛽, and 𝜃 are unknown parameters to be estimated. System (2.10) exhibits a chaotic attractor for 𝛼=36,𝛽=20, and 𝜃=3. Figure 2 displays the chaotic attractor of the fractional-order Lü chaotic system with the order 𝑞=0.9.

(a)

(b)

(c)

(d)

Figure 2 

Chaotic attractors of fractional-order chaotic Lü system with order 𝑞=0.9.

To obtain our results, the following Lemma is presented. For a given autonomous linear system of fractional order𝐷^𝑞_∗𝑥=𝐴𝑥, (2.11) with 𝑥(0)=𝑥₀, where 𝑥∈𝑅^𝑛 is the state vector, one has the following Lemma.

Lemma 2.1 (see [49]). System (2.11) is(i)asymptotically stable if and only if  |arg(𝜆_𝑖(𝐴))|>𝛼𝜋/2,𝑖=1,2,3,…, where arg(𝜆_𝑖(𝐴)) denotes the argument of the eigenvalue 𝜆_𝑖 of A. In this case, the component of the state decays towards 0 like 𝑡^−𝛼;(ii)stable if and only if either it is asymptotically stable or those critical eigenvalues that satisfy |arg(𝜆_𝑖(𝐴))|=𝛼𝜋/2 have geometric multiplicity one. Figure 3 illustrates the stability region of the fractional-order system.

Figure 3 

Stability region of the fractional-order system.

3. Adaptive Projective Synchronization

In the section, we will discuss adaptive projective synchronization behavior between fractional-order chaotic Chen system and fractional-order chaotic Lü system with fully unknown parameters. Assume that fractional-order chaotic Chen system with three unknown parameters is the drive system and the response system is fractional-order chaotic Lü system, which are given as follows, respectively,𝐷^𝑞_∗𝑥₁(𝑡)=𝑎𝑥₂−𝑥₁,𝐷^𝑞_∗𝑥₂(𝑡)=(𝑐−𝑎)𝑥₁−𝑥₁𝑥₃+𝑐𝑥₂,𝐷^𝑞_∗𝑥₃(𝑡)=𝑥₁𝑥₂−𝑏𝑥₃,𝐷^𝑞_∗𝑦₁(𝑡)=𝛼𝑦₂−𝑦₁+𝑢₁,𝐷^𝑞_∗𝑦₂(𝑡)=−𝑦₁𝑦₃+𝛽𝑦₂+𝑢₂,𝐷^𝑞_∗𝑦₃(𝑡)=𝑦₁𝑦₂−𝜃𝑦₃+𝑢₃, (3.1)(3.2) where 𝑥_𝑖 and 𝑦_𝑖(𝑖=1,2,3) stand for state variables of the master system and the slave one, respectively; 𝑎,𝑏,𝑐,𝛼,𝛽, and 𝜃 are unknown parameters, which need to be estimated; 𝑢₁,𝑢₂, and 𝑢₃ are nonlinear controllers to be designed later.

Definition 3.1. Systems (3.1) and (3.2) are adaptive projective synchronization if there exists a scaling matrix 𝜎 such that lim_𝑡→∞‖𝑒(𝑡)‖=lim_𝑡→∞𝑦−𝜎𝑥=0, (3.3) where ‖⋅‖ is the Euclidean norm and 𝜎=diag(𝜎₁,𝜎₂,𝜎₃).
It follows from (3.1)–(3.3) that we have the following error dynamical system: 𝐷^𝑞_∗𝑒₁(𝑡)=𝛼𝑦₂−𝑦₁−𝜎₁𝑎𝑥₂−𝑥₁+𝑢₁,𝐷^𝑞_∗𝑒₂(𝑡)=−𝑦₁𝑦₃+𝛽𝑦₂−𝜎₂(𝑐−𝑎)𝑥₁+𝜎₂𝑥₁𝑥₃−𝜎₂𝑐𝑥₂+𝑢₂,𝐷^𝑞_∗𝑒₃(𝑡)=𝑦₁𝑦₂−𝜃𝑦₃−𝜎₃𝑥₁𝑥₂+𝜎₃𝑏𝑥₃+𝑢₃, (3.4) where 𝑒₁=𝑦₁−𝜎₁𝑥₁,𝑒₂=𝑦₂−𝜎₂𝑥₂,and𝑒₃=𝑦₃−𝜎₃𝑥₃. Our aim is to find suitable control laws 𝑢_𝑖(𝑖=1,2,3) with a parameter estimation update law such that the response system (3.2) synchronizes the drive system (3.1) up to a scaling factor 𝜎.

To this end, the following criterion is proposed to ensure that the response system (3.2) effectively synchronizes the drive system (3.1) up to a scaling factor 𝜎.

