Adaptive Feedback Control for Chaos Control and Synchronization for New Chaotic Dynamical System

This paper investigates the problem of chaos control and synchronization for new chaotic dynamical system and proposes a simple adaptive feedback control method for chaos control and synchronization under a reasonable assumption. In comparison with previous methods, the present control technique is simple both in the form of the controller and its application. Based on Lyapunov’s stability theory, adaptive control law is derived such that the trajectory of the new system with unknown parameters is globally stabilized to the origin. In addition, an adaptive control approach is proposed to make the states of two identical systems with unknown parameters asymptotically synchronized. Numerical simulations are shown to verify the analytical results.


Introduction
Chaos control and synchronization methods were first addressed by Ott et al. 1 and Pecora and Carroll 2 in the beginning of the decade of 1990.Along with these concepts came the idea of chaotic encryption.
Nowadays, different techniques and methods have been proposed to achieve chaos control and chaos synchronization such as linear and nonlinear feedback control 3-10 .Most of them are based on exactly knowing the system structure and parameters.But in practical situations, some or all of the system's parameters are unknown.Moreover, these parameters change from time to time.Therefore, the derivation of an adaptive controller for the control and synchronization of chaotic systems in the presence of unknown system parameters is an important issue 11-21 .In recent years, chaos synchronization has received special interests due to its potential applications in secure communications 22-25 , biological systems 26 , circuits 27 , lasers 28 , and so forth.In operation, a chaotic system exhibits irregular behavior and produces broadband, noise-like signals, thus, it is thought to use in secure communications.
In this work we investigate the problem of chaos control and synchronization for new chaotic dynamical system and propose a simple adaptive feedback control method for chaos control and synchronization under a reasonable assumption.In comparison with previous methods, the present control technique is simple both in the form of the controller and its application.Based on Lyapunov's stability theory, adaptive control law is derived such that the trajectory of the new system with unknown parameters is globally stabilized to the origin.In addition, an adaptive control approach is proposed to make the states of two identical systems with unknown parameters asymptotically synchronized.Numerical simulations are shown to verify the analytical results.
The object of this work is chaos control and synchronization of two identical new chaotic dynamical systems 29 with adaptive feedback and application in secure communication.The new system 29 is described by where a, b, and c are three unknown parameters.This system exhibits a chaotic attractor at the parameter values a 10, b 16, and c −1 see Figure 1 .
The paper is organized as follows.In Section 2, we propose the main results of this paper.In Section 3, the adaptive feedback control method is applied to control of new attractor with unknown parameters and numerical simulations are presented to show the effectiveness of the proposed method.In Section 4, the adaptive feedback control method is applied to synchronization of two identical new attractor and numerical simulations are presented for verifying the effectiveness of the proposed method.We conclude the paper in Section 5.

Main Results
In this section, we investigate the problem of chaos control by modifying the previous method 30 and propose the main results of this paper.
Let a chaotic system be given as . ., f n x T : R n → R n is a smooth nonlinear vector function.Without loss of the generality, let x e 0 is an equilibrium point of the system 2.1 .To describe the new design and analysis, the following assumption is needed.

Assumption 2.1. There exists a nonsingular coordinate transformation y
T x, such that system 2.1 can be rewritten as

