Research Article  Open Access
Nuo Jia, Tao Wang, "Generation and Modified Projective Synchronization for a Class of New Hyperchaotic Systems", Abstract and Applied Analysis, vol. 2013, Article ID 804964, 11 pages, 2013. https://doi.org/10.1155/2013/804964
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 timevarying 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 (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 [10–13]. In comparison with lowdimensional 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 noiselike 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 higherdimensional 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], timedelay 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 timevarying 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 timevarying 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 timevarying 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 where , , , and are parameters to be tuned and , , , and are state variables. , are nonlinear continuous functions, and is a linear continuous function, which can be set to obtain different hyperchaotic systems. We set as nonlinear quadratic functions and as , , , or to show systems with relatively simple forms. Subsequently, six different nonlinear systems are listed in Table 1.

In the following, the dynamics of the system (a) with , , and is illustrated as an example, which is described as In case of , , , and , 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 saddlenode point and unstable saddlefocus point, respectively.

Furthermore, the chaotic attractor is shown in Figure 1. Combined with calculated Lyapunov Exponents (LEs) , , , , and Lyapunov Dimension (LD) , 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 , , and 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 and chaotic when . The Largest Lyapunov exponent is when , 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 , , and when , , and in Figure 3. It can be concluded from Figures 2 and 3 that the six new different systems are all hyperchaotic when .
3. Modified Projective Synchronization of General Chaotic Systems with the Same Structure to the New Hyperchaotic Systems
3.1. The Preliminaries
Consider the following driveresponse systems where , are state variables of the drive system (3) and the response system (4), respectively, and are continuous nonlinear vector functions, and 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 , such that , namely, (), where is scaling matrixand 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 equals to , and ( 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 dimensional continuous chaotic (hyperchaotic) system as drive system in the form of where is the state vector, is a continuous nonlinear vector function, is a continuous function matrix, and 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 is the state vector, is a continuous vector function, is a continuous function matrix, 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 . 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 TimeVarying 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 where and are parameter estimate vectors of the unknown parameter vectors , , to get the driveresponse systems to be in MPS with any arbitrarily given scaling factors. Namely, , together with , for .
Theorem 3. For given nonzero scaling factors (), the driveresponse systems (5) and (6) achieve MPS if the control vector and the parameter update laws are given as where , , and is a known positive definite matrix.
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, 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 timevarying parameters which are also frequently encountered in practical applications. Suppose parameter vectors , of driveresponse systems are both time varying and bounded, which means if one denote , the nominal constant vectors of , , respectively, and , 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.
Theorem 4. For given nonzero scaling factors (), the driveresponse systems (5) and (6) with timevarying parameters and diagonal and can achieve MPS if the nonlinear control strategy is designed as where is a known positive definite matrix and denote a vector with elements , .
Proof. According to (16), we rewrite the error system (7) as Constructing a Lyapunov function and differentiate with respect to time along the solution of (17), we have It results in because Based on Lyapunov stability theory, the system (17) converges to as , 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 driveresponse systems, which have the forms where , , , and and , , , and are unknown system parameters which need to be estimated. Their vector forms can be, respectively, described as where 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 , , , , and . Let , , , and , then the driveresponse systems are hyperchaotic. In addition, set the initial states of the driveresponse systems to be , , , and and , , , and , respectively, set the initial states of the estimated parameter errors to be , , , and , and set the scaling matrix to be . 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 and display an antisynchronization phenomenon, finally converges to half the value of , and show synchronization behavior, and converges four times the value of , 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.
Furthermore, suppose that the parameters are time varying, remain the drive system (20) and set system () to be response system which is expressed as with the vector form where is the controller to be determined. Set nominal values of , , , and to be , , , and , set timevarying parameters of the driveresponse systems to be , , , and and , , , and , respectively, set the upper bound to be , set positive definite matrix to be , set the initial states of the driveresponse systems to be , , , and and , , , , respectively, and set the scaling matrix to be , then according to (16), the controller can be described as 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.
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 PreResearch Foundations of Harbin Normal University, China (Grant no. 11XYG05 and no. 12XYS04), and academic backbone program foundation for youth by Harbin Normal University (Grant no. KGB201222).