Theorem 3.2. For a given constant scaling matrix 𝜎=diag(𝜎₁,𝜎₂,𝜎₃) and any initial conditions, projective synchronization between systems (3.1) and (3.2) will occur by the following adaptive controllers: 𝑢₁=(𝛼−1)𝑒₁+𝑦₃−𝛼𝑒₂−𝑦₂𝑒₃+𝛼𝜎₁𝑥₁−𝜎₂𝑥₂+𝑎𝜎₁𝑥₂−𝑥₁,𝑢₂=−𝛽+1𝑒₂+𝜎₁𝜎₃−𝜎₂𝑥₁𝑥₃−𝜎₂𝑥₂𝛽−𝑐̃+𝜎₂𝑥₁(),𝑢𝑐̃−𝑎₃=𝜃−1𝑒₃+𝜎₃−𝜎₁𝜎₂𝑥₁𝑥₂+𝜎₃𝑥₃𝜃−𝑏̃, (3.5) and the parameter update rules for six unknown parameters 𝑎,𝑏,𝑐,𝛼,𝛽, and 𝜃𝐷^𝑞_∗𝑎=𝜎₁𝑥₁−𝜎₁𝑥₂𝑒₁+𝜎₂𝑥₁𝑒₂,𝐷^𝑞_∗𝑏̃=𝜎₃𝑥₃𝑒₃,𝐷^𝑞_∗𝑐̃=−𝜎₂𝑥₁+𝜎₂𝑥₂𝑒₂,𝐷^𝑞_∗𝛼=𝑦₂−𝑦₁𝑒₁,𝐷^𝑞_∗𝛽=𝑦₂𝑒₂,𝐷^𝑞_∗𝜃=−𝑦₃𝑒₃, (3.6) where 𝑒_𝑎=𝑎−𝑎,𝑒_𝑏=𝑏̃−𝑏,𝑒_𝑐=𝑐̃−𝑐,𝑒_𝛼=𝛼−𝛼,𝑒_𝛽=𝛽−𝛽,𝑒_𝜃=𝜃−𝜃,𝑎,𝑏̃,𝑐,̃𝛼,𝛽, and 𝜃 are the parameter estimations of 𝑎,𝑏,𝑐,𝛼,𝛽, and 𝜃, respectively, and 𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝛼,𝑒_𝛽, and 𝑒_𝜃 are the corresponding parameter errors.

Proof. Substituting (3.5) into (3.4) leads to the following error system: 𝐷^𝑞_∗𝑒₁(𝑡)=−𝑒₁+𝑦₃𝑒₂−𝑦₂𝑒₃+𝜎₁𝑥₂−𝜎₁𝑥₁𝑒_𝑎+𝑦₁−𝑦₂𝑒_𝛼,𝐷^𝑞_∗𝑒₂(𝑡)=−𝑦₃𝑒₁−𝑒₂−𝜎₁𝑥₁𝑒₃−𝜎₂𝑥₁𝑒_𝑎+𝜎₂𝑥₁+𝜎₂𝑥₂𝑒_𝑐−𝑦₂𝑒_𝛽,𝐷^𝑞_∗𝑒₁(𝑡)=𝑦₂𝑒₁+𝜎₁𝑥₁𝑒₂−𝑒₃−𝜎₃𝑥₃𝑒_𝑏+𝑦₃𝑒_𝜃. (3.7)
Combining (3.6) with (3.7), one has 𝐷^𝑞_∗𝑒₁,𝐷^𝑞_∗𝑒₂,𝐷^𝑞_∗𝑒₃,𝐷^𝑞_∗𝑒_𝑎,𝐷^𝑞_∗𝑒_𝑏,𝐷^𝑞_∗𝑒_𝑐,𝐷^𝑞_∗𝑒_𝛼,𝐷^𝑞_∗𝑒_𝛽,𝐷^𝑞_∗𝑒_𝜃^𝑇=𝐴𝑒₁,𝑒₂,𝑒₃,𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝛼,𝑒_𝛽,𝑒_𝜃^𝑇, (3.8) where 𝐴=−1𝑦₃−𝑦₂𝑀00𝑦₁−𝑦₂00−𝑦₃−1−𝜎₁𝑥₁−𝜎₂𝑥₁0𝑁0−𝑦₂0𝑦₂𝜎₁𝑥₁−10−𝜎₃𝑥₃000𝑦₃−𝑀𝜎₂𝑥₁000000000𝜎₃𝑥₃𝑦0000000−𝑁0000000₂−𝑦₁000000000𝑦₂000000000−𝑦₃000000𝑀=𝜎₁𝑥₂−𝜎₁𝑥₁,𝑁=𝜎₂𝑥₁+𝜎₂𝑥₂., (3.9)
Suppose 𝜆 is one of the eigenvalues of matrix 𝐴 and the corresponding noneigenvector is 𝜀=(𝜀₁,𝜀₂,𝜀₃,𝜀₄,𝜀₅,𝜀₆,𝜀₇,𝜀₈,𝜀₉)^𝑇, that is, 𝐴𝜀=𝜆𝜀. (3.10) Take conjugate transpose (𝐻) on both sides of (3.10); one obtains (𝐴𝜀)^𝑇=𝜆𝜀^𝐻. (3.11) Using (3.10) multiplied left by (1/2)𝜀^𝐻 plus (3.11) multiplied right by (1/2)𝜀, we derive 𝜀^𝐻121𝐴+2𝐴^𝐻1𝜀=2𝜆+𝜆𝜀^𝐻𝜀. (3.12) From (3.12), we have 12𝜆+𝜆=𝜀^𝐻(1/2)𝐴+(1/2)𝐴^𝐻𝜀𝜀^𝐻𝜀. (3.13) By substituting 𝐴 into the right of (3.13), we can obtain 12𝜆+𝜆=1𝜀^𝐻𝜀𝜀^𝐻−1000000000−1000000000−1000000000000000000000000000000000000000000000000000000000000𝜀. (3.14)
From (3.14), we have 𝜆+𝜆≤0, that is, any eigenvalue of matrix 𝐴 satisfies (𝜆≥𝜋arg)2>𝛼𝜋2(𝛼<1). (3.15) Based on the stability theory of fractional-order systems (Lemma 2.1), the equilibrium point in error system (3.8) is asymptotically stable, that is, lim_𝑡→∞‖𝑒(𝑡)‖=lim_𝑡→∞𝑦−𝜎𝑥=0, (3.16) which implies that the adaptive projective synchronization between systems (3.1) and (3.2) can be achieved.
If we choose the scaling matrix 𝜎 as 𝐼 and −𝐼 (𝐼 is an 𝑛×𝑛 identity matrix), then we have the following Corollaries 3.3 and 3.4 to guarantee system (3.1) synchronizes system (3.2) in complete and inverse manner, respectively.

