Generation and Modified Projective Synchronization for a Class of New Hyperchaotic Systems

Abstract

A class of new hyperchaotic systems with different nonlinear terms is proposed, and the existence of hyperchaos is exhibited by calculating their Lyapunov exponent spectrums. Then the universal theories on modified projective synchronization (MPS) of the systems with general form which linearly depends on unknown parameters or time-varying parameters, are investigated by presenting an adaptive control strategy together with parameter update laws and a nonlinear control scheme based on Lyapunov stability theory. Subsequently, the presented control methods are applied to achieve MPS of the new hyperchaotic systems, and their effectiveness is illustrated by numerical simulations.

1. Introduction

Since the pioneer work by Pecora and Corroll in 1990 [1], chaos synchronization, which refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy) [2], has become an active research subject for its extensive potential application in physics, secure communication, chemical reactor, biological networks, and so on. Up to now, many different types of synchronization have been presented such as complete synchronization [1], phase synchronization [3], lag synchronization [4], generalized synchronization [5], and projective synchronization [6]. Among them, projective synchronization is the most noticeable one with the essence that the drive and response systems could be synchronized up to a scaling factor 𝛼 (a proportional relation), because it has some topological invariants, such as Lyapunov exponents (LEs) and fractional dimensions which is understood well, and it could be used to extend binary digital to variety M-ary digital communications for getting more secure and faster communications. By the way, generalized projective synchronization is its extension in general classes of chaotic systems including nonpartially linear systems [7]. Recently, a new synchronization termed as modified projective synchronization (MPS) [8] was presented, of which the different scaling factors in a scaling matrix 𝐻 can be arbitrarily designed to different state variables. It can be seen that MPS encompasses the complete synchronization, antisynchronization, and projective synchronization when scaling matrix 𝐻 equals to 𝐼, −𝐼, and 𝛼𝐼 (𝛼 is a constant), respectively. Consequently, it has more broad prospect in practical applications.

A lot of work has been done around these different chaos synchronization phenomena, which can be summarized to two aspects as generation of chaotic systems and synchronization schemes to achieve chaos synchronization. On the one hand, since Rössler first introduced the hyperchaotic dynamical system in 1979 [9], some hyperchaotic systems are constructed by adding state feedback to 3D chaotic systems such as Lorenz system, Chen system, and Lü system, and have been investigated to some degree [10–13]. In comparison with low-dimensional chaotic system, hyperchaotic system with higher than or equal to four dimension has two or more positive Lyapunov exponents, richer and more complex dynamical behaviors which appears in more directional separation of phase orbits, and much wider application. So, from a practical point of view, some scholars devote to applying hyperchaotic system to generate more unpredictable and noise-like chaotic signals and considering synchronization between two hyperchaotic systems or between chaotic system and hyperchaotic system [14–20]. However, it is still an interesting task to derive 4D or higher-dimensional hyperchaotic systems.

On the other hand, various synchronization schemes have been proposed such as linear and nonlinear feedback synchronization method [21–24], adaptive synchronization method [25–31], time-delay feedback method [32, 33], backstepping control method [34, 35], sliding mode control method [36, 37], and impulsive synchronization method [38, 39]. Among them, adaptive control and nonlinear control methods are often used to solve the problems on synchronization of systems with unknown parameters or time-varying parameters, which are usually encountered in practical applications. However, most of them mentioned above have concentrated on achieving complete synchronization of low-dimensional and identical chaotic systems, while synchronization schemes for identical or nonidentical hyperchaotic systems have not been investigated extensively enough. As far as we know within our range, there is few literatures on MPS of nonidentical hyperchaotic systems. Therefore, designing effective control schemes to achieve MPS of two hyperchaotic systems with unknown parameters or time-varying parameters is an interesting and challenging job for both theory and practical applications.

