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Theoretical Investigations on the Multistability, Quasiperiodicity and Synchronization of the Driven Chua’s Circuit

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Abstract

Analysis of the dynamics of chaotic systems through explicit analytical solutions has been a recent topic of research importance. The chaotic and synchronization dynamics of second-order non-autonomous electronic circuit systems has been studied analytically, while the analytical dynamics of third-order non-autonomous chaotic systems is yet to be studied. In this paper, we develop explicit analytical solutions for the state equations of a third-order, non-autonomous nonlinear electronic circuit system, namely the driven Chua’s circuit. The dynamics of the system is studied using the analytical solutions for different set of system parameters, and the existence of the phenomena of multistability of attractors, period-doubling scenario, antimonotonicity and quasiperiodicity are reported. Analytical solutions are also developed to study the evolution of certain types of synchronization such as phase, phase-lag and complete synchronization observed in coupled non-identical and identical systems. Analytical studies on the chaotic and synchronization dynamics of a third-order, non-autonomous electronic circuit system are reported in the literature for the first time.

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The datasets generated during the current study are not publicly available but are available from the corresponding author on reasonable request.

References

  1. K.A. Abro, A. Atangana, J.F. Gomez-Aguilar, Ferromagnetic chaos in thermal convection of fluid through fractal–fractional differentiations. J. Therm. Anal. Calorim. 147, 8461–8473 (2022)

    Article  Google Scholar 

  2. A.K. Alomari, M.S.M. Noorani, R. Nazar, Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system. Commun. Nonlinear Sci. Numer. Simul. 14(5), 2336–2346 (2009)

    Article  MATH  Google Scholar 

  3. A. Arulgnanam, A. Prasad, K. Thamilmaran, M. Daniel, Analytical study and experimental confirmation of sna through poincaré maps in a quasiperiodically forced electronic circuit. Int. J. Bifurcat. Chaos 25(08), 1530020 (2015)

    Article  MATH  Google Scholar 

  4. A. Arulgnanam, A. Prasad, K. Thamilmaran, M. Daniel, Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation. Chaos Solitons Fract. 75, 96–110 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Arulgnanam, K. Thamilmaran, M. Daniel, Chaotic dynamics with high complexity in a simplified new nonautonomous nonlinear electronic circuit. Chaos Solitons Fract. 42, 2246–2253 (2009)

    Article  MATH  Google Scholar 

  6. A. Arulgnanam, K. Thamilmaran, M. Daniel, A rich spectrum of dynamical phenomenon in a forced parallel LCR circuit with a simple nonlinear element. Chin. J. Phys. 53(3), 060702 (2014)

    MathSciNet  Google Scholar 

  7. L.O. Chua, L. Kocarev, K. Eckert, M. Itoh, Experimental chaos synchronization in Chua’s circuit. Int. J. Bifurc. Chaos 2(3), 705–708 (1992)

    Article  MATH  Google Scholar 

  8. L.O. Chua, L. Kocarev, K. Eckert, M. Itoh, Chaos synchronization in Chua’s circuit. Int. J. Circuits Syst. Comput. 3(1), 93–108 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. L.O. Chua, M. Komuro, T. Matsumoto, The double scroll family. IEEE Trans. Circuits Syst. 33(11), 1072–1118 (1986)

    Article  MATH  Google Scholar 

  10. A. Coronel-Escamilla, J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.R. Razo-Hernández, A.A. Alderremy, S. Aly, Dynamics and synchronization of a fractional conformable neural network with power-law. Eur. Phys. J. Spec. Top. 231, 1771–1788 (2022)

    Article  Google Scholar 

  11. T. Dong, A. Wang, X. Qiao, Dynamics of a coupled Chua’s circuit with lossless transmission line. Circuits Syst. Signal Process. 40, 1962–1985 (2021)

    Article  MATH  Google Scholar 

  12. R. Genesio, A. Tesi, A harmonic balance approach for chaos prediction: Chua’s circuit. Int. J. Bifurcat. Chaos 2(1), 61–79 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Genesio, A. Tesi, Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(3), 531–548 (1992)

