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Generalized combination complex synchronization of new hyperchaotic complex Lü-like systems
- Cuimei Jiang^{1} and
- Shutang Liu^{1}Email author
https://doi.org/10.1186/s13662-015-0490-y
© Jiang and Liu; licensee Springer. 2015
Received: 17 November 2014
Accepted: 29 April 2015
Published: 11 July 2015
Abstract
In this paper, a new hyperchaotic complex system is presented and its dynamical properties are discussed by phase portraits, bifurcation diagrams, and the Lyapunov exponents spectra. Noticeably, based on two drive complex systems and one response complex system with different dimensions, we propose generalized combination complex synchronization and design a general controller. Additionally, we investigate generalized combination complex synchronization between real systems and complex systems via two complex scaling matrices. Two examples, which include two chaotic complex systems driving one new hyperchaotic complex system and two new hyperchaotic complex systems driving one chaotic real system, are shown to demonstrate the effectiveness and feasibility of the schemes.
Keywords
- hyperchaotic complex systems
- chaotic attractors
- Lyapunov exponents
- generalized combination complex synchronization
1 Introduction
In 1982, Fowler et al. [1] proposed the complex Lorenz equations, which is the pioneering work in the domain of complex systems. After that, chaotic and hyperchaotic complex systems have been extensively studied owing to their important applications in physical systems, image processing and in particular in secure communication [2–4]. And researchers presented many chaotic and hyperchaotic complex systems, such as the complex Lorenz system [5], the complex Chen system [6], the complex Lü system [6], the hyperchaotic complex Lorenz system [7], the hyperchaotic complex Lü system [8], and so on. Compared with chaotic systems, the behavior of hyperchaotic complex systems is more complex and richer. Hence, when applying the hyperchaotic complex systems to secure communication, it is better to increase the complexity and the security of the transmitted information.
On the other hand, with the development of complex systems, synchronization of chaotic complex systems has gained a great deal of attentions. Some synchronization schemes of chaotic real systems were extended to the complex space, such as complete synchronization [9], anti-synchronization [10, 11], projective synchronization [12], etc. Recently, many authors have studied some new kinds of synchronization for complex dynamical systems, for example, complex complete synchronization [13], complex projective synchronization [14], complex modified projective synchronization [15, 16], and so forth. Since complex variables increase the diversity and the security of the transmitted signals, these synchronization methods of chaotic complex systems have potential applications in secure communication and image processing.
However, most of the above-mentioned works mainly focus on the usual drive-response synchronization which has one drive system and one response system. To improve the ability of anti-attack and anti-translated of the transmitted information, Luo et al. [17] proposed combination synchronization which has two drive real systems and one response real system. Subsequently, Wu and Fu [18] studied increased-order and reduced-order combination synchronization in the real space concerning two specific examples. Soon afterwards, Zhou et al. [19] introduced combination synchronization to the complex space and carried out synchronization of three identical or different nonlinear complex hyperchaotic systems. Very recently, Sun et al. [20] investigated combination complex synchronization between two drive chaotic complex systems and one response chaotic complex system. These synchronization schemes occur in chaotic complex systems with the same dimensions.
To the best of our knowledge, there are few papers discussing combination synchronization among two drive systems and one response system with different dimensions in the complex space. As a matter of fact, for nonlinear systems with different dimensions, a lot of synchronization phenomena exist in reality, especially in the chemical and biological sciences. For instance, we can observe the physiological synchronization phenomena between higher-dimensional and lower-dimensional thalamic neurons as well as between the circulatory and respiratory systems [21]. Therefore, it is meaningful and valuable to study synchronization between two drive systems and one response system with different dimensions from the application point of view.
Inspired by the above discussion, we introduce a new hyperchaotic complex system to investigate generalized combination complex synchronization between two drive complex systems and one response complex system with different structures. Meanwhile, a general controller is designed to synchronize chaotic complex systems in the sense of generalized combination complex synchronization. By virtue of two complex scaling matrices, we establish a link between real chaos and complex chaos. The proposed generalized combination complex synchronization will contain complex projective synchronization, combination synchronization, and combination complex synchronization. Consequently, our work will extend the previous results.
The remainder of this paper is organized as follows. In Section 2, we present a hyperchaotic complex Lü-like system and study its dynamical properties including symmetry, equilibria and stability, Lyapunov exponents and fractal dimensions, as well as hyperchaotic attractors. Section 3 introduces generalized combination complex synchronization and designs a general controller. Two typical examples are treated to exhibit the effectiveness and correctness of the proposed methods. Finally, a concluding remark is given in Section 4.