Motivated by the aforementioned aspects, we first propose a class of new systems with different nonlinear terms and show the existence of hyperchaos in certain parameter ranges by calculating their Lyapunov exponent spectrums. After that, by presenting an adaptive control strategy and a nonlinear control scheme based on Lyapunov stability theory, the theories on MPS of the systems with general form which linearly depends on unknown parameters or time-varying parameters, respectively, are investigated. Finally, the presented control methods are applied to achieve MPS of the new hyperchaotic systems, and their effectiveness is illustrated by numerical simulations. The organization of this paper is as follows. In Section 2, a class of new hyperchaotic systems is constructed and the existence of hyperchaos is shown. In Section 3, theories on MPS of the systems with general form are given. At last, MPS of our presented hyperchaotic systems together with numerical simulations is shown in Section 4.

2. The Description of a Class of New Hyperchaotic Systems

There are two important requisites to obtain hyperchaos. One that is the minimal dimension of the phase space that embeds a hyperchaotic attractor should be at least four, and the other that is the number of terms in the coupled equations giving rise to instability should be at least two, of which at least one should have a nonlinear function [9]. According to the two points, a class of new 4D hyperchaotic systems is proposed by modifying the nonlinear terms of the Lorenz system and adding state feedback to it. They are described as 𝑥̇₁=𝑎𝑥₂−𝑥₁,𝑥̇₂=𝑏𝑥₁−10𝑥₁𝑥₃+𝑥₂+𝑥₄,𝑥̇₃=−𝑐𝑥₃+10𝑇𝑥₁,𝑥₂,𝑥̇₄=−𝑑𝑄𝑥₁,𝑥₂,𝑥₃,𝑥₄+𝑅𝑥₁,𝑥₂,𝑥₃,𝑥₄, (1) where 𝑎, 𝑏, 𝑐, and 𝑑 are parameters to be tuned and 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₄ are state variables. 𝑇(𝑥₁,𝑥₂), 𝑅(𝑥₁,𝑥₂,𝑥₃,𝑥₄) are nonlinear continuous functions, and 𝑄(𝑥₁,𝑥₂,𝑥₃,𝑥₄) is a linear continuous function, which can be set to obtain different hyperchaotic systems. We set 𝑇,𝑅 as nonlinear quadratic functions and 𝑄 as 𝑥₁, 𝑥₂, 𝑥₃, or 𝑥₄ to show systems with relatively simple forms. Subsequently, six different nonlinear systems are listed in Table 1.

Table 1 

New hyperchaotic systems.

In the following, the dynamics of the system (a) with 𝑇=𝑥²₁, 𝑄=𝑥₂, and 𝑅=𝑥₂𝑥₄ is illustrated as an example, which is described as 𝑥̇₁=𝑎𝑥₂−𝑥₁,𝑥̇₂=𝑏𝑥₁−10𝑥₁𝑥₃+𝑥₂+𝑥₄,𝑥̇₃=−𝑐𝑥₃+10𝑥²₁,𝑥̇₄=−𝑑𝑥₂+𝑥₂𝑥₄. (2) In case of 𝑎=20, 𝑏=35, 𝑐=3, and 𝑑=10, it has four equilibrium points, and the types of which can be determined by calculating the eigenvalues of the Jacobian matrices, respectively. The detailed descriptions are shown in Table 2, where USNP and USFP mean unstable saddle-node point and unstable saddle-focus point, respectively.

Table 2 

The related descriptions on the equilibrium points of system (2) with 𝑎=20, 𝑏=35, 𝑐=3, and 𝑑=10.

Furthermore, the chaotic attractor is shown in Figure 1. Combined with calculated Lyapunov Exponents (LEs) 𝜆₁=1.0677, 𝜆₂=0.0994, 𝜆₃=0, 𝜆₄=−23.1526, and Lyapunov Dimension (LD) 3.0498, respectively, we can say that system (2) is hyperchaotic. These analyses suggest that system (2) has rich dynamics with fixed parameters. In order to give the existence of chaos and hyperchaos in different parameter ranges, the Lyapunov exponent spectrum versus parameters 𝑎, 𝑏, 𝑐, and 𝑑 for the first three LEs 𝜆₁, 𝜆₂, and 𝜆₃ is shown, respectively, in Figure 2, where it can be clearly seen how it evolves from negative to positive values. It is noted that the system (2) is hyperchaotic when 𝑏∈(12,75) and chaotic when 𝑏∈(75,2000). The Largest Lyapunov exponent is 𝜆=10.105 when 𝑏=2000, which suggests it has a big chaotic range and complex dynamics.

