Skip to main content
SpringerLink
Log in

Dynamics of a Coupled Chua’s Circuit with Lossless Transmission Line

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper proposes a coupled-circuit system composed of two Chua’s circuits with lossless transmission lines. By applying the Kirchhoff’s voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availability Statement

The data used to support the findings of this study are included in this article.

References

  1. A. Ahamed, M. Lakshmanan, Discontinuity induced Hopf and Neimark–Sacker bifurcations in a memristive Murali–Lakshmanan–Chua circuit. Int. J. Bifurc. Chaos 27(06), 1730021 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ö. Atan, Effects of variable-order passive circuit element in Chua circuit. Circuits Syst. Signal Process. 30(11), 4943–4968 (2019)

    Google Scholar 

  3. V. Arnold, Lectures on partial differential equations (Springer, Berlin, 2013)

    Google Scholar 

  4. B. Bao, T. Jiang, G. Wang et al., Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 89(2), 1157–1171 (2017)

    Article  Google Scholar 

  5. B. Bao, P. Jiang, H. Wu et al., Complex transient dynamics in periodically forced memristive Chua’s circuit. Nonlinear Dyn. 79(4), 2333–2343 (2015)

    Article  MathSciNet  Google Scholar 

  6. M. Borah, P. Singh, B. Roy, Improved chaotic dynamics of a fractional-order system, its chaos-suppressed synchronization and circuit implementation. Circuits Syst. Signal Process. 35(6), 1871–1907 (2016)

    Article  Google Scholar 

  7. M. Chen, Q. Xu, Y. Lin et al., Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit. Nonlinear Dyn. 87(2), 789–802 (2017)

    Article  Google Scholar 

  8. S.S. De Sarkar, S. Chakraborty, Nonlinear dynamics of a class of derivative controlled Chua’s circuit. Int. J. Dyn. Control 6(2), 827–834 (2018)

    Article  MathSciNet  Google Scholar 

  9. T. Dong, Q. Zhang, Dynamics of a hybrid circuit system with lossless transmission line. IEEE Access 8, 92969–92976 (2020)

    Article  Google Scholar 

  10. T. Dong, T. Huang, Neural cryptography based on complex-valued neural network. IEEE Trans. Neural Netw. Learn. Syst. (2019). https://doi.org/10.1109/TNNLS.2019.2955165

  11. T. Dong, A. Wang, X. Liao, Impact of discontinuous antivirus strategy in a computer virus model with the point to group. Appl. Math. Model. 40(4), 3400–3409 (2016)

    Article  MathSciNet  Google Scholar 

  12. T. Dong, Q. Zhang, Stability and oscillation analysis of a gene regulatory network with multiple time delays and diffusion rate. IEEE Trans. Nanobiosci. 19, 285–298 (2020)

    Article  Google Scholar 

  13. T. Dong, L. Xia, Stability and Hopf bifurcation of a reaction–diffusion neutral neuron system with time delay. Int. J. Bifurc. Chaos 27(14), 1750214 (2017)

    Article  MathSciNet  Google Scholar 

  14. T. Dong, L. Xia, Spatial temporal dynamic of a coupled reaction-diffusion neural network with time delay. Cogn. Comput. 11(2), 212–226 (2019)

    Article  Google Scholar 

  15. R. Euzébio, J. Llibre, Zero-Hopf bifurcation in a Chua system. Nonlinear Anal. Real World Appl. 37, 31–40 (2017)

    Article  MathSciNet  Google Scholar 

  16. Q. He, Y. Pu, B. Yu, X. Yuan, Scaling fractal-chuan fractance approximation circuits of arbitrary order. Circuits Syst. Signal Process. 38(11), 4933–4958 (2019)

    Article  Google Scholar 

  17. J. Hale, S.M.V. Lunel, Introduction to functional differential equations. Appl. Math. Sci. (1933)

  18. X. Hu, J. Xia, Z. Wang, X. Song, H. Shen, Robust distributed state estimation for Markov coupled neural networks under imperfect measurements. J. Franklin Inst. 357(1), 2420–2436 (2020)

    Article  MathSciNet  Google Scholar 

  19. B. Hassard, N. Kazarinoff, Y. Wan, Theory and applications of Hopf bifurcation (Cambridge University Press, Cambridge, 1981)

    MATH  Google Scholar 

  20. J. Jin, L.V. Zhao, Low voltage low power fully integrated chaos generator. J. Circuits Syst. Comput. 27(10), 1850155 (2018)