2 A new hyperchaotic complex Lü-like system
2.1 Symmetry and invariance
Note that the symmetry of system (4): It is symmetric about the \(m_{5}\)-axis, which means it is invariant for the coordinate transformation of \((m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})\rightarrow (-m_{1},-m_{2}, -m_{3},-m_{4},m_{5},-m_{6})\).
2.2 Dissipation
2.3 Equilibria and stability
2.4 Lyapunov exponents and fractal dimensions
2.5 Hyperchaotic behavior and attractors
Next, we calculate numerically the values of the parameters of (4) at which chaotic attractors exist under the above conditions. Now we consider the following two cases.
3 Generalized combination complex synchronization
The aim of this section is to present generalized combination complex synchronization and design a general controller. Then two simulation examples are given to verify the effectiveness of the schemes.
3.1 Scheme of generalized combination complex synchronization
Remark 1
Many chaotic and hyperchaotic complex systems can be described by (5), such as the complex Lorenz system, the complex Chen system, the complex Lü system, the hyperchaotic complex Lorenz system, the hyperchaotic complex Lü system, etc.
The definition of generalized combination complex synchronization is introduced below.
Definition 1
Remark 2
If the dimensions of the two drive systems (5) and (6) are equal to that of the response system (7), i.e., \(n=n_{1}=n_{2}\), then the proposed synchronization will be combination complex synchronization.
Remark 3
If the scaling matrix \(M_{1}=O_{n\times{n_{1}}}\) or \(M_{2}=O_{n\times{n_{2}}}\), then we can achieve complex projective synchronization. If \(M_{1}^{i}=O_{n\times{n_{1}}}\) or \(M_{2}^{i}=O_{n\times{n_{2}}}\), then combination synchronization can be carried out. If \(M_{1}=O_{n\times{n_{1}}}\) and \(M_{2}=O_{n\times{n_{2}}}\), then the synchronization problem will be turned into a chaos control problem.
Remark 4
Definition 1 can be applicable to three or more chaotic complex systems. Additionally, drive systems and response systems can be identical or different.
The following lemma is useful in this paper.
Lemma 1
[25]
For a matrix \(D\in{\mathbb{C}^{n\times{n}}}\), all of the real parts of whose eigenvalues are negative, i.e., \(\operatorname{Re}(\lambda_{i}(D))<0\) (\(i=1,2,\ldots ,n\)), then \(\lim_{t\rightarrow{\infty}}\exp(Dt)=0\).
Theorem 1
Proof
In the following, we investigate generalized combination complex synchronization between real chaos and complex chaos. Now we consider two cases which include two real systems driving one complex system and two complex systems driving one real system.
Corollary 1
Corollary 2
In addition, from Theorem 1, some corollaries can easily be obtained and their proofs are omitted.
Corollary 3
Corollary 4
3.2 Numerical examples
In this subsection, we provide two examples to illustrate the feasibility and effectiveness of the proposed schemes. Firstly, synchronization between two 3-dimensional chaotic complex systems and a 4-dimensional new hyperchaotic complex system is studied.
3.2.1 Synchronization between two drive chaotic complex systems and a response hyperchaotic complex system
3.2.2 Synchronization between two drive hyperchaotic complex systems and a response chaotic real system
4 Conclusions
In this work, we firstly introduce a new hyperchaotic complex system and study its dynamical behavior. The dynamical properties of this new system are identified by using phase portraits, bifurcation diagrams, and the Lyapunov exponents spectra. Secondly, we propose generalized combination synchronization between three different dimensional chaotic complex systems. In this proposed scheme, two drive systems and one response system can be synchronized to two complex scaling matrices which are non-square matrices. Besides, a general controller is designed to achieve generalized combination complex synchronization. Through this scheme, generalized combination synchronization between real chaos and complex chaos can be investigated by virtue of two complex scaling matrices. It is worth mentioning that there are various types of synchronization are special cases from our definition, which are complex projective synchronization, combination synchronization, and combination complex synchronization. Therefore, the obtained results extend many existing results in the literature. Moreover, many problems with unknown parameters and external disturbances exist in practical chaotic synchronization. Consequently, we will make an endeavor to investigate robust generalized combination complex synchronization considering unknown parameters and external disturbances in our future work.
Declarations
Acknowledgements
The authors would like to thank the editors and anonymous referees for their constructive comments and suggestions. The research was supported by the National Natural Science Foundation of China (numbers 61273088, 10971120) and the Natural Science Foundation of Shandong Province (number ZR2010FM010).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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