Figure 1 

The chaotic attractor of system (2) with 𝑎=20,𝑏=35,𝑐=3,and𝑑=10.

Figure 2 

The Lyapunov exponent spectrum of system (2) versus parameters 𝑎, 𝑏, 𝑐, and 𝑑 for the first three LEs.

The similar analyses for the other five systems can also be gotten naturally. To highlight the existence of hyperchaos, we only exhibit the Lyapunov exponent spectrums of the systems (b)–(f) versus 𝑏 for 𝜆₁, 𝜆₂, and 𝜆₃ when 𝑎=20, 𝑐=3, and 𝑑=10 in Figure 3. It can be concluded from Figures 2 and 3 that the six new different systems are all hyperchaotic when 𝑏∈(20,60).

Figure 3 

The Lyapunov exponent spectrum of system (b)–(f) versus 𝑏 for 𝜆1,𝜆2, and 𝜆3 when 𝑎=20, 𝑐=3, and 𝑑=10.

3. Modified Projective Synchronization of General Chaotic Systems with the Same Structure to the New Hyperchaotic Systems

3.1. The Preliminaries

Consider the following drive-response systems ̇(𝑥),̇𝑦,𝑥=𝑓𝑦=𝑔+𝑢𝑡,𝑥,𝑦 (3)(4) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛, 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑅^𝑛 are state variables of the drive system (3) and the response system (4), respectively, 𝑓∶𝑅^𝑛→𝑅^𝑛 and 𝑔∶𝑅^𝑛→𝑅^𝑛 are continuous nonlinear vector functions, and 𝑢(𝑡,𝑥,𝑦)=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝑅^𝑛 is the control vector for synchronization.

Definition 1. For the drive system (3) and the response system (4), they are said to be modified projective synchronization (MPS) if there exists a nonzero constant matrix 𝐻=diag(ℎ₁,ℎ₂,…,ℎ_𝑛)∈𝑅^𝑛×𝑛, such that lim_𝑡→∞‖𝑦−𝐻𝑥‖=0, namely, lim_𝑡→∞|𝑦_𝑖−ℎ_𝑖𝑥_𝑖|=0 (𝑖=1,2,…,𝑛), where 𝐻 is scaling matrixand ℎ₁,ℎ₂,…,ℎ_𝑛 are nonzero scaling factors which could be a predefined value or any desired value to be directed by a feedback control.

Remark 2. When the scaling matrix 𝐻 equals to 𝐼,−𝐼, and 𝛼𝐼 (𝛼 is a constant), respectively, it means complete synchronization, antisynchronization, and projective synchronization, respectively.

Aiming at considering MPS between two of the systems (a)–(f), we first investigate the theories on MPS of general chaotic systems with the same structure to them in this part. Consider an 𝑛-dimensional continuous chaotic (hyperchaotic) system as drive system in the form of 𝑥̇=𝑓(𝑥)+𝐹(𝑥)𝜃₁, (5) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 is the state vector, 𝑓∶𝑅^𝑛→𝑅^𝑛 is a continuous nonlinear vector function, 𝐹∶𝑅^𝑛→𝑅^𝑛×𝑛 is a continuous function matrix, and 𝜃₁∈𝑅^𝑛 is a parameter vector. It can be seen that the nonlinear dynamical system (5) linearly depends on the parameter vector, and systems (a)–(f) all have the same system structure, so do many well-known hyperchaotic systems, such as hyperchaotic Lorenz, Lü systems, and Rössler system. Accompanied with the drive system (5), a controlled response system is given by 𝑦̇=𝑔𝑦+𝐺𝑦𝜃₂+𝑢, (6) where 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑅^𝑛 is the state vector, 𝑔∶𝑅^𝑛→𝑅^𝑛 is a continuous vector function, 𝐺∶𝑅^𝑛→𝑅^𝑛×𝑛 is a continuous function matrix, 𝜃₂∈𝑅^𝑛 is a parameter vector, and 𝑢∈𝑅^𝑛 is the control vector to be determined. Let 𝑒=𝑦−𝐻𝑥 denote the error state vector, thus the error dynamical system has the form 𝑒̇=𝑦̇−𝐻𝑥̇=𝑔𝑦+𝐺𝑦𝜃₂−𝐻𝑓(𝑥)−𝐻𝐹(𝑥)𝜃₁()()𝜃+𝑢=𝑄𝑒,𝑥+𝐺𝑒,𝑥₂−𝐻𝐹(𝑥)𝜃₁+𝑢, (7) where 𝑄(𝑒,𝑥)=𝑔(𝑒+𝐻𝑥)−𝐻𝑓(𝑥) and 𝐻=diag(ℎ₁,ℎ₂,…,ℎ_𝑛). So the global and asymptotical stability of system (7) means that systems (5) and (6) achieve MPS.

