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Stability analysis of a class of electronic circuits based on thermodynamic principles part II: analysis of chaos in Chua’s circuit

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Abstract

This is the second part of a two-part manuscript focused on the analysis of electronic circuits employing extended irreversible thermodynamics. In this contribution, internal entropy production is used as a Lyapunov function for a thermodynamically consistent model of Chua’s circuit to address stability properties of an isolated, isothermal system containing it. In order to achieve this, simpler electronic circuits are first analyzed with an increasing complexity in construction. Here an RC-nonlinear resistor circuit and an RLC-nonlinear resistor circuit are considered as a preamble of the study of Chua’s circuit. In this manner, the entropy production approach allows conservative and dissipative phenomena and their interactions to be identified and contrasted with those of simpler study cases. Identification of recurrent terms in the dynamics of the entropy production function leads to the description of elemental interactions among electronic components. Analytical results are supported with numerical simulations at specific conditions. Chaos and other complex behaviors, like limit cycles, are analyzed and described with entropic and energetic perspectives. Apart from the determination of their stability, entropic descriptions for nodes, saddles and foci observed in the study cases and their vicinities are presented. It is observed through an extended Gibbs free energy analysis that the dissipation in limit cycles equals the build-up of energy. On the contrary, the behavior of changes of the state variables in a strange attractor is energetically asynchronous and unbalanced.

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References

  1. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MathSciNet  Google Scholar 

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

  3. Chua, L.O., Komuro, M., Matsumoto, T.: The double scroll family. Parts I and II. IEEE Trans. Circuits Syst. 33(11), 1072–1118 (1986)

    Article  Google Scholar 

  4. Chua, L.O.: A zoo of strange attractors from the canonical Chua’s circuits. In: Proceedings of the 35th Midwest Symposium on Circuits and Systems, 1992, pp. 916–926. IEEE (1992)

  5. Chua, L.O., Wu, C.W., Huang, A., Zhong, G.Q.: A universal circuit for studying and generating chaos. I. Routes to chaos. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 40(10), 732–744 (1993)

    Article  MathSciNet  Google Scholar 

  6. Chua, L.O., Wu, C.W., Huang, A., Zhong, G.Q.: A universal circuit for studying and generating chaos II Strange attractors. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 40(10), 745–761 (1993)

    Article  MathSciNet  Google Scholar 

  7. Chua, L.O.: Chua’s circuit: ten years later. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 77(11), 1811–1822 (1994)

  8. Dedieu, H., Kennedy, M.P., Hasler, M.: Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 40(10), 634–642 (1993)

  9. Peng, G., Min, F.: Multistability analysis, circuit implementations and application in image encryption of a novel memristive chaotic circuit. Nonlinear Dyn. 90(3), 1607–1625 (2017)

    Article  Google Scholar 

  10. Swathy, P.S., Thamilmaran, K.: An experimental study on SC-CNN based canonical Chua’s circuit. Nonlinear Dyn. 71(3), 505–514 (2013)

  11. Radwan, A.G., Soliman, A.M., El-Sedeek, A.L.: An inductorless CMOS realization of Chua’s circuit. Chaos Solitons Fractals 18(1), 149–158 (2003)

  12. Kılıc, R., Çam, U., Alçı, M., Kuntman, H., Uzunhisarcıklı, E.: Realization of inductorless Chua’s circuit using FTFN-based nonlinear resistor and inductance simulator. Frequenz 58, 37–40 (2004)

  13. Bao, B., Wang, N., Chen, M., Xu, Q., Wang, J.: Inductor-free simplified Chua’s circuit only using two-op-amp-based realization. Nonlinear Dyn. 84(2), 511–525 (2016)

  14. Zhong, G.Q., Ayrom, F.: Experimental confirmation of chaos from Chua’s circuit. Int. J. Circuit Theory Appl. 13(1), 93–98 (1985)

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

    Article  MathSciNet  Google Scholar 

  16. Matsumoto, T., Chua, L.O., Tokumasu, K.: Double scroll via a two-transistor circuit. IEEE Trans. Circuits Syst. 33(8), 828–835 (1986)

    Article  MathSciNet  Google Scholar 

  17. Kennedy, M.P.: Robust op amp realization of Chua’s circuit. Frequenz 46(3–4), 66–80 (1992)

  18. Senani, R., Gupta, S.S.: Implementation of Chua’s chaotic circuit using current feedback op-amps. Electron. Lett. 34(9), 829–830 (1998)