2.2
where z 1 y 1 , y 2 , . . ., y r T ∈ R r , z 2 y r 1 , y r 2 , . . ., y n T ∈ R n−r , the second equation satisfies ż2 g 2 0, z 2 , with the vector function g 2 z 1 , z 2 being smooth in a neighborhood of z 1 0, and the subsystem ż2 g 2 0, z 2 is uniformly exponentially stable about the origin z 2 0 for all z.Remark 2.2.It should be pointed out that not all the finite dimensional chaotic systems are given as 2.2 in their original forms.Therefore, we should make a nonsingular coordinate transformation T , which can adjust the array order of the variables x 1 , x 2 , . . ., x n to make the original systems in the new form have the form of 2.2 .Thus, Assumption 2.1 is reasonable, and system 2.2 is very general, which contains most well-known finite dimensional chaotic systems.
Remark 2.3.The vector function g 2 z 1 , z 2 being smooth in a neighborhood of z 1 0, that is, there is a positive constant λ 0 locally such that ||g 2 z 1 , z 2 − g 2 0, z 2 || ≤ λ 0 ||z 1 ||.And the subsystem ż2 g 2 0, z 2 is uniformly exponentially stable about the origin z 2 0 for all z, which implies that there are a Lyapunov function V 0 z 2 and two positive numbers respectively.Since the system 2.1 is chaotic and g 2 z 1 , z 2 is smooth function, there exists a positive number λ 3 , such that In order to stabilize the chaotic orbits in 2.1 to its equilibrium point x e 0, we add the following adaptive feedback controller to system 2.1 and the controlled system 2.1 is as follows: where the controller u u 1 , u 2 The feedback gain k 1 is adapted according to the following update law: where γ is an arbitrary positive constant, in general, we select γ 1.
Let the systems 2.2 and 2.4 be the augment systems, and introduce a Lyapunov function where L λ 3 α , α ≥ λ 2 0 λ 1 /4.Next, we give the following main result.
Theorem 2.4.Starting from any initial values of the augment system, the orbits of the augment system x t , k 1 t T converge to x e , k 0 T as t → ∞, where k 0 is a negative constant depending on the initial value.That is to say, the adaptive feedback controller stabilizes the chaotic orbits to its equilibrium point x e 0.
Proof.Differentiating the function V along the trajectories of the augment system, we obtain

2.7
Obviously, V 0 if and only if z i 0, i 1, 2, then the set E { z, k 1 | V z 0} {0} is the largest invariant set for the augment system.According to the well-known LaSalle invariance principle, z i 0, i 1, 2, which implies that x i 0, i 1, 2, . . ., n, thus, Theorem 2.4 is obtained.
Remark 2.5.In general, n − r ≥ 1, where n − r is the dimension of the variable z 2 .One the other hand, we stabilize the first subsystem ż1 g 1 z 1 , z 2 by applying the previous method 30 .Therefore, the controllers obtained in this paper are simpler than those controllers obtained by the previous method in general case or the same to those controllers obtained by the previous method even in the worst case n − r 0. Accordingly, the present method is a modification of the previous method.
Remark 2.6.If x e / 0 is an equilibrium point of the chaotic system 1.1 , then we make the coordinate transformation y x − x e , which make the original chaotic system 1.1 with new variable y y 1 , y 2 , . . ., y n has the equilibrium point y e 0. That is to say, this method can be also easily utilized whatever x e is origin or not.

Adaptive Feedback Control Method for Controling New Attractor
In this section, we apply the above technique to control the new chaotic system 29 .Now, we rewrite system 1.1 as the following:

3.1
It is easy to know the fact that if x 2 0 the following two dimensional subsystem of the system 3.1 : which is uniformly exponentially stable about the origin x 1 0, x 3 0 for all x 1 , x 3 , then there exists a nonsingular coordinate transformation y T x, that is, y 1 x 2 , y 2 x 1 , y 3 x 3 , which can make system 3.1 with new variable y has the form of system 2.2 , and the new system with controller u k 1 y ẏ3 y 1 y 2 cy 3 , k1 − γ y 1 2 .

Mathematical Problems in Engineering
Now, we define a Lyapunov function as where L is a sufficiently big constant.It is clear that the Lyapunov function V e is a positive definite function.Now, taking the time derivative of 3.4 , then we get Obviously, to ensure that the origin of the system 3.1 is asymptotically stable, the matrix P should be positive definite, which implies that V is negative definite under the condition L ≥ a b 2 /4a then dV e /dt ≤ 0. According to Theorem 2.4, the origin of system 3.3 is asymptotically stable.