3.2. MPS between Systems (5) and (6) with Unknown or Time-Varying Parameters

Usually, the system parameters are partially or entirely unknown in advance in practical applications, and adaptive controller is often used to solve the problem for its adaptive ability. So one of our objects is to design an adaptive synchronization scheme with parameter update laws 𝑢=𝑢𝑥,𝑦,𝜃₁,𝜃₂,̇𝜃₁=𝜃₁𝑥,𝑦,𝜃₁,𝜃₂,̇𝜃₂=𝜃₂𝑥,𝑦,𝜃₁,𝜃₂, (8) where 𝜃₁ and 𝜃₂ are parameter estimate vectors of the unknown parameter vectors 𝜃₁, 𝜃₂, to get the drive-response systems to be in MPS with any arbitrarily given scaling factors. Namely, ‖𝑦−𝐻𝑥‖→0, together with 𝜃₁→𝜃₁, 𝜃₂→𝜃₂ for 𝑡→∞.

Theorem 3. For given nonzero scaling factors ℎ_𝑖≠0 (𝑖=1,2,…,𝑛), the drive-response systems (5) and (6) achieve MPS if the control vector and the parameter update laws are given as 𝑢=−𝑄(𝑒,𝑥)−𝐺(𝑒,𝑥)𝜃₂+𝐻𝐹(𝑥)𝜃₁̇−𝑀𝑒,𝜃₁=−𝜃₁+𝐹^𝑇(𝑥)̇𝐻𝑒,𝜃₂=−𝜃₂−𝐺^𝑇(𝑒,𝑥)𝑒, (9)(10)(11) where 𝜃_𝑖=𝜃_𝑖−𝜃_𝑖, 𝑖=1,2, and 𝑀 is a known positive definite matrix.

Proof. Substitute (9) into the error system (7), we get 𝑒̇=−𝑀𝑒−𝐻𝐹(𝑥)𝜃₁+𝐺(𝑒,𝑥)𝜃₂. (12) Construct a Lyapunov function 1𝑉=2𝑒^𝑇1𝑒+2𝜃^𝑇₁𝜃₁+12𝜃^𝑇₂𝜃₂, (13) and differentiate 𝑉 with respect to time along the solution of (12). It yields 𝑉̇(𝑡)=𝑒^𝑇𝑒̇+𝜃̇^𝑇₁𝜃₁+𝜃̇^𝑇₂𝜃₂=𝑒^𝑇−𝑀𝑒−𝐻𝐹(𝑥)𝜃₁+𝐺(𝑒,𝑥)𝜃₂+−𝜃₁+𝐹^𝑇(𝑥)𝐻𝑒^𝑇𝜃₁+−𝜃₂−𝐺^𝑇(𝑒,𝑥)𝑒^𝑇𝜃₂=−𝑀𝑒^𝑇𝑒−𝜃^𝑇₁𝜃₁−𝜃^𝑇₂𝜃₂≤0, (14) namely, 𝑉̇ is negative definite. It results in that the drive-response systems (5) and (6) achieve MPS according to Lyapunov stability theorem.

