Complexity
Volume 2017 (2017), Article ID 5186714, 10 pages
https://doi.org/10.1155/2017/5186714

Research Article

Stabilization and Synchronization of Memristive Chaotic Circuits by Impulsive Control

Limin Zou,¹ Yang Peng,² Yuming Feng,³ and Zhengwen Tu³

¹School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
²School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100, China
³Key Laboratory of Intelligent Information Processing and Control, Chongqing Three Gorges University, Chongqing 404100, China

Correspondence should be addressed to Limin Zou; limin-zou@163.com

Received 14 April 2017; Revised 26 June 2017; Accepted 6 November 2017; Published 5 December 2017

Academic Editor: Sigurdur F. Hafstein

Copyright © 2017 Limin Zou et al. 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.

Abstract.

The purpose of this note is to study impulsive control and synchronization of memristor based chaotic circuits shown by Muthuswamy. We first establish a less conservative sufficient condition for the stability of memristor based chaotic circuits. After that, we discuss the effect of errors on stability. Meanwhile, we also discuss impulsive synchronization of two memristor based chaotic systems. Our results are more general and more applicable than the ones shown by Yang, Li, and Huang. Finally, several numerical examples are given to show the effectiveness of our methods.

1. Introduction

The memristor was postulated as the fourth circuit element by Chua [1, 2] and was realized by HP Labs [2, 3]. Memristor has widely potential applications in electronic circuits, computer memory, reconfigurable computing, and so on [4–7]. Recently, implementation of memristor based chaotic circuits is an active topic of research. For example, some memristor based chaotic circuits were proposed by Itoh and Chua [8] and Muthuswamy and Kokate [9]. Memristor based chaotic circuits have applications in many fields; for example, a memristor based chaotic circuit for image encryption was proposed by Lin and Wang [10].

In practical applications, impulsive control has some advantages: for example, impulsive control provides an efficient way in dealing with systems especially which cannot endure continuous control inputs. During the last several decades, impulsive control theory has attracted considerable attention because impulsive control method can be employed in many fields, such as the stabilization and synchronization of chaotic systems [11–14] and complex dynamical networks [15–19]. For more results on impulsive control and its applications, the reader is referred to [11, 20, 21] and the references therein.

Recently, complex dynamical systems are receiving much attention, and there is no exception for chaotic systems. Muthuswamy [22] provided a practical implementation of a memristor based chaotic circuit. By applying impulsive control theory, Yang et al. [23] obtained some sufficient conditions for the asymptotic stabilization and synchronization of the memristor based chaotic system shown in [22].

In this note, we shall also consider the asymptotic stabilization and synchronization of memristor based chaotic circuits, as in [22]. We first derive a less conservative sufficient condition for the stability of the memristor based chaotic circuits shown by Muthuswamy [22]. In many practical applications, we cannot guarantee the systems without any error due to the limit of equipment and technology. For this reason, we discuss the effect of errors on stability in this note. Meanwhile, we also discuss impulsive synchronization of two memristor based chaotic systems. Compared with the results shown in [23], our methods are more general and more applicable. Finally, we give some numerical examples which show the effectiveness of our methods.

2. Memristor Based Chaotic Circuit and Its Equivalent Form

The equations for the memristor based chaotic circuit presented in [22] are described by

where

The author of [22] chose a cubic nonlinearity for the function:

and so the memductance function is given by

We choose the system parameters as

which make system (1) chaotic [22, 23]. Figure 1 shows the chaotic phenomenon of this system with the initial condition .

Figure 1: The state trajectory of the memristor based chaotic circuit shown in (1).

In the sequel, we mainly adopt the notation and terminology in [23]. To analyze the asymptotic stabilization of system (1), we let

Then the memristor based chaotic circuit (1) can be rewritten as

which is equivalent to

By decomposing the linear and nonlinear parts of the memristor based chaotic circuit system in (8), we can rewrite it as

where

The impulsively controlled memristor based chaotic circuit is given by

where denote the moments when impulsive control occurs and is impulsive control gain. Without loss of generality, we assume that .

3. Impulsive Control of the Memristor Based Chaotic Circuit Shown by Muthuswamy

In this section, we design impulsive control for the memristor based chaotic circuit shown by Muthuswamy.

Theorem 1. Let be the largest eigenvalue of and suppose that is the largest eigenvalues of . Then the origin of impulsive control system (11) is asymptotically stable if

Proof. Let us construct the following Lyapunov function:

It is easy to verify that conditions 1 and 4 of Theorem   in [11] are satisfied. When , we have

Hence, condition 2 of Theorem   in [11] is satisfied with

From the fact that

and is finite, we know that there exists a such that , which implies that for all . When , we have

Hence, condition 3 of Theorem   in [11] is satisfied with

It follows from Theorem   in [11] that the asymptotic stability of the impulsive control system (11) is implied by that of the following comparison system:

It follows from Theorem  3.1.4 in [11] that if

is satisfied, then the origin of (11) is asymptotically stable. This completes the proof.