Corollary 3.3. If the adaptive controllers are chosen as 𝑢₁=(𝛼−1)𝑒₁+𝑦₃−𝛼𝑒₂−𝑦₂𝑒₃+𝑥₁−𝑥₂(),𝑢𝛼−𝑎₂=−𝛽+1𝑒₂−𝑥₂𝛽−𝑐̃+𝑥₁(),𝑢𝑐̃−𝑎₃=𝜃−1𝑒₃+𝑥₃𝜃−𝑏̃, (3.17) and update laws of parameters are chosen as 𝐷^𝑞_∗𝑎=𝑥₁−𝑥₂𝑒₁+𝑥₁𝑒₂,𝐷^𝑞_∗𝑏̃=𝑥₃𝑒₃,𝐷^𝑞_∗𝑐̃=−𝑥₁+𝑥₂𝑒₂,𝐷^𝑞_∗𝛼=𝑦₂−𝑦₁𝑒₁,𝐷^𝑞_∗𝛽=𝑦₂𝑒₂,𝐷^𝑞_∗𝜃=−𝑦₃𝑒₃, (3.18) then complete synchronization between systems (3.1) and (3.2) will be achieved for any initial values, where 𝑒_𝑎=𝑎−𝑎,𝑒_𝑏=𝑏̃−𝑏,𝑒_𝑐=𝑐̃−𝑐,𝑒_𝛼=𝛼−𝛼,𝑒_𝛽=𝛽−𝛽,𝑒_𝜃=𝜃−𝜃,𝑎,𝑏̃,𝑐,̃𝛼,𝛽, and 𝜃 are the parameter estimations of 𝑎,𝑏,𝑐,𝛼,𝛽, and 𝜃, respectively, and 𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝛼,𝑒_𝛽, and 𝑒_𝜃 are the corresponding parameter errors.