The other object is to achieve MPS between chaotic systems with time-varying parameters which are also frequently encountered in practical applications. Suppose parameter vectors 𝜃₁=𝜃₁(𝑡), 𝜃₂=𝜃₂(𝑡) of drive-response systems are both time varying and bounded, which means if one denote 𝜃^∼₁, 𝜃^∼₂ the nominal constant vectors of 𝜃₁, 𝜃₂, respectively, and Θ₁, Θ₂ known upper bounds, then 𝜃₁−𝜃^∼₁≤Θ₁,𝜃₂−𝜃^∼₂≤Θ₂. (15) In addition, since as far as we know, most hyperchaotic systems in the existing literatures such as hyperchaotic Lorenz system, hyperchaotic Lü system, and Rössler system, as well as the class of new systems (1) proposed here have the vectorial form (5) with diagonal 𝐹(𝑥), MPS of this kind of general hyperchaotic systems is discussed in the following theorem. It is noted that 𝐴=(|𝑎_𝑖𝑗|)_𝑛×𝑛 denotes a matrix, each element of which is the absolute value of corresponding element of matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, and 𝐴_𝑖𝑖 denotes its element at the cross of the 𝑖th row and the 𝑖th column.

Theorem 4. For given nonzero scaling factors ℎ_𝑖≠0 (𝑖=1,2,…,𝑛), the drive-response systems (5) and (6) with time-varying parameters and diagonal 𝐹(𝑥) and 𝐺(𝑒,𝑥) can achieve MPS if the nonlinear control strategy is designed as 𝑢=−𝑃𝑒−𝑄(𝑒,𝑥)−𝐺(𝑒,𝑥)𝜃^∼₂+𝐻𝐹(𝑥)𝜃^∼₁−𝐺(𝑒,𝑥)Θ₂sgn(𝑒)−𝐻𝐹(𝑥)Θ₁sgn(𝑒), (16) where 𝑃 is a known positive definite matrix and sgn(𝑒) denote a vector with elements sgn(𝑒_𝑖), 𝑖=1,2,…,𝑛.

Proof. According to (16), we rewrite the error system (7) as 𝑒̇=−𝑃𝑒−𝐻𝐹(𝑥)𝜃₁−𝜃^∼₁+𝐺(𝑒,𝑥)𝜃₂−𝜃^∼₂−𝐺(𝑒,𝑥)Θ₂sgn(𝑒)−𝐻𝐹(𝑥)Θ₁sgn(𝑒). (17) Constructing a Lyapunov function 𝑉=𝑒^𝑇𝑒/2 and differentiate 𝑉 with respect to time along the solution of (17), we have 𝑉̇(𝑡)=𝑒^𝑇𝑒̇=𝑒^𝑇−𝑃𝑒−𝐻𝐹(𝑥)𝜃₁−𝜃^∼₁+𝐺(𝑒,𝑥)𝜃₂−𝜃^∼₂−𝐺(𝑒,𝑥)Θ₂sgn(𝑒)−𝐻𝐹(𝑥)Θ₁sgn(𝑒)=−𝑒^𝑇𝑃𝑒−𝑒^𝑇𝐻𝐹(𝑥)𝜃₁−𝜃^∼₁+𝑒^𝑇𝐺(𝑒,𝑥)𝜃₂−𝜃^∼₂−𝑒^𝑇𝐺(𝑒,𝑥)Θ₂sgn(𝑒)−𝑒^𝑇𝐻𝐹(𝑥)Θ₁sgn(𝑒)≤−𝑒^𝑇𝑃𝑒+𝑒^𝑇𝐻𝐹(𝑥)𝜃₁−𝜃^∼₁−𝑒^𝑇𝐻𝐹(𝑥)Θ₁(𝑒)+𝑒sgn_𝑇𝐺(𝑒,𝑥)𝜃₂−𝜃^∼₂−𝑒^𝑇𝐺(𝑒,𝑥)Θ₂sgn(𝑒)≤−𝑒^𝑇𝑃𝑒+𝑒^𝑇𝐻𝐹(𝑥)Θ₁−𝑒^𝑇𝐻𝐹(𝑥)Θ₁sgn(𝑒)+𝑒^𝑇𝐺(𝑒,𝑥)Θ₂−𝑒^𝑇𝐺(𝑒,𝑥)Θ₂sgn(𝑒). (18) It results in 𝑉̇=−𝑒^𝑇𝑃𝑒≤0 because 𝑒^𝑇𝐻𝐹(𝑥)Θ₁sgn(𝑒)=Θ₁^𝑛_𝑖=1𝐻_𝑖𝑖𝐹(𝑥)_𝑖𝑖𝑒_𝑖sgn(𝑒)=Θ₁^𝑛_𝑖=1𝐻_𝑖𝑖𝐹(𝑥)_𝑖𝑖𝑒_𝑖=𝑒^𝑇𝐻𝐹(𝑥)Θ₁,𝑒^𝑇𝐺(𝑒,𝑥)Θ₂sgn(𝑒)=Θ₂^𝑛_𝑖=1𝐺(𝑒,𝑥)_𝑖𝑖𝑒_𝑖sgn𝑒_𝑖=Θ₂^𝑛_𝑖=1𝐺(𝑒,𝑥)_𝑖𝑖𝑒_𝑖=𝑒^𝑇𝐺(𝑒,𝑥)Θ₂. (19) Based on Lyapunov stability theory, the system (17) converges to 𝑂(0,0,0,0) as 𝑡→∞, which means that the two hyperchaotic systems achieve MPS asymptotically. This completes the proof.