    Article  MATH  Google Scholar 

  14. T. Kapitaniak, L.O. Chua, Hyperchaotic attractors of unidirectionally-coupled Chua’s circuits. Int. J. Bifurcat. Chaos 4(2), 477–482 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. M.P. Kennedy, Robust op amp realization of Chua’s circuit. Frequenz 46(3), 66–80 (1992)

    Google Scholar 

  16. J. Li, H. Jahanshahi, S. Kacar, Y. Chu, J.F. Gómez-Aguilar, N.D. Alotaibi, K.H. Alharbi, On the variable-order fractional memristor oscillator: data security applications and synchronization using a type-2 fuzzy disturbance observer-based robust control. Chaos Solitons Fract. 145, 110681 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Liao, On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147(2), 499–513 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Z. Liu, Strange nonchaotic attractors from periodically excited Chua’s circuit. Int. J. Bifurcat. Chaos 11(01), 225–230 (2001)

    Article  Google Scholar 

  19. O. Martínez-Fuentes, J.J. Montesinos-Garcia, J.F. Gomez-Aguilar, Generalized synchronization of commensurate fractional-order chaotic systems: applications in secure information transmission. Digital Signal Process. 126, 103494 (2022)

    Article  Google Scholar 

  20. T. Matsumoto, A chaotic attractor from Chua’s circuit. IEEE Trans. Circuits Syst. 31(12), 1055–1058 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. T. Matsumoto, L.O. Chua, M. Komuro, The double scroll. IEEE Trans. Circuits Syst. 32(8), 797–818 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. K. Murali, M. Lakshmanan, Bifurcation and chaos of the sinusoidally—driven Chua’s circuit. Int. J. Bifurcat. Chaos 01(02), 369–384 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Murali, M. Lakshmanan, Effect of sinusoidal excitation on the Chua’s circuit. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 39(4), 264–270 (1992)

    Article  MATH  Google Scholar 

  24. K. Murali, M. Lakshmanan, Chaotic dynamics of the driven Chua’s circuit. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 40(11), 836–840 (1993)

    Article  MATH  Google Scholar 

  25. K. Murali, M. Lakshmanan, L.O. Chua, The simplest dissipative nonautonomous chaotic circuit. IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 41, 462–463 (1994)

    Article  MATH  Google Scholar 

  26. M. Pecora, T.L. Carroll, Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)

    Article  Google Scholar 

  28. M. Pecora, T.L. Carroll, A. Johnson, J. Mar, J.F. Heagy, Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 7, 520–543 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4196 (1997)

    Article  MATH  Google Scholar 

  30. N.S. Singh, Some applications of special trans function theory in physics. Pramana 95, 32 (2021)

    Article  Google Scholar 

  31. G. Sivaganesh, An analytical study on the synchronization of Murali–Lakshmanan–Chua circuits. Chin. Phys. Lett. 32, 010503 (2015)

    Article  Google Scholar 

  32. G. Sivaganesh, A. Arulgnanam, A.N. Seethalakshmi, Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually coupled simple nonlinear electronic circuits. Chaos Solitons Fract. 113, 294–307 (2018)

    Article  MathSciNet  Google Scholar 

  33. G. Sivaganesh, A. Arulgnanam, A.N. Seethalakshmi, Generalized analytical solutions for secure transmission of signals using a simple communication scheme with numerical and experimental confirmation. Chin. J. Phys. 62, 72–85 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Sivaganesh, K. Srinivasan, A. Arulgnanam, Analytical studies on the dynamics of higher-dimensional nonlinear circuit systems. Pramana 96, 185 (2022)

    Article  Google Scholar 

  35. J.E. Solis-Pérez, J. Betancourt-Vera, J.F. Gomez-Aguilar, A.A. Alderremy, S. Aly, Non-integer order chaotic systems: numerical analysis and their synchronization scheme via m-backstepping technique. Eur. Phys. J. Spec. Top. 231, 1931–1968 (2022)

