Chaos Control and Hybrid Projective Synchronization of a Novel Chaotic System

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 [1–8]. 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 [15–17], 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 [18–20], active control [21–23], adaptive control [24–26], 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:𝑥𝑥̇₂̇₁=𝑎𝑥₁−𝑥₂𝑥₃,𝑥̇₃=−𝑏𝑥₂+𝑥₁𝑥₃,=−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.

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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 as𝑢_𝑖=−𝑘𝑥_𝑖,𝑢_𝑗=0,𝑗≠𝑖,𝑖,𝑗=1,2,3,(2.1) where feedback gain 𝑘 is adapted according to the following update law:𝑘̇=𝑘₁𝑥^2𝑖,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=𝑢₃=0,𝑘̇=𝑘₁𝑥²¹,𝑘(0)=0,𝑘₁>0,(2.3) and the controlled system is considered as𝑥̇₁𝑥̇=₂𝑎𝑥₁−𝑥₂𝑥₃+𝑢₁,𝑥̇₃=−𝑏𝑥₂+𝑥₁𝑥₃,=−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 𝑉(𝑡)=1243𝑥²¹+𝑥²²+𝑥²³+12𝑘₁𝑘−𝑘^∗²,(2.5) where 𝑘^∗ is a sufficiently large constant to be determined. Then, it is positive definite and 𝑉̇(𝑡)=43𝑥¹𝑥̇₁+𝑥₂𝑥̇₂+𝑥₃𝑥̇₃+1=−𝑘_∗𝑘₁𝑘−𝑘^∗𝑘̇−43𝑎+31𝑘𝑥²¹−𝑏𝑥²²−𝑐𝑥²³=−𝑥^𝑇𝑃𝑥,(2.6) where 𝑥=⎡⎢⎢⎣𝑥𝑥¹₂𝑥₃⎤⎥⎥⎦,𝑃=⎡⎢⎢⎢⎣𝑘^∗−4003𝑎+𝑘300𝑏00𝑐⎤⎥⎥⎥⎦.(2.7) We can choose the undetermined 𝑘^∗>4𝑎/3−𝑘/3 so that the symmetric matrix 𝑃 is positive definite. Then, 𝑉̇(𝑡) is negative semidefinite since 𝑎>0,𝑏>0,𝑐>0. According to ^𝑡0𝜆_min(𝑃)‖𝑥‖²𝑑𝑡≤^𝑡0𝑥^𝑇𝑃𝑥𝑑𝑡=−^𝑡0𝑉̇(𝑡)𝑑𝑡≤𝑉(0),(2.8) where 𝜆_min(𝑃) is the smallest eigenvalue of 𝑃,𝑥 and 𝑑𝑥/𝑑𝑡 are both bounded. It follows that 𝑉̇(𝑡) is uniformly continuous. Based on Barbalat's Lemma, 𝑉̇(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 𝑘̇=𝑘₁𝑥²¹.

