Finite-time multi-switching synchronization behavior for multiple chaotic systems with network transmission mode

doi:10.1016/j.jfranklin.2018.01.027

Journal of the Franklin Institute

Volume 355, Issue 5, March 2018, Pages 2892-2911

https://doi.org/10.1016/j.jfranklin.2018.01.027 Get rights and content

Abstract

By considering network transmission mode, this paper addresses the finite-time multi-switching synchronization problem for two kinds of multiple chaotic systems. For multiple same-order chaotic systems, we construct the general switching rules and analyze the existence of switching cases. The presented schemes guarantee the states of each derive system to be finite-timely synchronized with the desired states of every respond system in the different transmission paths and switching sequences. For multiple different order chaotic systems, we analyze a special multi-switching hybrid synchronization behavior, where part of the states are completely synchronized and the others belong to combination synchronization. Moveover, the easily verifiable criterion is derived for such synchronization. Finally, numerical examples are given to show the effectiveness of the presented theoretical results.

Introduction

It is well known that many fruitful results on synchronization of multiple chaotic systems (MCSs) have been continuously presented. And some complex synchronization mechanisms also were investigated. For example, Grassi studied projective synchronization among MCSs with series connection [1]. Various kinds of synchronization for MCSs with ring connection have been studied in [2], [3], [4]. Sun et al. [5] and Luo et al. [6] discussed respectively combination synchronization among three real or complex chaotic systems. These works have provided more important theoretical and application values than the pervious model for single drive and single response system. Therefore, it is meaningful to explore more complex synchronization modes for MCSs. In [7], Sun et al. presented the transmission synchronization of multi-systems for the first time, which had shown a class of complex synchronization mechanism, and may be helpful to improve the security of secret signals for multilateral communications. And subsequently, Chen et al. further extended this innovation to the multiple uncertain chaotic systems [8], [9], [10]. Until now, this topic are still open and challenging.

In [11], Ucar et al. proposed multi-switching synchronization mode for the first time. One of the main advantages is its multi-switching property, which increased the complexity and diversity of chaos synchronization in a preset multi-switching manner. Afterwards, some important results have been reported including complete synchronization [12], [13], switched function synchronization [14], [15], reduced order hybrid synchronization [16] etc. However, all mentioned works still belong to the scope of two chaotic or hyberchaotic systems. Recently, some further discussions on multi-switching synchronization among three chaotic systems have been concerned, e.g., multi-switching combination synchronization [17], switched projective synchronization [18] and so on. Nevertheless, there ignores multi-switching feature in signal transmission process for multi-systems, and few efforts were illustrated on the research of multi-switching synchronization with any switched transmission paths step by step. This motivates us to carry out the study of this paper.

From an application point of view, faster stability is one of the important issues needed to be discussed and also one great challenge in achieving chaos synchronization. In the past decades, finite-time control technique [19] has been applied successfully to synchronize complex networks [20], [21], [22], [23], neural networks [24], [25] and multi-agent systems [26]. For chaos synchronization, finite-time synchronization (FTS) for two real same-order or different-order chaotic systems was studied in [27], [28]. Hereafter, FTS of two uncertain complex systems was mainly analyzed in [29]. Extended to multi-systems, Sun et al. investigated FTS among four real chaotic systems with combination mode [30], and three complex systems were also used in [31]. Chen et al. focused on investigating FTS for multiple different-order chaotic systems with transmission mode [32]. Zhang et al. proposed a class of global FTS schemes for different dimensional chaotic systems, which displayed a faster convergence performance in chaos synchronization [33]. Unfortunately, seldom authors considered multi-switching mechanism for multi-systems in the finite time. Hence, the above discussions motivate us to carry out this study on designing the effective FTS scheme among multiple systems with multi-switching rules and transmission mode.

By combining the above analysis into account, this paper focus on the finite-time multi-switching synchronization (FTMSS) among MCSs with transmission mode and switched sequences. The main contributions are given as follows:

(1)
FTMSS for MCSs is discussed for the first time. Both same-order and different-order systems are involved, which can help to illustrate the different synchronization mechanism. Meanwhile, it is worth pointing out that there exist various kinds of multi-switching behaviors in transmission synchronization.
(2)
The general forms of multi-switching rules are established to define FTMSS of two kinds of MCSs with transmission mode. The corresponding schemes and stability criteria are presented to achieve such synchronization.
(3)
For different order multi-systems, the existence of multi-switching hybrid synchronization, involving combination synchronization and projective synchronization, is analyzed.