Corollary 3.4. If the adaptive controllers are chosen as 𝑢₁=(𝛼−1)𝑒₁+𝑦₃−𝛼𝑒₂−𝑦₂𝑒₃−𝑥₁−𝑥₂(),𝑢𝛼−𝑎₂=−𝛽+1𝑒₂+2𝑥₁𝑥₃+𝑥₂𝛽−𝑐̃−𝑥₁(),𝑢𝑐̃−𝑎₃=𝜃−1𝑒₃−2𝑥₁𝑥₂−𝑥₃𝜃−𝑏̃, (3.19) and update laws of parameters are chosen as 𝐷^𝑞_∗𝑎=𝑥₂−𝑥₁𝑒₁−𝑥₁𝑒₂,𝐷^𝑞_∗𝑏̃=−𝑥₃𝑒₃,𝐷^𝑞_∗𝑐̃=𝑥₁+𝑥₂𝑒₂,𝐷^𝑞_∗𝛼=𝑦₂−𝑦₁𝑒₁,𝐷^𝑞_∗𝛽=𝑦₂𝑒₂,𝐷^𝑞_∗𝜃=−𝑦₃𝑒₃, (3.20) then inverse synchronization between systems (3.1) and (3.2) will be obtained for any initial values, where 𝑒_𝑎=𝑎−𝑎,𝑒_𝑏=𝑏̃−𝑏,𝑒_𝑐=𝑐̃−𝑐,𝑒_𝛼=𝛼−𝛼,𝑒_𝛽=𝛽−𝛽,𝑒_𝜃=𝜃−𝜃,𝑎,𝑏̃,𝑐,̃𝛼,𝛽, and 𝜃 are the parameter estimations of 𝑎,𝑏,𝑐,𝛼,𝛽, and 𝜃, respectively, and 𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝛼,𝑒_𝛽, and 𝑒_𝜃 are the corresponding parameter errors.

4. Numerical Simulations

In this section, numerical simulations will be given to verify theoretical results obtained in the previous section. The true values of the “unknown” parameters of two uncertain systems are taken as 𝑎=35,𝑏=3,𝑐=28 and 𝛼=36,𝛽=20,𝜃=3. For this numerical simulation, the initial values of the drive and the response systems are chosen as (𝑥₁(0),𝑥₂(0),𝑥₃(0))=(0.3,0.4,0.5) and (𝑦₁(0),𝑦₂(0),𝑦₃(0))=(0.4,0.5,0.6), respectively. Hence the initial values of the error system are (𝑒₁(0),𝑒₂(0),𝑒₃(0))=(0.1,0.1,0.1), the initial estimated parameters (𝑎(0),𝑏̃(0),𝑐(̃0),𝛼(0),𝛽(0),𝜃(0))=(32,2,25,33,22,5). Without loss of generality, the scaling matrix 𝜎 is chosen as diag(2,−3,4) and 𝑞=0.9. Figure 4 displays the synchronization error states between systems (3.1) and (3.2), respectively. Figure 5 shows the signals after synchronization. Figures 6 and 7 show that the parameter errors 𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝛼,𝑒_𝛽,𝑒_𝜃 converge to 0, which means the estimated values of unknown parameters 𝑎,𝑏̃,𝑐,̃𝛼,𝛽,𝜃 converge to 𝑎=25,𝑏=3,𝑐=28,𝛼=36,𝛽=20,and𝜃=3 as 𝑡→∞, respectively. Figure 8 illustrates the projective chaotic attractors of drive system (3.1) and response system (3.2).

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Figure 4 

Error states of the drive system (3.1) and the response system (3.2).

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Figure 5 

Signals 𝑥1(𝑡) versus 𝑦1(𝑡), 𝑥2(𝑡) versus 𝑦2(𝑡), and 𝑥3(𝑡) versus 𝑦3(𝑡) after synchronization.

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Figure 6 

Parameter errors in the drive system (3.1), 𝑒𝑎=𝑎−𝑎, 𝑒𝑏=𝑏̃−𝑏, 𝑒𝑎=𝑏̃−𝑏.

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Figure 7 

Parameter errors in the response system (3.2), 𝑒𝛼=𝛼−𝛼, 𝑒𝛽=𝛽−𝛽, 𝑒𝜃=𝜃−𝜃.

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Figure 8 

The projective attractors of drive system (3.1) and response system (3.2) (𝜎1=2,𝜎2=−3,𝜎3=4).

5. Conclusion

In this paper, adaptive projective synchronization of different chaotic systems with fully unknown parameters is presented. Based on the stability theory of fractional-order differential equations, a suitable adaptive controller and parameter update laws are designed to ensure fractional-order Chen system synchronize with fractional-order Lü system up to a scaling factor 𝜎. Also the corresponding results are obtained to judge complete synchronization and inverse synchronization between fractional-order Chen system and fractional-order Lü system, respectively. Meanwhile, results could be extended to other fractional-order chaotic or hyperchaotic systems, such as fractional-order Lorenz system and fractional-order Rössler system, fractional-order hyperchaotic Chen system, and fractional-order hyperchaotic Lü 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), the Specialized Research Fund for the Doctoral Program of Higher Education of China (nos. 20093401120001; 102063720090013), the Natural Science Foundation of Anhui Province (no. 11040606M12), the Natural Science Foundation of Anhui Education Bureau (no. KJ2010A035), PhD. Candidate Academic Foundation of Ministry of Education of China and the 211 project of Anhui University.