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 system𝑥̇=𝐴𝑥+𝐺(𝑥),(2.9) where 𝑥=(𝑥₁,𝑥₂,𝑥₃)^𝑇∈𝑅³ is the state vector, 𝐴∈𝑅^3×3 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 [1–8] 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 systems𝑦̇=𝑔𝑥̇(=𝑦)𝑓+(𝑢𝑥)(𝑥,,𝑦),((33..12)) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇,𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑅^𝑛 are state variables of the drive system (3.1) and the response system (3.2), respectively, 𝑓∶𝑅^𝑛→𝑅^𝑛 and 𝑔∶𝑅^𝑛→𝑅^𝑛 are nonlinear vectorial functions, 𝑢(𝑥,𝑦)=(𝑢₁(𝑥,𝑦),𝑢₂(𝑥,𝑦),…,𝑢_𝑛(𝑥,𝑦))^𝑇 is the nonlinear control vector. If there exists a nonzero constant matrix 𝛼=diag(𝛼₁,𝛼₂,…,𝛼_𝑛) such that lim_𝑡→∞|𝑦−𝛼𝑥|=0, namely, lim_𝑡→∞|𝑦_𝑖−𝛼_𝑖𝑥_𝑖|=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 𝛼_𝑖 are equal to 1 and the two chaotic systems are said to be in antisynchronization when all values of 𝛼_𝑖 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:𝑦𝑦̇₂̇₁=𝑎𝑦₁−𝑦₂𝑦₃+𝑢₁,𝑦̇₃=−𝑏𝑦₂+𝑦₁𝑦₃+𝑢₂,=−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 aslim_𝑡→∞||𝑦_𝑖−𝛼_𝑖𝑥_𝑖||=0,𝑖=1,2,3.(3.4) Define the state error vector as 𝑒=𝑦−𝛼𝑥, namely,𝑒𝑒₁₂=𝑦₁−𝛼₁𝑥₁,𝑒₃=𝑦₂−𝛼₂𝑥₂,=𝑦₃−𝛼₃𝑥₃,(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 as𝑒𝑒₂̇₁̇=𝑎𝑒₁−𝑒₂𝑒₃−𝛼₂𝑥₂𝑒₃−𝛼₃𝑥₃𝑒₂−𝛼₂𝛼₃−𝛼₁𝑥₂𝑥₃+𝑢₁,𝑒₃̇=−𝑏𝑒₂+𝑒₁𝑒₃+𝛼₁𝑥₁𝑒₃+𝛼₃𝑥₃𝑒₁+𝛼₁𝛼₃−𝛼₂𝑥₁𝑥₃+𝑢₂,=−𝑐𝑒₃+𝑒¹𝑒₂3+𝛼¹𝑥₁𝑒₂3+𝛼²𝑥₂𝑒₁3+𝛼¹𝛼₂−𝛼₃𝑥₁𝑥₂3+𝑢³.(3.6) The target is to find a controller such that the state errors satisfylim_𝑡→∞𝑒₁(𝑡)=0,lim_𝑡→∞𝑒₂(𝑡)=0,lim_𝑡→∞𝑒₃(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:𝑢𝑢₁₂=𝛼₂𝛼₃−𝛼₁𝑥₂𝑥₃,𝑢₃=−𝛼₁𝛼₃−𝛼₂𝑥₁𝑥₃,=−𝛼¹𝛼₂−𝛼₃𝑥₁𝑥₂3−2𝛼¹𝑥₁𝑒₂3−𝑒¹𝑒₃3−𝑘𝑒³,(3.8) where 𝑘 is the feedback gain and is adapted according to the following update law:𝑘̇=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 𝑉₁(𝑡)=1213𝑒²¹+13𝑒²²+𝑒²³+12𝑘₂𝑘−𝑘^∗².(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 𝑒𝑒₂̇₁̇=𝑎𝑒₁−𝑒₂𝑒₃−𝛼₂𝑥₂𝑒₃−𝛼₃𝑥₃𝑒₂,𝑒₃̇=−𝑏𝑒₂+𝑒₁𝑒₃+𝛼₁𝑥₁𝑒₃+𝛼₃𝑥₃𝑒₁,=−𝑐𝑒₃−𝛼¹𝑥₁𝑒₂3+𝛼²𝑥₂𝑒₁3−𝑘𝑒³.