Table of Contents Author Guidelines Submit a Manuscript
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

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

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 [47]. 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 [1114] and complex dynamical networks [1519]. 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 bywhere 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 toBy decomposing the linear and nonlinear parts of the memristor based chaotic circuit system in (8), we can rewrite it aswhere The impulsively controlled memristor based chaotic circuit is given bywhere 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 aswhere 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 aswhere 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 byNote 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).

References

  1. L. O. Chua, “Memristor—the missing circuit element,” IEEE Transactions on Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971. View at Publisher · View at Google Scholar
  2. “Memristor-Wikipedia,” https://en.wikipedia.org/wiki/Memristor.
  3. D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, pp. 80–83, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. Y. V. Pershin, S. La Fontaine, and M. Di Ventra, “Erratum: Memristive model of amoeba learning,” Physical Review E, vol. 82, no. 1, 2010. View at Publisher · View at Google Scholar
  5. A. Afifi, A. Ayatollahi, and F. Raissi, “Implementation of biologically plausible spiking neural network models on the memristor crossbar-based CMOS/nano circuits,” in Proceedings of the ECCTD 2009 - European Conference on Circuit Theory and Design Conference Program, pp. 563–566, Turkey, August 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. J. Borghetti, G. S. Snider, P. J. Kuekes, J. J. Yang, D. R. Stewart, and R. S. Williams, “‘Memristive’ switches enable “stateful” logic operations via material implication,” Nature, vol. 464, no. 7290, pp. 873–876, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. S. H. Jo, T. Chang, I. Ebong, B. B. Bhadviya, P. Mazumder, and W. Lu, “Nanoscale memristor device as synapse in neuromorphic systems,” Nano Letters, vol. 10, no. 4, pp. 1297–1301, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Itoh and L. O. Chua, “Memristor oscillators,” International Journal of Bifurcation and Chaos, vol. 18, no. 11, pp. 3183–3206, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. B. Muthuswamy and P. Kokate, “Memristor-based chaotic circuits,” IETE Technical Review, vol. 26, no. 6, pp. 415–426, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. Z. Lin and H. Wang, “Image encryption based on chaos with PWL memristor in Chua's circuit,” in Proceedings of the 2009 International Conference on Communications, Circuits and Systems (ICCCAS), pp. 964–968, Milpitas, Ca, USA, July 2009. View at Publisher · View at Google Scholar
  11. T. Yang, Impulsive Control Theory, Springer, Berlin , Germany, 2001. View at MathSciNet
  12. T. Yang, L.-B. Yang, and C.-M. Yang, “Impulsive control of Lorenz system,” Physica D: Nonlinear Phenomena, vol. 110, no. 1-2, pp. 18–24, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. Sun, Y. Zhang, and Q. Wu, “Impulsive control for the stabilization and synchronization of Lorenz systems,” Physics Letters A, vol. 298, no. 2-3, pp. 153–160, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Z. Ai and C. Chen, “Asymptotic stability analysis and design of nonlinear impulsive control systems,” Nonlinear Analysis: Hybrid Systems, vol. 24, pp. 244–252, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. C. Li, X. Yu, Z.-W. Liu, and T. Huang, “Asynchronous impulsive containment control in switched multi-agent systems,” Information Sciences, vol. 370, pp. 667–679, 2016. View at Publisher · View at Google Scholar · View at Scopus
  16. X. Yang, J. Cao, and J. Qiu, “Pth moment exponential stochastic synchronization of coupled memristor-based neural networks with mixed delays via delayed impulsive control,” Neural Networks, vol. 65, pp. 80–91, 2015. View at Publisher · View at Google Scholar · View at Scopus
  17. X. Yang and J. Lu, “Finite-time synchronization of coupled networks with Markovian topology and impulsive effects,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 61, no. 8, pp. 2256–2261, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. Q. Song, H. Yan, Z. Zhao, and Y. Liu, “Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects,” Neural Networks, vol. 79, pp. 108–116, 2016. View at Publisher · View at Google Scholar
  19. Q. Song, H. Yan, Z. Zhao, and Y. Liu, “Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays,” Neural Networks, vol. 81, pp. 1–10, 2016. View at Publisher · View at Google Scholar · View at Scopus
  20. W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, USA, 2006. View at MathSciNet
  21. T. Ma, Impulsive Control and Synchronization, Science Press, Beijing, China, 2016.
  22. B. Muthuswamy, “Implementing memristor based chaotic circuits,” International Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 1335–1350, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. S. J. Yang, C. D. Li, and T. W. Huang, “Impulsive control and synchronization of memristor-based chaotic circuits,” International Journal of Bifurcation and Chaos, vol. 24, no. 12, Article ID 1450162, 12 pages, 2014. View at Publisher · View at Google Scholar
  24. T. D. Ma and F. Y. Zhao, “Impulsive stabilization of a class of nonlinear system with bounded gain error,” Chinese Physics B, vol. 23, no. 12, Article ID 120504, 2014. View at Publisher · View at Google Scholar · View at Scopus