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Mathematical Problems in Engineering
Volume 2011, Article ID 452671, 13 pages
http://dx.doi.org/10.1155/2011/452671
Research Article

Chaos Control and Hybrid Projective Synchronization of a Novel Chaotic System

Tao Wang,1,2 Kejun Wang,2 and Nuo Jia1,2

1College of Mathematics Science, Harbin Normal University, Harbin 150025, China
2College of Automation, Harbin Engineering University, Harbin 150001, China

Received 22 August 2010; Revised 8 February 2011; Accepted 14 March 2011

Academic Editor: Massimo Scalia

Copyright © 2011 Tao Wang 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

Adaptive feedback controllers based on Lyapunov's direct method for chaos control and hybrid projective synchronization (HPS) of a novel 3D chaotic system are proposed. Especially, the controller can be simplified ulteriorly into a single scalar one to achieve complete synchronization. The HPS between two nearly identical chaotic systems with unknown parameters is also studied, and adaptive parameter update laws are developed. Numerical simulations are demonstrated to verify the effectiveness of the control strategies.

1. Introduction

Since chaotic attractors were found by Lorentz in 1963, many chaotic systems have been constructed, such as the Lorentz system, Chen system, and Lü system [18]. Because of the potential applications in engineering, the study of chaotic systems has attracted more and more researchers' attention. Chaos control and synchronization of chaotic systems have been interesting research fields since the pioneering work of Ott, Grebogi, and Yorke and the seminal work of Pecora and Carroll, which are simultaneously reported in 1990. Ever since, various types of synchronization phenomena have been found such as complete synchronization [9, 10], phase synchronization [11, 12], partial synchronization [13], generalized synchronization [14], projective synchronization [1517], and so forth. They are applied in many fields, such as secure communication, neural networks, optimization of nonlinear system performance, ecological systems, modeling brain activity, system identification and pattern recognition, and so on.

Recently, hybrid projective synchronization (HPS) was proposed. It can be considered as an extension of projective synchronization because complete synchronization and anti-synchronization are both its special cases. It is worthy of study because the response signals can be any proportional to the drive signals by adjusting the factors and it can be used to extend binary digital to variety M-nary digital communications for achieving fast communication. However, the controllers based on different control methods in the existing literatures, such as nonlinear feedback control [1820], active control [2123], adaptive control [2426], and so forth, are mostly vectorial and they are difficult to be put into practice. So, the controllers which are simple, efficient, and easy to implement are required to be designed for both chaos control and HPS between two chaotic systems.

Chen-Lee system [27] is a new 3D chaotic system which was proposed by Chen and Lee. It takes the following form:̇𝑥1=𝑎𝑥1𝑥2𝑥3,̇𝑥2=𝑏𝑥2+𝑥1𝑥3,̇𝑥3=𝑐𝑥3+𝑥1𝑥23,(1.1) where 𝑥1,𝑥2,𝑥3 are state variables, 𝑎,𝑏,𝑐 are positive constant parameters, and 0<𝑎<𝑏+𝑐 to guarantee the system to generate chaos. System (1.1) is symmetrical about three coordinate axes, 𝑥1,𝑥2,𝑥3, respectively, and these symmetries persist for all values of the system parameters. This chaotic system is robust to various small perturbations due to its highly symmetric structure, and it is dissipative. Its chaotic attractor is shown in Figure 1 for 𝑎=5,𝑏=10, and 𝑐=3.8. Due to the fact that the new system has five equilibrium points, some larger chaotic regions, and more complex bifurcation behaviors compared with Lorenz system, Chen system [1], and Lü system [2], it may have good application prospects. So we study chaos control and HPS of the new chaotic system motivated by the idea of designing simple and efficient controller for application. Firstly, the system converges to its unstable equilibrium point by designing an adaptive linear feedback controller which only includes single-state variable. Moreover, the control method proposed is generalized to a class of chaotic systems. Secondly, HPS between two identical systems is studied based on Lyapunov's direct method. An adaptive nonlinear feedback vectorial controller is derived to guarantee HPS, which can degenerate into a single scalar one in the case of complete synchronization. Furthermore, the problem of adaptive hybrid projective synchronization between two nearly identical novel chaotic systems with unknown parameters is also studied, and adaptive parameter update laws are developed. Finally, numerical simulation results illustrate the effectiveness of the proposed control strategies. The approaches in our paper have certain significance for reducing the cost and complexity for controller implementation.

fig1
Figure 1: The chaotic attractor of system (1.1) with 𝑎=5,𝑏=10,𝑐=3.8.

