- Research
- Open Access
Sandwich synchronization of memristor-based hyperchaos systems with time delays
- Hongjuan Wu^{1}Email author,
- Jiang Xiong^{1},
- Xiang Hu^{1},
- Yuming Feng^{1} and
- Liangliang Li^{2}
https://doi.org/10.1186/s13662-017-1451-4
© The Author(s) 2018
- Received: 14 August 2017
- Accepted: 14 December 2017
- Published: 16 January 2018
Abstract
In this paper, a memristor-based hyperchaotic system is introduced. Considering time delays between the drive system and the response system in the process of synchronization, this paper designs one kind of flexible sandwich controller, which includes a rest in the sandwich structure, to realize the synchronization between two memristor-based hyperchaotic systems. Based on Lyapunov stability theory, matrix inequality, sandwich control and considering time delays, the exponential synchronization conditions for the memristor-based hyperchaotic systems with time delays via sandwich control are given. Finally, simulation results are displayed to verify the effectiveness and feasibility of this method.
Keywords
- memristor-based
- hyperchaotic system
- time delays
- synchronization
- sandwich control
1 Introduction
Memristor as the fourth fundamental circuit element was first proposed by Chua [1] in 1971 based on logical symmetry arguments, and it was realized by Hewlett-Packard [2] research team in 2008. This passive electronic device has generated unprecedented worldwide interest because of its potential applications in signal processing, programmable logic, control system, neural network, brain-computer interface [3], etc.
Recently, the research on memristor-based circuits is becoming a hot topic. A lot of memristor oscillator systems have been used in generating signals which are found in satellite communications, radio, switching power supply, etc. [4–10]. With the potential memristor applications, it is necessary to do some deep research on the related nonlinear memristor-based oscillator systems [11–13]. Itoh and Chua [14] derived several nonlinear oscillators from Chua’s oscillators by replacing Chua’s diodes with memristors. Bao et al. [15, 16] studied the complicated dynamical behaviors of the memristor oscillators. Although various memristor-based chaotic systems have been researched in recent years [17–19], the research of synchronization between two memristor-based hyperchaotic systems is rarely reported. Because the synchronization of the memristor-based chaotic systems is a challenging problem [20–23], chaotic behavior, especially the hyperchaotic behavior that has more than one positive Lyapunov exponent, may be uncoordinated and unpredictable.
Sandwich control is one kind of discontinuous control. It can be used in many industrial fields [24]. It could include many subsystems that are continuous. Feng et al. [25] studied the sandwich structure control system that includes two continuous controls and an impulsive control in each period and applied it to control Chua’s oscillator. While this paper will talk about another kind of flexible sandwich structure, which is different from [25]. In each period of this sandwich control system, the first and third parts of the control system are continuous controls, which may be continuous controls with different control gains. Between these two parts, there is a rest. This kind of sandwich control structure is very suitable for these systems that cannot be controlled continuously all the time.
In this paper, we apply this kind of sandwich control to ensure the synchronization between two memristor-based hyperchaotic systems. We pay attention to time delays between the drive system and the response system when we control the error system [26–29], because there are always some transmission time delays between the drive system and the response system in the real environment. Based on Lyapunov stability theory, matrix inequality, sandwich control and considering time delays, the exponential synchronization conditions for the memristor-based hyperchaotic systems with time delays via sandwich control are given.
2 The fourth-order memristor-based hyperchaotic system
Remark 1
Although various memristor-based chaotic systems have been researched extensively in recent years, the research of memristor-based hyperchaotic systems is rarely reported and investigated directly. Thus the hyperchaotic system (5) is important for understanding of memristor-based hyperchaotic systems.
3 Synchronization of the memristor-based hyperchaotic systems with time delays
Remark 2
In the real environment, there are always some time delays between the drive system and the response system. Thus considering time delays between the drive system and the response system in the process of synchronization is of great practical significance.
Remark 3
The sandwich control put forward by this paper is a general model, which can be used as a prototype of other discontinuous controls that include more than two continuous controls with different control gains in each period.
Lemma 1
([32])
Next, we will find the proper T, \(I_{1}\), \(I_{2}\), \(\theta_{1}\), \(\theta _{2}\), \(s_{1}\), \(s_{2}\), \(s_{3}\) to ensure the synchronization between drive system (6) and response system (8). In other words, if the stability of error system (11) can be guaranteed, drive system (6) and response system (8) can realize synchronization.
Theorem 1
- (1)
\(A+A^{T}+2I_{1}E+\varepsilon_{1} BB^{T}+\varepsilon_{1}^{-1} \tilde{L}^{2}E+s_{1}E\le 0\),
- (2)
\(A+A^{T}+\varepsilon_{2}BB^{T}+\varepsilon_{2}^{-1}\tilde{L}^{2}E-s _{2}E\le 0\),
- (3)
\(A+A^{T}+2I_{2}E+\varepsilon_{3} BB^{T}+\varepsilon_{3}^{-1} \tilde{L}^{2}E+s_{3}E\le 0\),
- (4)
\({{s}_{1}}{{\theta }_{1}}-{{s}_{2}}{{\theta }_{2}}+{{s}_{3}}(1- {{\theta }_{1}}-{{\theta }_{2}})>0\), where L̃ is the largest Lipschitz coefficient, then error system (11) is exponentially stable. That is, the exponential synchronization between system (6) and system (8) with time delays will be realized.
Proof
Therefore, we get that
Case 1. When \(n=0\), then
Similarly, we get that
Case 2. When \(n=1\), then
By induction, we get the following.
Case \(m+1\). When \(n=m\), then
Therefore, in this situation, for any \(t>0\), if \({{s}_{1}}{{\theta } _{1}}-{{s}_{2}}{{\theta }_{2}}+{{s}_{3}}(1-{{\theta }_{1}}-{{\theta } _{2}})>0\), error system (11) is exponentially stable, which implies system (6) and system (8) with time delays can realize exponential synchronization. □
Corollary 1
4 Simulation results
5 Conclusions
In this paper, the characteristics of a memristor-based hyperchaotic system have been discussed. Based on Lyapunov stability theory, matrix inequality, sandwich control and considering time delays, this paper designed one type of sandwich controller and applied it to realize the exponential synchronization between two memristor-based hyperchaotic systems with transmission time delays. Simulation results were given to verify the effectiveness of this method.
6 Competing interests
The authors declare that they have no competing interests.
Declarations
Acknowledgements
The authors sincerely thank the referees for their helpful suggestions, which greatly improved the paper. This research is supported by Chongqing Municipal Key Laboratory of Institutions of Higher Education (Grant No. [2017]3) and Program of Chongqing Development and Reform Commission (Grant No. 2017[1007]).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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