4. MPS of the New Hyperchaotic Systems and Numerical Simulations

The presented theories are applied to MPS between two of the new hyperchaotic systems in this part. Set system (c) and system (f) as drive-response systems, which have the forms 𝑥̇₁=𝑎₁𝑥₂−𝑥₁,𝑥̇₂=𝑏₁𝑥₁−10𝑥₁𝑥₃+𝑥₄+𝑥₂,𝑥̇₃=−𝑐₁𝑥₃+10𝑥₁𝑥₂,𝑥̇₄=−𝑑₁𝑥₂+𝑥₂𝑥₄,𝑦̇₁=𝑎₂𝑦₂−𝑦₁+𝑢₁,𝑦̇₂=𝑏₂𝑦₁−10𝑦₁𝑦₃+𝑦₄+𝑦₂+𝑢₂,𝑦̇₃=−𝑐₂𝑦₃+10𝑦₁𝑦₂+𝑢₃,𝑦̇₄=−𝑑₂𝑦₁+𝑦₁𝑦₃+𝑢₄, (20)(21) where 𝑎₁, 𝑏₁, 𝑐₁, and 𝑑₁ and 𝑎₂, 𝑏₂, 𝑐₂, and 𝑑₂ are unknown system parameters which need to be estimated. Their vector forms can be, respectively, described as 𝑥̇₁𝑥̇₂𝑥̇₃𝑥̇₄=0−10𝑥₁𝑥₃+𝑥₄+𝑥₂10𝑥₁𝑥₂𝑥₂𝑥₄+𝑥₂−𝑥₁0000𝑥₁0000−𝑥₃0000−𝑥₂𝑎₁𝑏₁𝑐₁𝑑₁,𝑦̇₁𝑦̇₂𝑦̇₃𝑦̇₄=0−10𝑦₁𝑦₃+𝑦₄+𝑦₂10𝑦₁𝑦₂𝑦₁𝑦₃+𝑦₂−𝑦₁0000𝑦₁0000−𝑦₃0000−𝑦₁𝑎₂𝑏₂𝑐₂𝑑₂+𝑢₁𝑢₂𝑢₃𝑢₄, (22)(23) where (𝑢₁,𝑢₂,𝑢₃,𝑢₄)^𝑇 is the controller to be determined. Let positive definite matrix 𝑀 be 𝑀=1000010100100001, (24) then according to (9), (10), and (11), we get the controller 𝑢₁=−𝑒₁−𝑎̇₂𝑒₂−𝑒₁+ℎ₂𝑥₂−ℎ₁𝑥₁+𝑎̇₁ℎ₁𝑥₂−𝑥₁,𝑢₂=−2𝑒₂−𝑒₄+10𝑒₁+ℎ₁𝑥₁𝑒₃+ℎ₃𝑥₃−𝑒₄+ℎ₄𝑥₄−10ℎ₂𝑥₁𝑥₃+ℎ₂𝑥₄−̇̂𝑏₂𝑒₁+ℎ₁𝑥₁+̇̂𝑏₁ℎ₂𝑥₁,𝑢₃=−𝑒₃−10𝑒₁+ℎ₁𝑥₁𝑒₂+ℎ₂𝑥₂+10ℎ₃𝑥₁𝑥₂+𝑐₂̇̂𝑒₃+ℎ₃𝑥₃−𝑐₁̇̂ℎ₃𝑥₃,𝑢₄=−𝑒₄−𝑒₁+ℎ₁𝑥₁𝑒₃+ℎ₃𝑥₃+ℎ₄𝑥₂𝑥₄+̇𝑑₂𝑒₁+ℎ₁𝑥₁−̇𝑑₁ℎ₄𝑥₂, (25) with the parameter update laws 