2. Chaos Control of the Novel Chaotic System

In this section, chaotic system (1.1) will be controlled to its unstable equilibrium point 𝑂(0,0,0) via an adaptive linear feedback controller which only includes one-state variable. The controller can be designed as𝑢𝑖=𝑘𝑥𝑖,𝑢𝑗=0,𝑗𝑖,𝑖,𝑗=1,2,3,(2.1) where feedback gain 𝑘 is adapted according to the following update law:̇𝑘=𝑘1𝑥2𝑖,𝑖=1,2,3,𝑘(0)=0,𝑘1>0.(2.2) According to (2.1) and (2.2), the controller associated with adaptive update law can be chosen as𝑢1=𝑘𝑥1,𝑢2=𝑢3̇=0,𝑘=𝑘1𝑥21,𝑘(0)=0,𝑘1>0,(2.3) and the controlled system is considered aṡ𝑥1=𝑎𝑥1𝑥2𝑥3+𝑢1,̇𝑥2=𝑏𝑥2+𝑥1𝑥3,̇𝑥3=𝑐𝑥3+𝑥1𝑥23.(2.4) Then, we have the following theorem on stabilizing the origin of system.

Theorem 2.1. The controlled chaotic system (2.4) will globally and asymptotically converge to the unstable equilibrium point 𝑂(0,0,0) under the controller with the update law (2.3).

Proof. Introducing a candidate Lyapunov function as 1𝑉(𝑡)=243𝑥21+𝑥22+𝑥23+12𝑘1𝑘𝑘2,(2.5) where 𝑘 is a sufficiently large constant to be determined. Then, it is positive definite and ̇4𝑉(𝑡)=3𝑥1̇𝑥1+𝑥2̇𝑥2+𝑥3̇𝑥3+1𝑘1𝑘𝑘̇𝑘𝑘=431𝑎+3𝑘𝑥21𝑏𝑥22𝑐𝑥23=𝑥𝑇𝑃𝑥,(2.6) where 𝑥𝑥=1𝑥2𝑥3𝑘,𝑃=4𝑎3+𝑘3000𝑏000𝑐.(2.7) We can choose the undetermined 𝑘>4𝑎/3𝑘/3 so that the symmetric matrix 𝑃 is positive definite. Then, ̇𝑉(𝑡) is negative semidefinite since 𝑎>0,𝑏>0,𝑐>0. According to 𝑡0𝜆min(𝑃)𝑥2𝑑𝑡𝑡0𝑥𝑇𝑃𝑥𝑑𝑡=𝑡0̇𝑉(𝑡)𝑑𝑡𝑉(0),(2.8) where 𝜆min(𝑃) is the smallest eigenvalue of 𝑃,𝑥 and 𝑑𝑥/𝑑𝑡 are both bounded. It follows that ̇𝑉(𝑡) is uniformly continuous. Based on Barbalat's Lemma, ̇𝑉(𝑡)0 as 𝑡. So, the system (1.1) converges to 𝑂(0,0,0) as 𝑡 tends to infinity under the controller with the update law (2.3). This completes the proof.

