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

Abstract

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.

1. 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 ,̇𝑥̇=𝑎𝑦−𝑥𝑦̇=𝑏𝑥−𝑥𝑧,𝑧=𝑥𝑦+𝑐𝑧, (1.1) 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).

Figure 1 

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

2. 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 𝑥̇=𝑓(𝑥), (2.1) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛,𝑓(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),…,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=𝑇𝑥, such that system (2.1) can be rewritten as 𝑧₁̇=𝑔₁𝑧₁,𝑧₂𝑧₂̇=𝑔₂𝑧₁,𝑧₂, (2.2) where 𝑧₁=(𝑦₁,𝑦₂,…,𝑦_𝑟)^𝑇∈𝑅^𝑟,𝑧₂=(𝑦_𝑟+1,𝑦_𝑟+2,…,𝑦_𝑛)^𝑇∈𝑅^𝑛−𝑟, the second equation satisfies 𝑧₂̇=𝑔₂(0,𝑧₂), with the vector function 𝑔₂(𝑧₁,𝑧₂) being smooth in a neighborhood of 𝑧₁=0, and the subsystem 𝑧₂̇=𝑔₂(0,𝑧₂) is uniformly exponentially stable about the origin 𝑧₂=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 (𝑥₁,𝑥₂,…,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 𝑔₂(𝑧₁,𝑧₂) being smooth in a neighborhood of 𝑧₁=0, that is, there is a positive constant 𝜆₀ (locally) such that ||𝑔₂(𝑧₁,𝑧₂)−𝑔₂(0,𝑧₂)||≤𝜆₀||𝑧₁||. And the subsystem 𝑧₂̇=𝑔₂(0,𝑧₂) is uniformly exponentially stable about the origin 𝑧₂=0 for all 𝑧, which implies that there are a Lyapunov function 𝑉₀(𝑧₂) and two positive numbers 𝜆₁,𝜆₂ such that 𝑉̇₀𝑧₂=𝜕𝑉₀𝑧₂𝜕𝑧₂𝑔₂0,𝑧₂≤−𝜆₁𝑧₂²,𝜕𝑉₀𝑧₂𝜕𝑧₂≤𝜆₂𝑧₂, (2.3) respectively. Since the system (2.1) is chaotic and 𝑔₂(𝑧₁,𝑧₂) is smooth function, there exists a positive number 𝜆₃, such that ‖𝑔₁(𝑧₁,𝑧₂)‖≤𝜆₃‖𝑧₁‖.

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: 𝑧₁̇=𝑔₁𝑧₁,𝑧₂+𝑢₁=𝑔₁𝑧₁,𝑧₂+𝑘₁𝑧₁,𝑧₂̇=𝑔₂𝑧₁,𝑧₂+𝑢₂=𝑔₂𝑧₁,𝑧₂, (2.4) where the controller 𝑢=(𝑢₁,𝑢₂)^𝑇=𝑘₁𝑧=(𝑘₁𝑧₁,0)^𝑇. The feedback gain 𝑘₁ is adapted according to the following update law: 𝑘₁̇=−𝛾𝑧₁², (2.5) 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 1𝑉=2𝑧^𝑇₁𝑧₁+𝑉₀𝑧₂+1𝑘2𝛾¹+𝐿², (2.6) where 𝐿=(𝜆₃+𝛼),𝛼≥𝜆²₀𝜆₁/4. 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),𝑘₁(𝑡))^𝑇 converge to (𝑥_𝑒,𝑘₀)^𝑇 as 𝑡→∞, where 𝑘₀ 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 𝑉̇=𝑧^𝑇₁𝑧₁̇+𝜕𝑉₀𝑧₂𝜕𝑧₂𝑔₂𝑧₁,𝑧₂+1𝛾𝑘₁+𝐿𝑘₁̇=𝑧^𝑇₁𝑔₁𝑧₁,𝑧₂+𝑘₁𝑧₁+𝜕𝑉₀𝑧₂𝜕𝑧₂𝑔₂𝑧₁,𝑧₂−𝑘₁+𝐿𝑧^𝑇₁𝑧₁=𝑧^𝑇₁𝑔₁𝑧₁,𝑧₂−𝐿𝑧^𝑇₁𝑧₁+𝜕𝑉₀𝑧₂𝜕𝑧₂𝑔₂𝑧₁,𝑧₂−𝑔0,𝑧₂+𝜕𝑉₀𝑧₂𝜕𝑧₂𝑔0,𝑧₂≤𝜆₃𝑧^𝑇₁𝑧₁−𝜆₃+𝛼𝑧^𝑇₁𝑧₁+𝜆₀𝜆₂𝑧₁𝑧₂−𝜆₁𝑧₁²≤−𝛼𝑧^𝑇₁𝑧₁+𝜆₀𝜆₂𝑧₁𝑧₂−𝜆₁𝑧₂²≤0. (2.7)
Obviously, 𝑉̇=0 if and only if 𝑧_𝑖=0, 𝑖=1,2, then the set 𝐸={(𝑧,𝑘₁)∣𝑉̇(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 𝑧₂. One the other hand, we stabilize the first subsystem 𝑧₁̇=𝑔₁(𝑧₁,𝑧₂) 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=(𝑦₁,𝑦₂,…,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.