    Article  Google Scholar 

  36. K. Srinivasan, K. Thamilmaran, A. Venkatesan, Classification of bifurcations and chaos in Chua’s circuit with effect of different periodic forces. Int. J. Bifurcat. Chaos 19(06), 1951–1973 (2009)

    Article  MathSciNet  Google Scholar 

  37. K. Sugandha, P.P. Singh, Generation of a multi-scroll chaotic system via smooth state transformation. J. Comput. Electron. 21, 32 (2022)

    Article  Google Scholar 

  38. W. Szemplińska-Stupnicka, J. Rudowski, Neimark bifurcation, almost-periodicity and chaos in the forced van der pol-duffing system in the neighbourhood of the principal resonance. Phys. Lett. A 192(2), 201–206 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. K. Thamilmaran, M. Lakshmanan, Classification of bifurcations and routes to chaos in a variant of Murali–Lakshmanan–Chua circuit. Int. J. Bifurcat. Chaos 12, 783–813 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. K. Thamilmaran, M. Lakshmanan, K. Murali, Rich variety of bifurcations and chaos in a variant of Murali–Lakshmanan–Chua circuit. Int. J. Bifurcat. Chaos 10(7), 1175–1180 (2000)

    Article  Google Scholar 

  41. P.R. Venkatesh, A. Venkatesan, M. Lakshmanan, Analytical treatment for synchronizing chaos through unidirectional coupling and implementation of logic gates. Pramana 86, 1195–1207 (2016)

    Article  Google Scholar 

  42. E. Viera-Martin, J.F. Gomez-Aguilar, J.E. Solis-Perez, J.A. Hernández-Perez, V.H. Olivares-Peregrino, Anti-synchronization of a m-hopfield neural network with generalized hyperbolic tangent activation function. Eur. Phys. J. Spec. Top. 231, 1801–1814 (2022)

    Article  Google Scholar 

  43. S. Wiggins, Global Bifurcations and Chaos (Springer, New York, 1988)

    Book  MATH  Google Scholar 

  44. P. Xiong, H. Jahanshahi, R. Alcaraz, Y. Chu, J.F. Gomez-Aguilar, F.E. Alsaadi, Spectral entropy analysis and synchronization of a multi-stable fractional-order chaotic system using a novel neural network-based chattering-free sliding mode technique. Chaos Solitons Fract. 144, 110576 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  45. Y. Zhang, Y. Yang, Y. Zhang, Theoretical study of electric energy consumption for self-powered chaos signal generator. SCIENCE CHINA Technol. Sci. 57(6), 1063–1067 (2014)

    Article  Google Scholar 

  46. Z. Zhu, Z. Liu, Strange nonchaotic attractors of Chua’s circuit with quasiperiodic excitation. Int. J. Bifurcat. Chaos 07(01), 227–238 (1997)

    Article  MATH  Google Scholar 

  47. C.J. Zúñiga Aguilar, J.F. Gomez-Aguilar, H.M. Romero-Ugalde, R.F. Escobar-Jimenez, G. Fernandez-Anaya, Fawaz E. Alsaadi, Numerical solution of fractal-fractional mittag-leffler differential equations with variable-order using artificial neural networks. Eng. Comput. 38(3), 2669–2682 (2022)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Alagappa Chettiar Government College of Engineering and Technology, Karaikudi, Tamilnadu, 630 003, India

    G. Sivaganesh

  2. Department of Physics, Nehru Memorial College (Affiliated to Bharathidasan University, Tiruchirapalli, 620024, Tamilnadu, India), Puthanampatti, Tiruchirapalli, Tamilnadu, 621 007, India

    K. Srinivasan

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Sivaganesh, G., Srinivasan, K. Theoretical Investigations on the Multistability, Quasiperiodicity and Synchronization of the Driven Chua’s Circuit. Circuits Syst Signal Process 42, 3200–3228 (2023). https://doi.org/10.1007/s00034-022-02274-2

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