Numerical simulation demonstrates the performance of the system controlled by the proposed method. The system parameters are chosen to be 𝑎=5,𝑏=10, and 𝑐=3.8 so that the system (1.1) has a chaotic attractor. The initial conditions are set to be 𝑥1(0)=1,𝑥2(0)=2,𝑥3(0)=3. The initial condition of the adaptive feedback gain is set to be 𝑘(0)=0, and the constant coefficient 𝑘1 is set to be 20. Figure 2 shows the time responses of states 𝑥1,𝑥2,𝑥3 for the controlled system (2.4). It is observed that chaotic system is suppressed to its unstable equilibrium point 𝑂(0,0,0) under the single scalar controller 𝑢1=𝑘𝑥1 with the feedback gain adaptive update law ̇𝑘=𝑘1𝑥21.

fig2
Figure 2: State trajectories of the controlled chaotic system (2.4) when 𝑥1(0)=1,𝑥2(0)=1,𝑥3(0)=1,𝑘(0)=0, and 𝑘1=20.

The single scalar adaptive control strategy (2.1) with adaptive update law (2.2) proposed above can be applied to a class of general 3D chaotic systeṁ𝑥=𝐴𝑥+𝐺(𝑥),(2.9) where 𝑥=(𝑥1,𝑥2,𝑥3)𝑇𝑅3 is the state vector, 𝐴𝑅3×3 is a constant matrix, and 𝐺(𝑥)=(𝑔1(𝑥),𝑔2(𝑥),𝑔3(𝑥))𝑇 is a vector nonlinear term which satisfies𝑎1𝑥1𝑔1(𝑥)+𝑎2𝑥2𝑔2(𝑥)+𝑎3𝑥3𝑔3(𝑥)=0,(2.10) where 𝑎1,𝑎2,𝑎3 are constants. For example, the well-known Lorentz system, Chen system, Lü system, and so on [18] all have the forms and they all can be controlled by single scalar adaptive control method.

3. Hybrid Projective Synchronization

3.1. Hybrid Projective Synchronization by Adaptive Feedback Control Law

For two dynamical systemṡ𝑥=𝑓(𝑥),(3.1)̇𝑦=𝑔(𝑦)+𝑢(𝑥,𝑦),(3.2) where 𝑥=(𝑥1,𝑥2,,𝑥𝑛)𝑇,𝑦=(𝑦1,𝑦2,,𝑦𝑛)𝑇𝑅𝑛 are state variables of the drive system (3.1) and the response system (3.2), respectively, 𝑓𝑅𝑛𝑅𝑛 and 𝑔𝑅𝑛𝑅𝑛 are nonlinear vectorial functions, 𝑢(𝑥,𝑦)=(𝑢1(𝑥,𝑦),𝑢2(𝑥,𝑦),,𝑢𝑛(𝑥,𝑦))𝑇 is the nonlinear control vector. If there exists a nonzero constant matrix 𝛼=diag(𝛼1,𝛼2,,𝛼𝑛) such that lim𝑡|𝑦𝛼𝑥|=0, namely, lim𝑡|𝑦𝑖𝛼𝑖𝑥𝑖|=0,(𝑖=1,2,,𝑛), then the response system and the drive system are said to be in HPS. In particular, the drive-response systems achieve complete synchronization when all values of 𝛼𝑖 are equal to 1 and the two chaotic systems are said to be in antisynchronization when all values of 𝛼𝑖 are equal to −1.