The rest of this paper contains the following four sections. Section 2 presents FTMSS for MCSs, which including the same-order case and the different-order case. Section 3 gives some examples and analyze the results. Finally, Section 4 concludes the innovations and discusses future works.

Section snippets

Finite-time multi-switching synchronization for multi-systems

In accordance with multi-switching characteristics in transmission process, multiple same-order and different-order chaotic systems are exploited to investigate FTMSS, respectively.

Numerical simulations and analysis

Example 1

In this simulation, we choose the following unified chaotic systems, which have been used in [34] ${\begin{matrix} {\dot{x}}_{11} = (25 α_{1} + 10) (x_{12} - x_{11}), \\ {\dot{x}}_{12} = (28 - 35 α_{1}) x_{11} - x_{11} x_{13} + (29 α_{1} - 1) x_{12}, \\ {\dot{x}}_{13} = x_{11} x_{12} - \frac{(8 + α_{1})}{3} x_{13}, \end{matrix}$ ${\begin{matrix} {\dot{x}}_{j 1} = (25 α_{1} + 10) (x_{j 2} - x_{j 1}) + u_{j - 1, 1}, \\ {\dot{x}}_{j 2} = (28 - 35 α_{1}) x_{j 1} - x_{j 1} x_{j 3} + (29 α_{1} - 1) x_{j 2} + u_{j - 1, 2}, \\ {\dot{x}}_{j 3} = x_{j 1} x_{j 2} - \frac{(8 + α_{1})}{3} x_{j 3} + u_{j - 1, 3}, \end{matrix}$ where $j = 2, 3$ . α₁ is the parameter. The above systems are chaotic in the whole interval [0, 1]. Especially, the Lorenz and Chen systems belong to two extremes and the Lü system is a special case. According to

Conclusions

FTMSS of MCSs with transmission mode has been addressed in this paper. Both same-order and different-order cases have been discussed. By presenting general switching rules, the definitions of FTMSS have been given for the first time, and some criteria have been established to obtain the finite time stability of all errors systems. In particular, a special finite-time multi-switching hybrid synchronization has been mainly analyzed for MDOCSs. Simulations results have validated the effectiveness

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China, under grants 61403179, 61273012 and 61573102, and by the Applied Mathematics Enhancement Program (AMEP) of Linyi University, by a Project of the Postdoctoral Sustentation Fund of Jiangsu Province of China under Grant 1402042B, and by 2016 Visiting Scholar Program of China Scholarship Council. The work of J.H. Park was supported by Basic Science Research Programs through the National Research Foundation of Korea

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Finite-time multi-switching synchronization behavior for multiple chaotic systems with network transmission mode

Abstract

Introduction

Section snippets

Finite-time multi-switching synchronization for multi-systems

Numerical simulations and analysis

Conclusions

Acknowledgments

J. Frankl. Inst.

Commun. Nonlinear Sci. Numer. Simul.

Neurocomputing

Appl. Math. Comput.

Nonlinear Dyn.

SIAM J. Control Optim.

Commun. Nonlinear Sci. Numer. Simul.

J. Frankl. I.

Neurocomputing

Neurocomputing

J. Frankl. I.

Nonlinear Dyn.

IEEE Trans. Neur. Net. Lear. Syst.

Synchronization and anti-synchronization of n different coupled chaotic systems with ring connection

Int. J. Modern Phys. C

Combination complex synchronization of three chaotic complex systems

Nonlinear Dyn.

Combination synchronization of three classic chaotic systems using active backstepping design

Chaos

Transmission projective synchronization of multi-systems with non-delayed and delayed coupling via impulsive control

Chaos

Adaptive control of multiple chaotic systems with unknown parameters in two different synchronization modes

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Adaptive synchronization of multiple uncertain coupled chaotic systems via sliding mode control

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