Numerical Results
By using Maple 13 to solve the systems of differential equation 3.1 with the parameters are chosen to a 10, b 16, and c −1 in all simulations so that the new system exhibits a chaotic behavior if no control is applied see Figure 1 .The initial states of system 3.1 are y 1 0 1.5, y 2 0 −2, and y 3 0 3.2 and the initial value of the controller k 1 0 −1.
Figure 2: The new dynamical system 3.1 is driven to its stable equilibrium 0, 0, 0 asymptotically as t → ∞.When γ 1, the new system is driven to its stable equilibrium 0, 0, 0 asymptotically as t → ∞ are shown in Figure 2. The feedback gain k 1 tends to a negative constant as shown in Figure 3.

Adaptive Feedback Control Method for Synchronization of Two Identical New Attractors
In this section, we apply the adaptive feedback control technique for synchronization of two identical new chaotic systems 29 .For the new system 1.1 , the master or drive and slave or response systems are defined below, respectively,

4.2
For this purpose, the error dynamical system between the drive system 4.1 and response system 4.2 can be expressed by In order that two chaotic systems can be synchronized in the sense of PS, the following condition should be satisfied: It is easy to know the fact that if y 3 0 the following two-dimensional subsystem of system 4.3 : under the condition L > z 1 − a − b 2 /4a, then dV e /dt ≤ 0. Based on Lyapunov's stability theory, this translates to lim t → ∞ e t 0. Thus, the response system and drive systems are asymptotically synchronized by using adaptive feedback control method.According to Theorem 2.4, the origin of system 4.6 is asymptotically stable.

Numerical Results
By using Maple 13 to solve the systems of differential 4.1 , 4.2 , and 4.6 with the parameters are chosen to a 10, b 16 and c −1 in all simulations, so that the new system exhibits a chaotic behavior if no control is applied see Figure 1 .The initial states of the drive system are x 1 0 1.5, y 1 0 −2, and z 1 0 3.2, the initial values of the response system are x 2 0 1, y 2 0 −1, and z 2 0 2, and the initial value of the controller k 2 0 −1.When γ 1, the new system is driven to asymptotically synchronize as t → ∞ are shown in Figure 4.The feedback gain k 2 tends to a negative constant as shown in Figure 5.

Conclusions
In this paper, we present a simple adaptive feedback control method for chaos control and synchronization by modifying the previous method.Adaptive feedback control method is applied to control and synchronization of new chaotic dynamical system with known parameters.Numerical simulations are also given to validate the proposed synchronization approach.

Figure 1 :
Figure 1: The chaotic attractor of new dynamical system at a 10, b 16 and c −1.

1 Figure 3 :
Figure 3: The feedback gain k 1 tends to a negative constant as t → ∞.

e z ce z x 2 e x y 1 e y − k 2 L e 2 x be x e y − x 2 e x e z − z 1 e x e y k 2 e 2 x ae x e y − ae 2 y ce 2 z x 2 e x e z y 1 e y e z − k 2 e 2 x − Le 2 x − Le 2 x − ae 2 y ce 2 z a b − z 1 e x e y y 1 e y e z − Le 2 x z 1 ≤ − Le 2 x z 1
− a − b e x e y ae 2 y − ce 2 z − y 1 e y e z − a − b e x e y ae 2 y − ce 2 z − y 1 e y e z −e t Ae ≤ 0, 4.8 where e |e x |, |e y |, |e z | T is the states vector, and

Figure 4 : 7 Figure 5 :
Figure 4: The dynamics of synchronization errors states e x , e y , and e z of two identical new dynamical systems with adaptive feedback control.
5 which is uniformly exponentially stable about the origin x 3 0, z 3 0 for all x 3 , z 3 , then there exists a nonsingular coordinate transformation e x y 3 , e y x 3 , e z z 3 , which can make system 4.3 with new variable e has the form of system 2.2 , and the new system with controller u k 2 e k 2 e x , 0, 0 T is ėx be y − x 2 e z − z 1 e y k 2 e x , ėy a e x − e y , ėz ce z x 2 e x y 1 e y , x be y − x 2 e z − z 1 e y k 2 e x e y ae x − ae y