Function Projective Synchronization of a Class of Chaotic Systems with Uncertain Parameters

Abstract

This paper investigates the function projective synchronization of a class of chaotic systems with uncertain parameters. Based on Lyapunov stability theory, the nonlinear adaptive control law and the parameter update law are derived to make the state of two chaotic systems function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive scheme.

1. Introduction

Synchronization of chaotic systems has attracted considerable attention from scientists and engineers and has been investigated intensively both theoretically and experimentally over the last decade. A variety of methods have been introduced for the synchronization of chaotic systems which include complete synchronization, generalized synchronization, phase synchronization, lag synchronization, adaptive synchronization, time scale synchronization, intermittent lag synchronization, projective synchronization, and function projective synchronization, see [1–6] and references therein. Amongst all these synchronization schemes, it is worthwhile to mention that function projective synchronization has attracted much attention recently due to its potential application in secure communications. Function projective synchronization is the more general definition of projective synchronization. As compared with projective synchronization, function projective synchronization means that the master and slave systems could be synchronized up to a scaling function, but not a constant. This feature could be used to get more secure communication in application to secure communications, because it is obvious that the unpredictability of the scaling function in function projective synchronization can additionally enhance the security of communication. Motivated by the aforementioned reasons, this paper investigates function projective synchronization of recently developed chaotic systems with uncertain parameters using nonlinear adaptive controller. The rest of this paper is organized as follows. In Section 2, the nonlinear adaptive controller is designed to synchronize the chaotic system. In Section 3, numerical simulations are presented to demonstrate the effectiveness of the theoretical analysis.

2. Adaptive Control Scheme

The chaotic system introduced by Liu [7] recently is described by𝑥̇₁=𝑎𝑥₂−𝑥₁+𝑥₂𝑥₃,𝑥̇₂=𝑏𝑥₂−𝑐𝑥₁𝑥₃,𝑥̇₃=𝑔𝑥₂−ℎ𝑥₃, (2.1) where 𝑎,𝑏,𝑐,𝑔,ℎ are the system parameters, when 𝑎=1,𝑏=2.5,𝑐=1,𝑔=1,ℎ=4, the system is chaotic.

We take the system (2.1) as the master system, and the slave system with an adaptive control scheme is given by𝑦₁̇=𝑎₁𝑦₂−𝑦₁+𝑦₂𝑦₃+𝑢₁,𝑦₂̇=𝑏₁𝑦₂−𝑐₁𝑦₁𝑦₃+𝑢₂,𝑦₃̇=𝑔₁𝑦₂−ℎ₁𝑦₃+𝑢₃, (2.2) where 𝑎₁,𝑏₁,𝑐₁,𝑔₁,ℎ₁ are parameters of the slave system which needs to be estimated, and 𝑢₁,𝑢₂,𝑢₃ are the nonlinear controllers such that the two chaotic systems are function projective synchronized in the sense thatlim_𝑡→∞𝑦_𝑖−𝛼(𝑡)𝑥_𝑖=0,𝑖=1,2,3, (2.3) where 𝛼(𝑡) is the scaling function.

From system (2.1) and system (2.2), we get the error dynamical system which can be written as𝑒₁̇=𝑎₁𝑦₂−𝑦₁+𝑦₂𝑦₃+𝑢₁−𝑎𝛼(𝑡)𝑥₂−𝑥₁+𝑥₂𝑥₃−𝛼̇(𝑡)𝑥₁,𝑒₂̇=𝑏₁𝑦₂−𝑐₁𝑦₁𝑦₃+𝑢₂−𝑏𝛼(𝑡)𝑥₂−𝑐𝛼(𝑡)𝑥₁𝑥₃−𝛼̇(𝑡)𝑥₂,𝑒₃̇=𝑔₁𝑦₂−ℎ₁𝑦₃+𝑢₃−𝑔𝛼(𝑡)𝑥₂+ℎ𝛼(𝑡)𝑥₃−𝛼̇(𝑡)𝑥₃, (2.4) where 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝛼(𝑡)𝑥_𝑖(𝑡),𝑖=1,2,3.

In order to stabilize the error variables of system (2.3) at the origin, we propose the adaptive control law and the parameter update law for system (2.3) as follows:𝑢̇₁=−𝑎₁𝑦₂−𝑦₁+𝑦₂𝑦₃+𝛼̇(𝑡)𝑥₁+𝑎₁𝛼(𝑡)𝑥₂−𝑥₁+𝑥₂𝑥₃−𝑘₁𝑒₁,𝑢̇₂=−𝑏₁𝑦₂−𝑥₂𝛼(𝑡)+𝑐₁𝑦₁𝑦₃+𝑥₁𝑥₃𝛼(𝑡)+𝛼̇(𝑡)𝑥₂−𝑘₂𝑒₂,𝑢̇₃=−𝑔₁𝑦₂−𝑥₂𝛼(𝑡)+ℎ₁𝑦₃−𝑥₃𝛼(𝑡)+𝛼̇(𝑡)𝑥₃−𝑘₃𝑒₃, (2.5) and the update rules for the five uncertain parameters 𝑎₁,𝑏₁,𝑐₁,𝑔₁, and ℎ₁ are𝑎₁̇=𝑥₁−𝑥₂−𝑥₂𝑥₃𝛼(𝑡)𝑒₁−𝑘₄𝑒_𝑎,𝑏̇₁=−𝑥₂𝛼(𝑡)𝑒₂−𝑘₅𝑒_𝑏,𝑐₁̇=−𝑥₁𝑥₃𝛼(𝑡)𝑒₂−𝑘₆𝑒_𝑐,𝑔₁̇=−𝑥₂𝛼(𝑡)𝑒₃−𝑘₇𝑒_𝑔,ℎ̇₁=𝑥₃𝛼(𝑡)𝑒₃−𝑘₈𝑒_ℎ, (2.6) where 𝑘_𝑖>0(𝑖=1,2,…,8) and 𝑒_𝑎=𝑎₁−𝑎,𝑒_𝑏=𝑏₁−𝑏,𝑒_𝑐=𝑐₁−𝑐,𝑒_𝑔=𝑔₁−𝑔,𝑒_ℎ=ℎ₁−ℎ.