3. 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:𝑥̇₁=𝑎𝑥₂−𝑥₁,𝑥̇₂=𝑏𝑥₁−𝑥₁𝑥₃,𝑥̇₃=𝑥₁𝑥₂+𝑐𝑥₃. (3.1)

It is easy to know the fact that if 𝑥₂=0 the following two dimensional subsystem of the system (3.1): 𝑥̇₁=−𝑎𝑥₁𝑥̇₃=𝑐𝑥₃, (3.2) which is uniformly exponentially stable about the origin 𝑥₁=0,𝑥₃=0 for all 𝑥₁,𝑥₃, then there exists a nonsingular coordinate transformation 𝑦=𝑇𝑥, that is, 𝑦₁=𝑥₂,𝑦₂=𝑥₁,𝑦₃=𝑥₃, which can make system (3.1) with new variable 𝑦 has the form of system (2.2), and the new system with controller 𝑢=𝑘₁𝑦=(𝑘₁𝑦₁,0,0)^𝑇 is 𝑦₁̇=𝑏𝑦₂−𝑦₂𝑦₃+𝑘₁𝑦₁,𝑦₂̇=𝑎𝑦₁−𝑦₂,𝑦₃̇=𝑦₁𝑦₂+𝑐𝑦₃,𝑘₁̇=−𝛾𝑦₁². (3.3)

Now, we define a Lyapunov function as 1𝑉=2𝑦²₁+𝑦²₂+𝑦²₃+1𝑘2𝛾¹+𝐿², (3.4) 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 get𝑑𝑉(𝑒)𝑑𝑡=𝑦₁𝑦₁̇+𝑦₂𝑦₂̇+𝑦₃𝑦₃̇+1𝛾𝑘₁+𝐿𝐾₁̇=𝑦₁𝑏𝑦₂−𝑦₂𝑦₃+𝑘₁𝑦₁+𝑦₂𝑎𝑦₁−𝑎𝑦₂+𝑦₃𝑦₁𝑦₂+𝑐𝑦₃−𝑘₁+𝐿𝑦²₁=𝑏𝑦₂𝑦₁−𝑦₁𝑦₂𝑦₃+𝑘₁𝑦²₁+𝑎𝑦₁𝑦₂−𝑎𝑦²₂+𝑦₃𝑦₁𝑦₂+𝑐𝑦²₃−𝑘₁𝑦²₁−𝐿𝑦²₁=(𝑎+𝑏)𝑦₂𝑦₁−𝑎𝑦²₂+𝑐𝑦²₃−𝐿𝑦²₁=−𝑎𝑦²₂−(𝑎+𝑏)𝑦₂𝑦₁+𝐿𝑦²₁+𝑐𝑦²₃≤−𝑎𝑦²₂−(𝑎+𝑏)𝑦₂𝑦₁+𝐿𝑦²₁+𝑐𝑦²₃≤−𝑒^𝑇𝑃𝑒<0, (3.5) where 𝑒=[|𝑦₁|,|𝑦₂|,|𝑦₃|]^𝑇 is the states vector, and 𝐿𝑃=𝑎+𝑏20𝑎+𝑏2𝑎000𝑐. (3.6)

Obviously, to ensure that the origin of the system (3.1) is asymptotically stable, the matrix 𝑃 should be positive definite, which implies that 𝑉̇ 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.

3.1. 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 𝑦₁(0)=1.5,𝑦₂(0)=−2, and 𝑦₃(0)=3.2and the initial value of the controller 𝑘₁(0)=−1. 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 𝑘₁ tends to a negative constant as shown in Figure 3.

Figure 2 

The new dynamical system (3.1) is driven to its stable equilibrium (0,0,0) asymptotically as 𝑡→∞.

Figure 3 

The feedback gain 𝑘1 tends to a negative constant as 𝑡→∞.

4. 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, 𝑥̇₁=𝑎𝑦₁−𝑥₁,𝑦₁̇=𝑏𝑥₁−𝑥₁𝑧₁,𝑧₁̇=𝑥₁𝑦₁+𝑐𝑧₁,𝑥̇₂=𝑎𝑦₂−𝑥₂𝑦₂̇=𝑏𝑥₂−𝑥₂𝑧₂𝑧₂̇=𝑥₂𝑦₂+𝑐𝑧₂. (4.1)(4.2)

For this purpose, the error dynamical system between the drive system (4.1) and response system (4.2) can be expressed by 𝑥̇₃=𝑎𝑦₃−𝑥₃𝑦₃̇=𝑏𝑥₃−𝑥₂𝑧₃−𝑧₁𝑥₃𝑧₃̇=𝑐𝑧₃+𝑥₂𝑦₃+𝑦₁𝑥₃, (4.3) where 𝑥₃=𝑥₂−𝑥₁,𝑦₃=𝑦₂−𝑦₁,𝑧₃=𝑧₂−𝑧₁.

