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

and Applied Analysis 3


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  (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 Mary 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  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  equals to , −, and  ( 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][11][12][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][15][16][17][18][19][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][22][23][24], adaptive synchronization method [25][26][27][28][29][30][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 lowdimensional 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.

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  1.
In the following, the dynamics of the system (a) with  =  2  1 ,  =  2 , and  =  2  4 is illustrated as an example, which is described as In case of  = 20,  = 35,  = 3, and  = 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.
Furthermore, the chaotic attractor is shown in Figure 1.Combined with calculated Lyapunov Exponents (LEs)  1 = 1.0677,  2 = 0.0994,  3 = 0,  4 = −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 , , , and  for the first three LEs  1 ,  2 , and  3 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  ∈ (12, 75) and chaotic when  ∈ (75,2000).The Largest Lyapunov exponent is  = 10.105 when  = 2000, which suggests it has a big chaotic range and complex dynamics.
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  for  1 ,  2 , and  3 when  = 20,  = 3, and  = 10 in Figure 3.It can be concluded from Figures 2 and 3 that the six new different systems are all hyperchaotic when  ∈ (20, 60).

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 where  = ( 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 -dimensional continuous chaotic (hyperchaotic) system as drive system in the form of where  = ( 1 ,  2 , . . .,   )  ∈   is the state vector,  :   →   is a continuous nonlinear vector function,  :   →  × is a continuous function matrix, and  1 ∈   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 wellknown hyperchaotic systems, such as hyperchaotic Lorenz, Lü systems, and Rössler system.Accompanied with the drive system (5), a controlled response system is given by where  = ( 1 ,  2 , . . .,   )  ∈   is the state vector,  :   →   is a continuous vector function,  :   →  × is a continuous function matrix,  2 ∈   is a parameter vector, and  ∈   is the control vector to be determined.Let  = − denote the error state vector, thus the error dynamical system has the form where (, ) = ( + ) − () and  = diag(ℎ 1 , ℎ 2 , . . ., ℎ  ).So the global and asymptotical stability of system (7) means that systems ( 5) and ( 6) achieve MPS.

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  =  (, , θ1 , θ2 ) , θ 1 =  1 (, , θ1 , θ2 ) , where θ1 and θ2 are parameter estimate vectors of the unknown parameter vectors  1 ,  2 , to get the drive-response systems to be in MPS with any arbitrarily given scaling factors.Namely, ‖ − ‖ → 0, together with θ1 →  1 , θ2 →  2 for  → ∞.
Proof.Substitute (9) into the error system (7), we get Construct a Lyapunov function and differentiate  with respect to time along the solution of (12).It yields namely, V is negative definite.It results in that the driveresponse 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  1 =  1 (),  2 =  2 () of drive-response systems are both time varying and bounded, which means if one denote θ1 , θ2 the nominal constant vectors of  1 ,  2 , respectively, and Θ 1 , Θ 2 known upper bounds, then 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 (), MPS of this kind of general hyperchaotic systems is discussed in the following theorem.It is noted that  = (|  |) × denotes a matrix, each element of which is the absolute value of corresponding element of matrix  = (  ) × , and   denotes its element at the cross of the th row and the th column.
Proof.According to (16), we rewrite the error system (7) as 6 Abstract and Applied Analysis Constructing a Lyapunov function  =   /2 and differentiate  with respect to time along the solution of ( 17), we have It results in V = −   ≤ 0 because Based on Lyapunov stability theory, the system (17) converges to (0, 0, 0, 0) as  → ∞, which means that the two hyperchaotic systems achieve MPS asymptotically.This completes the proof.

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 where  1 ,  1 ,  1 , and  1 and  2 ,  2 ,  2 , and  2 are unknown system parameters which need to be estimated.Their vector forms can be, respectively, described as where ( 1 ,  2 ,  3 ,  4 )  is the controller to be determined.Let positive definite matrix  be  = ( then according to (9), (10), and (11), we get the controller with the parameter update laws where   =   −â  ,   =   − b ,   =   −ĉ  ,   =   − d , and  = 1, 2.

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.

Table 1 :
New hyperchaotic systems.where , , , and  are parameters to be tuned and  1 ,  2 ,  3 , and  4 are state variables.( 1 ,  2 ), ( 1 ,  2 ,  3 ,  4 ) are nonlinear continuous functions, and ( 1 ,  2 ,  3 ,  4 ) is a linear continuous function, which can be set to obtain different hyperchaotic systems.We set ,  as nonlinear quadratic functions and  as  1 ,  2 ,  3 , or  4 to show systems with relatively simple forms.Subsequently, six different nonlinear systems are listed in Table