References
 L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, “The synchronization of chaotic systems,” Physics Reports, vol. 366, no. 12, pp. 1–101, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, “Phase synchronization of chaotic oscillators by external driving,” Physica D, vol. 104, no. 34, pp. 219–238, 1997. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Physical Review Letters, vol. 78, no. 22, pp. 4193–4196, 1997. View at: Google Scholar
 N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, “Generalized synchronization of chaos in directionally coupled chaotic systems,” Physical Review E, vol. 51, no. 2, pp. 980–994, 1995. View at: Publisher Site  Google Scholar
 R. Mainieri and J. Rehacek, “Projective synchronization in threedimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042–3045, 1999. View at: Google Scholar
 G.H. Li, “Generalized projective synchronization of two chaotic systems by using active control,” Chaos, Solitons and Fractals, vol. 30, no. 1, pp. 77–82, 2006. View at: Publisher Site  Google Scholar
 G.H. Li, “Modified projective synchronization of chaotic system,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1786–1790, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 O. E. Rössler, “An equation for hyperchaos,” Physics Letters A, vol. 71, no. 23, pp. 155–157, 1979. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Wang and M. Wang, “A hyperchaos generated from Lorenz system,” Physica A, vol. 387, no. 14, pp. 3751–3758, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 T. Gao, G. Chen, Z. Chen, and S. Cang, “The generation and circuit implementation of a new hyperchaos based upon Lorenz system,” Physics Letters A, vol. 361, no. 12, pp. 78–86, 2007. View at: Publisher Site  Google Scholar
 Y. Li, W. K. S. Tang, and G. Chen, “Generating hyperchaos via state feedback control,” International Journal of Bifurcation and Chaos, vol. 15, no. 10, pp. 3367–3375, 2005. View at: Publisher Site  Google Scholar
 A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Physica A, vol. 364, pp. 103–110, 2006. View at: Publisher Site  Google Scholar
 M. T. Yassen, “Synchronization hyperchaos of hyperchaotic systems,” Chaos, Solitons and Fractals, vol. 37, no. 2, pp. 465–475, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 C.H. Chen, L.J. Sheu, H.K. Chen et al., “A new hyperchaotic system and its synchronization,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2088–2096, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Huang, “Chaos synchronization between two novel different hyperchaotic systems with unknown parameters,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 11, pp. 4174–4181, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. M. AlSawalha and M. S. M. Noorani, “Adaptive antisynchronization of two identical and different hyperchaotic systems with uncertain parameters,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 1036–1047, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. Zhu, “Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 557–561, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. E. Mahmoud, “Dynamics and synchronization of new hyperchaotic complex Lorenz system,” Mathematical and Computer Modelling, vol. 55, no. 78, pp. 1951–1962, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. Fu, “Robust adaptive modified function projective synchronization of different hyperchaotic systems subject to external disturbance,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2602–2608, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. Chen, X. Wu, and Z. Gui, “Global synchronization criteria for a class of thirdorder nonautonomous chaotic systems via linear state error feedback control,” Applied Mathematical Modelling, vol. 34, no. 12, pp. 4161–4170, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Du, Q. Zeng, C. Wang, and M. Ling, “Function projective synchronization in coupled chaotic systems,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 705–712, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H.H. Chen, G.J. Sheu, Y.L. Lin, and C.S. Chen, “Chaos synchronization between two different chaotic systems via nonlinear feedback control,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 12, pp. 4393–4401, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. H. Park, “Further results on functional projective synchronization of Genesiotesi chaotic system,” Modern Physics Letters B, vol. 23, no. 15, pp. 1889–1895, 2009. View at: Publisher Site  Google Scholar
 T. H. Lee and J. H. Park, “Adaptive functional projective lag synchronization of a hyperchaotic Rössler system,” Chinese Physics Letters, vol. 26, no. 9, Article ID 090507, 4 pages, 2009. View at: Publisher Site  Google Scholar
 J. Zheng, “A simple universal adaptive feedback controller for chaos and hyperchaos control,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2000–2004, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. H. Park, “Adaptive controller design for modified projective synchronization of GenesioTesi chaotic system with uncertain parameters,” Chaos, Solitons and Fractals, vol. 34, no. 4, pp. 1154–1159, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. H. Park, “Adaptive control for modified projective synchronization of a fourdimensional chaotic system with uncertain parameters,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 288–293, 2008. View at: Publisher Site  Google Scholar
 N. Jia and T. Wang, “Chaos control and hybrid projective synchronization for a class of new chaotic systems,” Computers and Mathematics with Applications, vol. 62, no. 12, pp. 4783–4795, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Mu and L. Pei, “Synchronization of the nearidentical chaotic systems with the unknown parameters,” Applied Mathematical Modelling, vol. 34, no. 7, pp. 1788–1797, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Zheng, “Adaptive modified function projective synchronization of unknown chaotic systems with different order,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5891–5899, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Wei, “Delayed feedback on the 3D chaotic system only with two stable nodefoci,” Computers and Mathematics with Applications, vol. 63, no. 3, pp. 728–738, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 T. Botmart, P. Niamsup, and X. Liu, “Synchronization of nonautonomous chaotic systems with timevarying delay via delayed feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1894–1907, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. Yu and H.X. Li, “Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 388–393, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. Chen, L. Chen, and W. Zhang, “Stabilization of parameters perturbation chaotic system via adaptive backstepping technique,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 101–109, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. P. Aghababa and A. Heydari, “Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1639–1652, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. H. Hosseinnia, R. Ghaderi, A. Ranjbar N., M. Mahmoudian, and S. Momani, “Sliding mode synchronization of an uncertain fractional order chaotic system,” Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1637–1643, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Hu, Y. Yang, and Z. Xu, “Impulsive control of projective synchronization in chaotic systems,” Physics Letters A, vol. 372, no. 18, pp. 3228–3233, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Hamiche, K. Kemih, M. Ghanes, G. Zhang, and S. Djennoune, “Passive and impulsive synchronization of a new fourdimensional chaotic system,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 4, pp. 1146–1154, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2013 Nuo Jia and Tao Wang. 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.