In order that two chaotic systems can be synchronized in the sense of PS, the following condition should be satisfied: lim_𝑡→∞𝑥₂−𝑥₁=lim_𝑡→∞𝑦₂−𝑦₁=lim_𝑡→∞𝑧₂−𝑧₁=0. (4.4)

It is easy to know the fact that if 𝑦₃=0 the following two-dimensional subsystem of system (4.3): 𝑥̇₃=−𝑎𝑥₃,𝑧₃̇=𝑐𝑧₃+𝑦₁𝑥₃, (4.5) which is uniformly exponentially stable about the origin 𝑥₃=0,𝑧₃=0 for all 𝑥₃,𝑧₃, then there exists a nonsingular coordinate transformation 𝑒_𝑥=𝑦₃,𝑒_𝑦=𝑥₃,𝑒_𝑧=𝑧₃, which can make system (4.3) with new variable 𝑒 has the form of system (2.2), and the new system with controller 𝑢=𝑘₂𝑒=(𝑘₂𝑒_𝑥,0,0)^𝑇 is 𝑒_𝑥̇=𝑏𝑒_𝑦−𝑥₂𝑒_𝑧−𝑧₁𝑒_𝑦+𝑘₂𝑒_𝑥,𝑒_𝑦̇=𝑎𝑒_𝑥−𝑒_𝑦,𝑒_𝑧̇=𝑐𝑒_𝑧+𝑥₂𝑒_𝑥+𝑦₁𝑒_𝑦,𝑘₂̇=−𝛾𝑒_𝑥². (4.6)

Let us consider the Lyapunov function 𝑉(𝑒) which is defined by 𝑉(𝑒)=12𝑒²_𝑥+𝑒²_𝑦+𝑒²_𝑧+1𝛾𝑘₂+𝐿², (4.7) 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 (4.7), then we get 𝑑𝑉(𝑒)𝑑𝑡=𝑒_𝑥𝑒_𝑥̇+𝑒_𝑦𝑒_𝑦̇+𝑒_𝑧𝑒_𝑧̇+1𝛾𝑘₂+𝐿𝑘̇₂=𝑒_𝑥𝑏𝑒_𝑦−𝑥₂𝑒_𝑧−𝑧₁𝑒_𝑦+𝑘₂𝑒_𝑥+𝑒_𝑦𝑎𝑒_𝑥−𝑎𝑒_𝑦+𝑒_𝑧𝑐𝑒_𝑧+𝑥₂𝑒_𝑥+𝑦₁𝑒_𝑦−𝑘₂+𝐿𝑒²_𝑥=𝑏𝑒_𝑥𝑒_𝑦−𝑥₂𝑒_𝑥𝑒_𝑧−𝑧₁𝑒_𝑥𝑒_𝑦+𝑘₂𝑒²_𝑥+𝑎𝑒_𝑥𝑒_𝑦−𝑎𝑒²_𝑦+𝑐𝑒²_𝑧+𝑥₂𝑒_𝑥𝑒_𝑧+𝑦₁𝑒_𝑦𝑒_𝑧−𝑘₂𝑒²_𝑥−𝐿𝑒²_𝑥=−𝐿𝑒²_𝑥−𝑎𝑒²_𝑦+𝑐𝑒²_𝑧+𝑎+𝑏−𝑧₁𝑒_𝑥𝑒_𝑦+𝑦₁𝑒_𝑦𝑒_𝑧=−𝐿𝑒²_𝑥+𝑧₁−𝑎−𝑏𝑒_𝑥𝑒_𝑦+𝑎𝑒²_𝑦−𝑐𝑒²_𝑧−𝑦₁𝑒_𝑦𝑒_𝑧≤−𝐿𝑒²_𝑥+𝑧₁−𝑎−𝑏𝑒_𝑥𝑒_𝑦+𝑎𝑒²_𝑦−𝑐𝑒²_𝑧−𝑦₁𝑒_𝑦𝑒_𝑧=−𝑒^𝑡𝐴𝑒≤0, (4.8) where 𝑒=[|𝑒_𝑥|,|𝑒_𝑦|,|𝑒_𝑧|]^𝑇 is the states vector, and 𝐴=−𝐿𝑎+𝑏−𝑧₁20𝑎+𝑏−𝑧₁2𝑦−𝑎−¹2𝑦0−¹2𝑐, (4.9) under the condition 𝐿>(𝑧₁−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.

4.1. 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 𝑥₁(0)=1.5,𝑦₁(0)=−2, and 𝑧₁(0)=3.2, the initial values of the response system are 𝑥₂(0)=1,𝑦₂(0)=−1, and 𝑧₂(0)=2, and the initial value of the controller 𝑘₂(0)=−1. When 𝛾=1, the new system is driven to asymptotically synchronize as 𝑡→∞ are shown in Figure 4. The feedback gain 𝑘₂ tends to a negative constant as shown in Figure 5.

Figure 4 

The dynamics of synchronization errors states 𝑒𝑥,𝑒𝑦,and𝑒𝑧 of two identical new dynamical systems with adaptive feedback control.

Figure 5 

The feedback gain 𝑘2 tends to a negative constant as 𝑡→∞.

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.

Acknowledgment

The authors would like to thank the Editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.