Remark 2. Let

and suppose that is the largest eigenvalues of . Yang et al. showed in [23] that the origin of impulsive control system (11) is asymptotically stable if

Since

our result is less conservative than Theorem of [23]. Meanwhile, our method is also simpler than Theorem of [23], because we do not need to calculate the supremum of .
In many practical applications, we cannot guarantee the impulses without any error due to the limit of equipment and technology. So we should take into account the influence of impulsive control gain errors on the systems. Motivated by the above discussions, we will study the stabilization of system (11) with bounded impulsive control gain error. The corresponding system can be described as

where is gain error, which is often time-varying and bounded. As pointed out in [21, 24], we can assume that ,  , and .

Theorem 3. Let be the largest eigenvalue of

where and suppose that is the largest eigenvalues of . Then the origin of impulsive control system (24) is asymptotically stable if

Proof. Let us construct the following Lyapunov function:

When , we have

where . The rest of proof is the same as that of Theorem 1, so we omit it here for simplicity. This completes the proof.

In many practical applications, the parameters ,  ,  , and   may also contain errors. In what follows, we will consider system (24) with parameter uncertainty. The corresponding system can be described as

where is the parametric uncertainty and has the following form: ,  ,  .

Theorem 4. Let be the largest eigenvalue of

where and suppose that is the largest eigenvalues of . Then the origin of impulsive control system (29) is asymptotically stable if

Proof. Let us construct the following Lyapunov function:

When , we have

where . The rest of proof is the same as that of Theorem 1, so we omit it here for simplicity. This completes the proof.

4. Impulsive Synchronization of the Memristor Based Chaotic Circuit Shown by Muthuswamy

In this section, we investigate impulsive synchronization of two memristor based chaotic circuits. Equation (9) is the driving system and the driven system is defined as

where is impulsive control gain and is the synchronization error. Then the error system of the impulsive synchronization is given by

Note that

where

The eigenvalues of are

From Figure 1, we know that the state variable of (11) is bounded and suppose that , .

Theorem 5. Let be the largest eigenvalue of and suppose that is the largest eigenvalues of . Then the origin of impulsive synchronization error system (35) is asymptotically stable if

Proof. Let us construct the following Lyapunov function:

When , we have

The rest of proof is the same as that of Theorem 1, which is omitted here for simplicity. This completes the proof.

5. Numerical Examples

In this section, some numerical examples are given to illustrate the effectiveness of our results. The initial condition of the system (11) is .

Example 1. It is easy to see that

and . In this example, we choose the impulsive control gain matrix as

Then, we have . By Theorem 1, we know that if

holds, then origin of impulsive control system (11) is asymptotically stable. To do this, we choose ; then we have

The simulation results with are shown in Figure 2.

Figure 2: The state trajectory of the controlled memristor based chaotic circuit.

Example 2. In this example, the coefficient matrix and the impulsive control gain matrix are the same as Example 1. Suppose that . Then, we have . By Theorem 3, we know that if

holds, then origin of impulsive control system (24) is asymptotically stable. To do this, we choose ,  ; then we have and so

The simulation results with are shown in Figure 3.

Figure 3: The state trajectory of the controlled memristor based chaotic circuit.

Example 3. In this example, the matrices , , and are the same as Example 2. For the sake of simplicity, is specified as

Then, we have . By Theorem 4, we know that if

holds, then origin of impulsive control system (29) is asymptotically stable. To do this, we choose ,  ; then we have ,   and so

The simulation results with are shown in Figure 4.

Figure 4: The state trajectory of the controlled memristor based chaotic circuit.

Example 4. In this example, the matrix is the same as Example 1. We choose the matrix as

Then, we have ,  . Meanwhile, we know that ,  . By Theorem 5, we know that if

holds, then the origin of impulsive synchronization error system (35) is asymptotically stable. To do this, we choose and so

The initial condition of the driving system (9) is also and the initial condition of the driven system (35) is . The simulation results with are shown in Figure 5.

Figure 5: Simulation results of synchronization errors.

6. Conclusion

In this note, we discuss impulsive control and synchronization of memristor based chaotic circuits shown by Muthuswamy [22]. Our first result is less conservative than Theorem of [23]. Meanwhile, we also discuss the effect of errors on stability, so our results are more general and more applicable than the ones shown in [23].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final version of this paper.

Acknowledgments

This work is funded by Chongqing Research Program of Basic Research and Frontier Technology (no. cstc2017jcyjAX0032), the National Natural Science Foundation of China under Grant no 11601047, Key Laboratory of Chongqing Municipal Institutions of Higher Education (Grant no. 20173), and project supported by Program of Chongqing Development and Reform Commission (Grant no. 20171007).

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