In this section, we study the hybrid projection synchronization of two identical chaotic systems. The response system corresponding to the drive system (1.1) is defined as follows:̇𝑦1=𝑎𝑦1𝑦2𝑦3+𝑢1,̇𝑦2=𝑏𝑦2+𝑦1𝑦3+𝑢2,̇𝑦3=𝑐𝑦3+𝑦1𝑦23+𝑢3,(3.3) where (𝑢1,𝑢2,𝑢3)𝑇 is the nonlinear control vector. System (1.1) and system (3.3) are in HPS as long aslim𝑡||𝑦𝑖𝛼𝑖𝑥𝑖||=0,𝑖=1,2,3.(3.4) Define the state error vector as 𝑒=𝑦𝛼𝑥, namely,𝑒1=𝑦1𝛼1𝑥1,𝑒2=𝑦2𝛼2𝑥2,𝑒3=𝑦3𝛼3𝑥3,(3.5) where 𝛼=diag(𝛼1,𝛼2,𝛼3) and 𝛼1,𝛼2,𝛼3 are different, desired scaling factors for HPS. The error dynamical system between system (1.1) and system (3.3) can be written aṡ𝑒1=𝑎𝑒1𝑒2𝑒3𝛼2𝑥2𝑒3𝛼3𝑥3𝑒2𝛼2𝛼3𝛼1𝑥2𝑥3+𝑢1,̇𝑒2=𝑏𝑒2+𝑒1𝑒3+𝛼1𝑥1𝑒3+𝛼3𝑥3𝑒1+𝛼1𝛼3𝛼2𝑥1𝑥3+𝑢2,̇𝑒3=𝑐𝑒3+𝑒1𝑒23+𝛼1𝑥1𝑒23+𝛼2𝑥2𝑒13+𝛼1𝛼2𝛼3𝑥1𝑥23+𝑢3.(3.6) The target is to find a controller such that the state errors satisfylim𝑡𝑒1(𝑡)=0,lim𝑡𝑒2(𝑡)=0,lim𝑡𝑒3(𝑡)=0,(3.7) then the global and asymptotical stability of system (3.6) means system (1.1) and (3.3) are in HPS. Choose the control functions 𝑢1,𝑢2, and 𝑢3 as follows:𝑢1=𝛼2𝛼3𝛼1𝑥2𝑥3,𝑢2𝛼=1𝛼3𝛼2𝑥1𝑥3,𝑢3𝛼=1𝛼2𝛼3𝑥1𝑥232𝛼1𝑥1𝑒23𝑒1𝑒33𝑘𝑒3,(3.8) where 𝑘 is the feedback gain and is adapted according to the following update law:̇𝑘=𝑘2𝑒21,𝑘(0)=0,𝑘2>0.(3.9) We have the following result.

Theorem 3.1. For any initial conditions, the drive system (1.1) and the response system (3.3) are globally and asymptotically hybrid projective synchronized by nonlinear feedback controller (3.8) with the update law (3.9).

Proof. Construct a candidate Lyapunov function 𝑉11(𝑡)=213𝑒21+13𝑒22+𝑒23+12𝑘2𝑘𝑘2.(3.10) It is clear that 𝑉1(𝑡) is a positive definite function. By applying the controller (3.8) to (3.6), the error dynamics can be written as ̇𝑒1=𝑎𝑒1𝑒2𝑒3𝛼2𝑥2𝑒3𝛼3𝑥3𝑒2,̇𝑒2=𝑏𝑒2+𝑒1𝑒3+𝛼1𝑥1𝑒3+𝛼3𝑥3𝑒1,̇𝑒3=𝑐𝑒3𝛼1𝑥1𝑒23+𝛼2𝑥2𝑒13𝑘𝑒3.(3.11) The time derivative of 𝑉1(𝑡) along the solution of error dynamical system (3.6) is as follows: ̇𝑉11(𝑡)=3𝑒1̇𝑒1+13𝑒2̇𝑒2+𝑒3̇𝑒3+1𝑘2𝑘𝑘̇𝑘𝑘=1𝑘3𝑎𝑒2113𝑏𝑒22(𝑐+𝑘)𝑒23=𝑒𝑇𝑃𝑒,(3.12) where 𝑒𝑒=1𝑒2𝑒3𝑘,𝑃=𝑎𝑘30𝑏003000𝑐+𝑘.(3.13) The symmetric matrix 𝑃 should be positive definite when 𝑃 satisfies the following conditions 𝑘𝑎𝑘3𝑏𝑘>0,𝑘𝑎/33𝑘>0,𝑏(𝑐+𝑘)𝑘𝑎/33>0.(3.14) Since 𝑎>0,𝑏>0,𝑐>0,𝑘0, the symmetric matrix 𝑃 is positive when 𝑘>𝑘+𝑎/3, then ̇𝑉1(𝑡) is negative semidefinite. Based on Barbalat's Lemma, ̇𝑉1(𝑡)0 as 𝑡. It follows that the error variables become zero as 𝑡 tends to infinity, namely, lim𝑡|𝑦𝑖𝛼𝑖𝑥𝑖|=0,𝑖=1,2,3. This means that the two chaotic systems (1.1) and (3.3) are in HPS under the controller (3.8).