Theorem 2.1. For the given scaling function 𝛼(𝑡), the function projective synchronization between the master system (2.1) and the slave system (2.2) can be achieved if the nonlinear controller (2.5) and the update law (2.6) are adopted.

Proof. Construct the following Lyapunov function: 1𝑉=2³_𝑖=1𝑒^𝑇_𝑖𝑒_𝑖+12𝑒²_𝑎+𝑒²_𝑏+𝑒²_𝑐+𝑒²_𝑔+𝑒²_ℎ. (2.7) Calculating the time derivative of 𝑉 along the trajectory of error system (2.4), we have 𝑉̇=𝑒₁(𝑡)𝑎₁𝑦₂−𝑦₁+𝑦₂𝑦₃+𝑢₁−𝑎𝛼(𝑡)𝑥₂−𝑥₁+𝑥₂𝑥₃−𝛼̇(𝑡)𝑥₁+𝑒₂(𝑡)𝑏₁𝑦₂−𝑐₁𝑦₁𝑦₃+𝑢₂−𝑏𝛼(𝑡)𝑥₂−𝑐𝛼(𝑡)𝑥₁𝑥₃−𝛼̇(𝑡)𝑥₂+𝑒₃(𝑡)𝑔₁𝑦₂−ℎ₁𝑦₃+𝑢₃−𝑔𝛼(𝑡)𝑥₂+ℎ𝛼(𝑡)𝑥₃−𝛼̇(𝑡)𝑥₃+𝑒_𝑎(𝑡)𝑥₁−𝑥₂−𝑥₂𝑥₃𝛼(𝑡)𝑒₁−𝑘₄𝑒_𝑎+𝑒_𝑏(𝑡)−𝑥₂𝛼(𝑡)𝑒₂−𝑘₅𝑒_𝑏+𝑒_𝑐(𝑡)−𝑥₁𝑥₃𝛼(𝑡)𝑒₂−𝑘₆𝑒_𝑐+𝑒_𝑔(𝑡)−𝑥₂𝛼(𝑡)𝑒₃−𝑘₇𝑒_𝑔+𝑒_ℎ(𝑡)𝑥₃𝛼(𝑡)𝑒₃−𝑘₈𝑒_ℎ=−𝐞^𝑇𝐊𝐞, (2.8) where 𝐞=[𝑒₁,𝑒₂,𝑒₃,𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝑔,𝑒_ℎ]^𝑇 and 𝐊=diag{𝑘₁,𝑘₂,𝑘₃,𝑘₄,𝑘₅,𝑘₆,𝑘₇,𝑘₈}.
It is clear that 𝑉̇ is negative definite and 𝑉̇=0 if and only if 𝐞(𝑡)=𝟎. According to the Lyapunov stability theorem, the function projective synchronization is achieved. This completes the proof.

3. Numerical Simulations

In this section, we will perform some numerical simulations to verify the effectiveness of the proposed adaptive synchronization controllers. The fourth-order Runge-Kutta method is used to solve the master system (2.1) and the slave system (2.2) with time step size 0.001. The initial conditions of the master system are 𝑥₁(0)=0.2,𝑥₂(0)=−1.2,𝑥₃(0)=0.6, and those of the slave system are 𝑦₁(0)=0.4,𝑦₂(0)=0.2,𝑦₃(0)=−1.8. Moreover, the true values of the uncertain parameters are 𝑎=1,𝑏=2.5,𝑐=1,𝑔=1,ℎ=4, and the initial values of the estimated parameters are chosen as 𝑎₁(0)=2,𝑏₁(0)=3,𝑐₁(0)=3,𝑔₁(0)=5,ℎ₁(0)=5, the scaling function is chosen as 𝛼(𝑡)=0.3+0.2sin(𝑡). Furthermore, the control gains are chosen as (𝑘₁,𝑘₂,𝑘₃,𝑘₄,𝑘₅,𝑘₆,𝑘₇,𝑘₈)=(0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2). The error variables 𝑒₁,𝑒₂,𝑒₃ which tend to zero with 𝑡→∞ are illustrated in Figure 1. Figure 2 shows that the estimated values of the uncertain parameters converge to 𝑎=1,𝑏=2.5,𝑐=1,𝑔=1,ℎ=4 as 𝑡→∞.

Figure 1 

Error dynamics between the master system (2.1) and the slave system (2.2).

Figure 2 

The time evolution of the estimated parameters.

4. Conclusion

In summary, this work has investigated the function projective synchronization of the new chaotic systems with uncertain parameters. Based on the Lyapunov stability theory, we have designed the adaptive synchronization controllers with corresponding parameter update laws to stabilize the error dynamics between the master and the slave systems. All the theoretical results are verified by numerical simulations to illustrate the effectiveness of the proposed adaptive function projective synchronization scheme.