(3.11) The time derivative of 𝑉₁(𝑡) along the solution of error dynamical system (3.6) is as follows: 𝑉₁̇(𝑡)=13𝑒¹𝑒₁̇+13𝑒²𝑒₂̇+𝑒₃𝑒₃̇+1=−𝑘_∗𝑘₂𝑘−𝑘^∗𝑘̇−𝑘−13𝑎𝑒²¹−13𝑏𝑒²²−(𝑐+𝑘)𝑒²³=−𝑒^𝑇𝑃𝑒,(3.12) where 𝑒=⎡⎢⎢⎣𝑒𝑒¹₂𝑒₃⎤⎥⎥⎦,𝑃=⎡⎢⎢⎢⎣𝑘^∗−𝑘−𝑎030𝑏0003𝑐+0𝑘⎤⎥⎥⎥⎦.(3.13) The symmetric matrix 𝑃 should be positive definite when 𝑃 satisfies the following conditions 𝑘^∗−𝑘−𝑎𝑏𝑘3>0,_∗−𝑘−𝑎/3𝑏(𝑐+𝑘)𝑘3_∗>0,−𝑘−𝑎/33>0.(3.14) Since 𝑎>0,𝑏>0,𝑐>0,𝑘≥0, the symmetric matrix 𝑃 is positive when 𝑘^∗>𝑘+𝑎/3, then 𝑉₁̇(𝑡) is negative semidefinite. Based on Barbalat's Lemma, 𝑉₁̇(𝑡)→0 as 𝑡→∞. It follows that the error variables become zero as 𝑡 tends to infinity, namely, lim_𝑡→∞|𝑦_𝑖−𝛼_𝑖𝑥_𝑖|=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 𝛼_𝑖=1,𝑖=1,2,3, two chaotic systems are in complete synchronization, the controller can be simplified ulteriorly into 𝑢₁=0,𝑢₂=0,𝑢₃=−2𝛼₁𝑥₁𝑒₂/3−𝑒₁𝑒₃/3−𝑘𝑒₃, 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:𝑦𝑦̇₂̇₁=𝑎𝑦̂₁−𝑦₂𝑦₃+𝑢₁,𝑦̇₃=−𝑏̂𝑦₂+𝑦₁𝑦₃+𝑢₂,=−𝑐̂𝑦₃+𝑦¹𝑦₂3+𝑢³,(3.15) where 𝑢_𝑖(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,̂𝑐̂ 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: 𝑒𝑒₂̇₁̇=𝑎𝑒₁+𝑎𝛼₁𝑥₁+𝑎𝑒₁−𝑒₂𝑒₃−𝛼₂𝑥₂𝑒₃−𝛼₃𝑥₃𝑒₂−𝛼₂𝛼₃−𝛼₁𝑥₂𝑥₃+𝑢₁,𝑒₃̇=−𝑏𝑒₂−𝑏𝛼₂𝑥₂−𝑏𝑒₂+𝑒₁𝑒₃+𝛼₁𝑥₁𝑒₃+𝛼₃𝑥₃𝑒₁+𝛼₁𝛼₃−𝛼₂𝑥₁𝑥₃+𝑢₂,=−𝑐𝑒₃−𝑐𝛼₃𝑥₃−𝑐𝑒₃+𝑒¹𝑒₂3+𝛼¹𝑥₁𝑒₂3+𝛼²𝑥₂𝑒₁3+𝛼¹𝛼₂−𝛼₃𝑥₁𝑥₂3+𝑢³,(3.16) where 𝑎=𝑎̂−𝑎,𝑏=𝑏̂−𝑏,𝑐=𝑐̂−𝑐 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 𝑎̇‌=−𝑒²¹−𝛼₁𝑥₁𝑒₁̇3,𝑏=𝑒²²+𝛼₂𝑥₂𝑒₂̇3,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)=𝑉₁(𝑡)+12𝑎²+𝑏²+𝑐²=1213𝑒²¹+13𝑒²²+𝑒²³+12𝑎²+𝑏²+𝑐²+12𝑘₂𝑘−𝑘^∗²,(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 𝑉₂̇(𝑡)=13𝑒¹𝑒₁̇+13𝑒²𝑒₂̇+𝑒₃𝑒₃̇+𝑎𝑎̇+𝑏𝑏̇+𝑐𝑐̇+1=−𝑘_∗𝑘₂𝑘−𝑘^∗𝑘̇−𝑘−13𝑎𝑒²¹−13𝑏𝑒²²−(𝑐+𝑘)𝑒²³=−𝑒^𝑇𝑃𝑒,(3.19) where 𝑒,𝑃 are defined in (3.13). If the conditions in (3.14) hold, 𝑉₂̇(𝑡) is negative semidefinite, it follows that 𝑉₂̇(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.

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.

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.

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).

Figure 6 

The HPS errors 𝑒1,𝑒2,𝑒3 of system (3.16) with unknown parameters of the response system.

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).