Remark 3.2. In case of 𝛼𝑖=1,𝑖=1,2,3, two chaotic systems are in complete synchronization, the controller can be simplified ulteriorly into 𝑢1=0,𝑢2=0,𝑢3=2𝛼1𝑥1𝑒2/3𝑒1𝑒3/3𝑘𝑒3, namely, it can be simplified to a single scalar form.

3.2. Adaptive Hybrid Projective Synchronization with Unknown Parameters of the Response System

In practical applications, the response system parameters are partially or entirely unknown in advance. Therefore, it is necessary to investigate the synchronization problem of chaotic systems with unknown system parameters. In the following, we adopt the adaptive control laws (3.8) to drive two nearly identical chaotic systems with the unknown response system parameters and different initial conditions in HPS. The drive system is designed as (1.1) and the response system is modeled as follows:̇𝑦1=̂𝑎𝑦1𝑦2𝑦3+𝑢1,̇𝑦2̂=𝑏𝑦2+𝑦1𝑦3+𝑢2,̇𝑦3=̂𝑐𝑦3+𝑦1𝑦23+𝑢3,(3.15) where 𝑢𝑖(𝑡)(𝑖=1,2,3) are the controllers to be designed. The drive system parameters 𝑎,𝑏,𝑐 are known, but the response system parameters ̂̂𝑎,𝑏,̂𝑐 which need to be identified are unknown. Define the error vector as (3.5), then the error dynamical system with unknown parameters is as follows: ̇𝑒1=𝑎𝑒1+𝑎𝛼1𝑥1+𝑎𝑒1𝑒2𝑒3𝛼2𝑥2𝑒3𝛼3𝑥3𝑒2𝛼2𝛼3𝛼1𝑥2𝑥3+𝑢1,̇𝑒2=𝑏𝑒2𝑏𝛼2𝑥2𝑏𝑒2+𝑒1𝑒3+𝛼1𝑥1𝑒3+𝛼3𝑥3𝑒1+𝛼1𝛼3𝛼2𝑥1𝑥3+𝑢2,̇𝑒3=𝑐𝑒3𝑐𝛼3𝑥3𝑐𝑒3+𝑒1𝑒23+𝛼1𝑥1𝑒23+𝛼2𝑥2𝑒13+𝛼1𝛼2𝛼3𝑥1𝑥23+𝑢3,(3.16) where 𝑎=̂𝑎𝑎,̂𝑏=𝑏𝑏,𝑐=̂𝑐𝑐 are the unknown error system parameters.

Theorem 3.3. For the given drive system (1.1), the response system (3.15), and the corresponding error dynamical system (3.16), if the adaptive control law (3.8) associated with (3.9) is applied to system (3.15) and estimated update laws of the response system parameters satisfy ̇𝑎=𝑒21𝛼1𝑥1𝑒13,̇𝑒𝑏=22+𝛼2𝑥2𝑒23,̇𝑐=𝑒23+𝛼3𝑥3𝑒3,(3.17) then the trivial solution of the error dynamical system (3.16) is globally and asymptotically stable so that systems (1.1) and (3.15) achieve HPS.

Proof. Construct a candidate Lyapunov function 𝑉2(𝑡)=𝑉11(𝑡)+2𝑎2+𝑏2+𝑐2=1213𝑒21+13𝑒22+𝑒23+12𝑎2+𝑏2+𝑐2+12𝑘2𝑘𝑘2,(3.18) where 𝑉1(𝑡) has been defined in (3.10). Taking the time derivative of 𝑉2(𝑡) along the solution of error system (3.16) and applying the control law (3.8) associated with (3.9) and parameter estimated update laws (3.17), we have ̇𝑉21(𝑡)=3𝑒1̇𝑒1+13𝑒2̇𝑒2+𝑒3̇𝑒3+𝑎̇𝑎+𝑏̇𝑏+𝑐̇1𝑐+𝑘2𝑘𝑘̇𝑘𝑘=1𝑘3𝑎𝑒2113𝑏𝑒22(𝑐+𝑘)𝑒23=𝑒𝑇𝑃𝑒,(3.19) where 𝑒,𝑃 are defined in (3.13). If the conditions in (3.14) hold, ̇𝑉2(𝑡) is negative semidefinite, it follows that ̇𝑉2(𝑡)0 as 𝑡 according to Barbalat's Lemma. So, the equilibrium point 𝑒1=0,𝑒2=0,𝑒3=0,𝑘=𝑘,𝑎=0,𝑏=0,𝑐=0 is globally and asymptotically stable. Thus, the two chaotic systems with unknown parameters are in HPS. This completes the proof.