    Article  Google Scholar 

  21. E. Kengne, A. Lakhssassi, Analytical study of dynamics of matter-wave solitons in lossless nonlinear discrete bi-inductance transmission lines. Phys. Rev. E 91(3), 032907 (2015)

    Article  MathSciNet  Google Scholar 

  22. X. Liao, Dynamical behavior of Chua’s circuit with lossless transmission line. IEEE Trans. Circuits Syst. I Regul. Pap. 63(10), 781–783 (2016)

    MathSciNet  Google Scholar 

  23. Q. Liu, X. Liao, Y. Liu et al., Dynamics of an inertial two-neuron system with time delay. Nonlinear Dyn. 58(3), 573 (2009)

    Article  MathSciNet  Google Scholar 

  24. A.N. Sharkovsky, Chaos from a time-delayed Chua’s circuit. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 40(10), 781–783 (1993)

    Article  Google Scholar 

  25. W.S. Sayed, A.G. Radwan, H.A.H. Fahmy et al., Software and hardware implementation sensitivity of chaotic systems and impact on encryption applications. Circuits Syst. Signal Process. 39, 5638–5655 (2020)

    Article  Google Scholar 

  26. S. Shahsavari, M. Saberi, A power-efficient CMOS active rectifier with circuit delay compensation for wireless power transfer systems. Circuits Syst. Signal Process. 38(2), 947–966 (2019)

    Article  Google Scholar 

  27. C. Shen, S. Yu, J. Lü et al., A systematic methodology for constructing hyperchaotic systems with multiple positive Lyapunov exponents and circuit implementation. IEEE Trans. Circuits Syst. I Regul. Pap. 61(3), 854–864 (2013)

    Article  Google Scholar 

  28. N. Stankevich, N. Kuznetsov, G. Leonov et al., Scenario of the birth of hidden attractors in the Chua circuit. Int. J. Bifurc. Chaos 27(12), 1730038 (2017)

    Article  MathSciNet  Google Scholar 

  29. Z. Taskiran, U. Ayten, H. Sedef, Dual-output operational transconductance amplifier-based electronically controllable memristance simulator circuit. Circuits Syst. Signal Process. 38(2), 26–40 (2019)

    Article  Google Scholar 

  30. C. Wang, J. Wei, Normal forms for NFDEs with parameters and application to the lossless transmission line. Nonlinear Dyn. 52(3), 199–206 (2008)

    Article  MathSciNet  Google Scholar 

  31. A. Wang, T. Dong, X. Liao, Event-triggered synchronization strategy for complex dynamical networks with the Markovian switching topologies. Neural Netw. 74, 52–57 (2016)

    Article  Google Scholar 

  32. Q. Xu, Y. Lin, B. Bao et al., Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos, Solitons Fractals 83, 186–200 (2016)

    Article  MathSciNet  Google Scholar 

  33. W. Yu, J. Cao, Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay. Nonlinear Anal. Theory Methods Appl. 62(1), 141–165 (2005)

    Article  MathSciNet  Google Scholar 

  34. J. Zhang, X. Liao, Synchronization and chaos in coupled memristor-based FitzHugh–Nagumo circuits with memristor synapse. AEU-Int. J. Electron. Commun. 75, 82–90 (2017)

    Article  Google Scholar 

  35. C. Zheng, T. Fernando, Analysis and generation of chaos using compositely connected coupled memristors. Chaos Interdiscip. J. Nonlinear Sci. 28(6), 063115 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Key Research and Development Project of China under Grant 2018AAA0100101, in part by Fundamental Research Funds for the Central Universities under Grant XDJK2020B009, in part by the Chongqing Technological Innovation and Application Project under Grant cstc2018jszx-cyzdX0171, in part by Chongqing Basic and Frontier Research Project under Grant cstc2019jcyj-msxm2105 and cstc2020jcyj-msxmX0139, in part by the Science and Technology Research Program of Chongqing Municipal Education Commission under Grant KJQN201900816, in part by Chongqing Social Science Planning Project under Grant 2019BS053.

Author information

Authors and Affiliations

  1. Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronics and Information Engineering, Southwest University, Chongqing, 400715, China

    Tao Dong, Aiqing Wang & Xing Qiao

Authors
  1. Tao Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aiqing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xing Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Dong.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, T., Wang, A. & Qiao, X. Dynamics of a Coupled Chua’s Circuit with Lossless Transmission Line. Circuits Syst Signal Process 40, 1962–1985 (2021). https://doi.org/10.1007/s00034-020-01563-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-020-01563-y

Keywords

Search

Navigation