𝑎̇‌₁=−𝑎₁+ℎ₁𝑥₂−𝑥₁𝑒₁,̇𝑏₁=−𝑏₁+ℎ₂𝑥₁𝑒₂,𝑐₁̇=−𝑐₁−ℎ₃𝑥₃𝑒₃,̇𝑑₁=−𝑑₁−ℎ₄𝑥₂𝑒₄,𝑎̇₂=−𝑎₂−𝑒₂−𝑒₁+ℎ₂𝑥₂−ℎ₁𝑥₁𝑒₁,̇𝑏₂=−𝑏₂−𝑒₁+ℎ₁𝑥₁𝑒₂,𝑐₂̇=−𝑐₂+𝑒₃+ℎ₃𝑥₃𝑒₃,̇𝑑₂=−𝑑₂+𝑒₁+ℎ₁𝑥₁𝑒₄, (26)(27) where 𝑎_𝑖=𝑎_𝑖−𝑎_𝑖, 𝑏_𝑖=𝑏_𝑖−𝑏_𝑖, 𝑐_𝑖=𝑐_𝑖−𝑐_𝑖, 𝑑_𝑖=𝑑_𝑖−𝑑_𝑖, and 𝑖=1,2. Let 𝑎₁=𝑎₂=20, 𝑏₁=𝑏₂=35, 𝑐₁=𝑐₂=3, and 𝑑₁=𝑑₂=10, then the drive-response systems are hyperchaotic. In addition, set the initial states of the drive-response systems to be 𝑥₁(0)=1, 𝑥₂(0)=1, 𝑥₃(0)=1, and 𝑥₄(0)=1 and 𝑦₁(0)=4, 𝑦₂(0)=5, 𝑦₃(0)=6, and 𝑦₄(0)=7, respectively, set the initial states of the estimated parameter errors to be 𝑎₁(0)=𝑎₂(0)=1, 𝑏₁(0)=𝑏₂(0)=1, 𝑐₁(0)=𝑐₂(0)=1, and 𝑑₁(0)=𝑑₂(0)=1, and set the scaling matrix to be 𝐻=diag(−1,0.5,1,4). Then the time response of the errors is shown in Figure 4. For further observations, the state trajectories of the two systems are depicted in Figure 5. It is exhibited that 𝑥₁ and 𝑦₁ display an antisynchronization phenomenon, 𝑦₂ finally converges to half the value of 𝑥₂, 𝑥₃ and 𝑦₃ show synchronization behavior, and 𝑦₄ converges four times the value of 𝑥₄, just as we intended. Moreover, the curves of estimated parameters are also shown in Figure 6. It can be concluded that the two systems achieve MPS successfully.