3.3. Numerical Simulations

To verify and demonstrate the effectiveness of the proposed adaptive control laws for HPS, we will display the numerical simulation results. Let the system parameters be 𝑎=5,𝑏=10, and 𝑐=3.8, the initial states of the drive system and the response system 𝑥1(0)=1,𝑥2(0)=2, and 𝑥3(0)=3 and 𝑦1(0)=5,𝑦2(0)=7, and 𝑦3(0)=5, respectively, the initial condition of the adaptive feedback gain 𝑘(0)=0, the constant coefficient 𝑘2=1, and the synchronization factors be 𝛼1=1,𝛼2=3, and 𝛼3=1. Fourth-order Runge-Kutta method is used to solve the system with time step size 0.001.

(1) The HPS errors between the two identical chaotic systems (1.1) and (3.3) are displayed in Figure 3. From these curves, we can see that each error converges to 0. For further observations, the state trajectories of the two systems are depicted in Figure 4. It is shown that variables 𝑥1 and 𝑦1 display a synchronization phenomenon, 𝑦2 finally converges to three times the value of 𝑥2, 𝑥3 and 𝑦3 show antisynchronization behavior. It is clear that the two chaotic systems achieve HPS. Figure 5 shows the curve of adaptive feedback control gain 𝑘 with time history.

fig3
Figure 3: The HPS errors 𝑒1,𝑒2,𝑒3 between the two identical chaotic systems (1.1) and (3.3) with 𝛼1=1,𝛼2=3, and 𝛼3=1.
fig4
Figure 4: State trajectories of the drive system and the response system, 𝑥1, and 𝑦1 with 𝛼1=1, 𝑥2, and 𝑦2 with 𝛼2=3, 𝑥3, and 𝑦3 with 𝛼3=1.
452671.fig.005
Figure 5: Adaptive feedback control gain 𝑘 with time history.

(2) Let the initial parameter errors of system (3.16) be 𝑎(0)=0,𝑏(0)=0,𝑐(0)=0. The evolution of the synchronized errors with unknown parameters of the response system is illustrated in Figure 6. It is shown that synchronized errors between the two chaotic systems converge to zero asymptotically. It means the two nearly identical chaotic systems are in HPS. Figure 7 illustrates the estimated curves of parameters of the response system (3.15) under the update law (3.17).

fig6
Figure 6: The HPS errors 𝑒1,𝑒2,𝑒3 of system (3.16) with unknown parameters of the response system.
452671.fig.007
Figure 7: Estimated curves of parameters of the response system (3.15) under the update law (3.17) with 𝑎(0)=0,𝑏(0)=0, and 𝑐(0)=0.

4. Conclusion

Chaos control and hybrid projective synchronization between two chaotic systems are addressed. The single scalar adaptive feedback control method is proposed and numerical simulation results are demonstrated to verify the effectiveness and efficiency of the control strategies. Based on the control method, single scalar adaptive feedback controllers overcome the shortcomings of the controllers in the existing literature. They have less parameters, simpler form, and are easier to implement compared with other controllers. They are valuable to be applied to the realization in engineering.

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable suggestions and comments which have led to the improvement of this paper. They also thanks the editor for his(her) kind consideration. This research was supported by the Scientific Research Fund of Heilongjiang Provincial Education Department of China (no. 11551140).

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