Figure 4 

The MPS errors 𝑒1, 𝑒2, 𝑒3, and 𝑒4 between the drive-response systems (20) and (21).

Figure 5 

State trajectories of the drive-response systems (20) and (21), 𝑥1 and 𝑦1 with ℎ1=−1, 𝑥2 and 𝑦2 with ℎ2=0.5, 𝑥3 and 𝑦3 with ℎ3=1, and 𝑥4 and 𝑦4 with ℎ4=4.

Figure 6 

Curves of the estimated parameters of the drive-response systems (20) and (21) under the update law (26) and (27).

Furthermore, suppose that the parameters are time varying, remain the drive system (20) and set system (𝑒) to be response system which is expressed as 𝑦̇₁=𝑎₂𝑦₂−𝑦₁+𝑢₁,𝑦̇₂=𝑏₂𝑦₁−10𝑦₁𝑦₃+𝑦₄+𝑦₂+𝑢₂,𝑦̇₃=−𝑐₂𝑦₃+10𝑦²₁+𝑢₃,𝑦̇₄=−𝑑₂𝑦₁+𝑦₁𝑦₃+𝑢₄, (28) with the vector form 𝑦̇₁𝑦̇₂𝑦̇₃𝑦̇₄=0−10𝑦₁𝑦₃+𝑦₄+𝑦₂10𝑦²₁𝑦₁𝑦₃+𝑦₂−𝑦₁0000𝑦₁0000−𝑦₃0000−𝑦₁𝑎₂𝑏₂𝑐₂𝑑₂+𝑢₁𝑢₂𝑢₃𝑢₄, (29) where (𝑢₁,𝑢₂,𝑢₃,𝑢₄)^𝑇 is the controller to be determined. Set nominal values of 𝑎, 𝑏, 𝑐, and 𝑑 to be 𝑎=20, 𝑏=35, 𝑐=3, and 𝑑=10, set time-varying parameters of the drive-response systems to be 𝑎₁=20+sin(𝑡), 𝑏₁=35+sin(𝑡), 𝑐₁=3+sin(𝑡), and 𝑑₁=10+sin(𝑡) and 𝑎₂=20+cos(𝑡), 𝑏₂=35+cos(𝑡), 𝑐₂=3+cos(𝑡), and 𝑑₂=10+cos(𝑡), respectively, set the upper bound to be Θ₁=Θ₂=2, set positive definite matrix to be 𝑃=𝑀, set the initial states of the drive-response systems to be 𝑥₁(0)=1, 𝑥₂(0)=1, 𝑥₃(0)=1, and 𝑥₄(0)=1 and 𝑦₁(0)=4, 𝑦₂(0)=5, 𝑦₃(0)=6, 𝑦₄(0)=7, respectively, and set the scaling matrix to be 𝐻=diag(−1,0.5,1,4), then according to (16), the controller can be described as 𝑢₁=−𝑒₁−20𝑦₂−𝑦₁+20ℎ₁𝑥₂−𝑥₁−𝑦₂−𝑦₁sgn𝑒₁−ℎ₁𝑥₂−𝑥₁sgn𝑒₁,𝑢₂=−2𝑒₂−𝑒₄−10𝑦₁𝑦₃+10ℎ₂𝑥₁𝑥₃−ℎ₂𝑥₄−ℎ₂𝑥₂−35𝑦₁+35ℎ₂𝑥₁−𝑦₁sgn𝑒₂−ℎ₂𝑥₁sgn𝑒₂,𝑢₃=−𝑒₃−10𝑦₁𝑦₂+10ℎ₃𝑥₁𝑥₂+3𝑦₃−3ℎ₃𝑥₃+𝑦₃sgn𝑒₃−ℎ₃𝑥₃sgn𝑒₃,𝑢₄=−𝑒₄−𝑦₁𝑒₃+ℎ₄𝑥₂𝑥₄+10𝑦₁−10ℎ₄𝑥₁+𝑦₁sgn𝑒₄−ℎ₄𝑥₂sgn𝑒₄. (30) Subsequently, the time response of the error systems is given in Figure 7, which suggests, the error vector converges to zero asymptotically and the control strategy for MPS is successful.

Figure 7 

The MPS errors 𝑒1,𝑒2,𝑒3, and 𝑒4 of drive-response systems (20) and (21) with time-varying parameters.

5. Conclusions

Generation and MPS for a class of new hyperchaotic systems are both considered in this paper. First, six new hyperchaotic systems with different nonlinear terms are derived and the existence of hyperchaos is exhibited by calculating their Lyapunov exponent spectrums. Second, the universal theories on MPS of general chaotic systems with the structure like that are investigated by presenting an adaptive control strategy together with parameter update laws and a nonlinear control scheme based on Lyapunov stability theory. Finally, the methods are applied to our proposed hyperchaotic systems, and numerical simulations demonstrate the effectiveness of the proposed synchronization schemes.

Acknowledgments

The authors would like to thank the reviewer for his helpful suggestions on the paper. This research is supported by Natural Science Foundation of Heilongjiang Province, China (Grant no. A201101), the Science and Technology Pre-Research Foundations of Harbin Normal University, China (Grant no. 11XYG-05 and no. 12XYS-04), and academic backbone program foundation for youth by Harbin Normal University (Grant no. KGB201222).