  19. Elwakil, A.S., Kennedy, M.P.: Improved implementation of Chua’s chaotic oscillator using current feedback op amp. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47(1), 76–79 (2000)

  20. Cruz, J.M., Chua, L.O.: A CMOS IC nonlinear resistor for Chua’s circuit. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 39(12), 985–995 (1992)

  21. Rodriguez-Vazquez, A., Delgado-Restituto, M.: CMOS design of chaotic oscillators using state variables: a monolithic Chua’s circuit. IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process. 40(10), 596–613 (1993)

  22. Zhong, G.Q.: Implementation of Chua’s circuit with a cubic nonlinearity. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41(12), 934–941 (1994)

  23. Eltawil, A.M., Elwakil, A.S.: Low-voltage chaotic oscillator with an approximate cubic nonlinearity. AEU-Int. J. Electron. C. 53(3), 11–17 (1999)

    Google Scholar 

  24. Srisuchinwong, B., San-Um, W.: A Chua’s chaotic oscillator based on a coarsely cubic-like CMOS resistor. In: Proc. Asia-Pacific Conf. Communications, pp. 47–49 (2007)

  25. O’Donoghue, K., Kennedy, M.P., Forbes, P.: A fast and simple implementation of Chua’s oscillator using a ’cubic-like’ Chua diode. In: Proc. European Conf. Circuit Theory and Design, vol. 2, pp. II/83–II/86 (2005)

  26. Munguía-Medina, S.J., García-Sandoval, J.P., González-Álvarez, A.: Stability analysis of a class of electronic circuits based on thermodynamic principles part I: analysis of limit cycles (submitted to Nonlinear Dynamics). Nonlinear Dyn. (2021)

  27. Demirel, Y.: Nonequilibrium Thermodynamics: Transport and Rate Processes in Physical, Chemical and Biological Systems, 4th edn. Elsevier, New York (2018)

    MATH  Google Scholar 

  28. Dammers, W.R., Tels, M.: Thermodynamic stability and entropy production in adiabatic stirred flow reactors. Chem. Eng. Sci. 29(1), 83–90 (1974)

    Article  Google Scholar 

  29. Landauer, R.: Stability and entropy production in electrical circuits. J. Stat. Phys. 13(1), 1–16 (1975)

    Article  Google Scholar 

  30. Ydstie, B.E.: Passivity based control via the second law. Comput. Chem. Eng. 26(7–8), 1037–1048 (2002)

    Article  Google Scholar 

  31. Jeltsema, D., Scherpen, J.M.A.: A power-based description of standard mechanical systems. Syst. Control Lett. 56(5), 349–356 (2007)

    Article  MathSciNet  Google Scholar 

  32. Favache, A., Dochain, D.: Thermodynamics and chemical systems stability: the CSTR case study revisited. J. Process Control 19(3), 371–379 (2009)

    Article  Google Scholar 

  33. Jeltsema, D., Scherpen, J.M.A.: Multidomain modeling of nonlinear networks and systems. IEEE Control. Syst. 29(4) (2009)

  34. Robinett, R.D., Wilson, D.G.: Exergy and irreversible entropy production thermodynamic concepts for nonlinear control design. Int. J. Exergy 6(3), 357–387 (2009)

    Article  Google Scholar 

  35. García-Canseco, E., Jeltsema, D., Ortega, R., Scherpen, J.M.A.: Power-based control of physical systems. Automatica 46(1), 127–132 (2010)

    Article  MathSciNet  Google Scholar 

  36. García-Sandoval, J.P., González-Álvarez, V., Calderón, C.: Stability analysis and passivity properties for a class of chemical reactors: internal entropy production approach. Comput. Chem. Eng. 75, 184–195 (2015)

    Article  Google Scholar 

  37. García-Sandoval, J.P., Dochain, D., Hudon, N.: Dissipative and conservative structures for thermo-mechanical systems. IFAC-PapersOnLine 48(8), 1057–1064 (2015)

    Article  Google Scholar 

  38. García-Sandoval, J.P., Hudon, N., Dochain, D., González-Álvarez, V.: Stability analysis and passivity properties of a class of thermodynamic processes: an internal entropy production approach. Chem. Eng. Sci. 139, 261–272 (2016)

    Article  Google Scholar 

  39. Alonso, A.A., Ydstie, B.E.: Stabilization of distributed systems using irreversible thermodynamics. Automatica 37(11), 1739–1755 (2001)

    Article  MathSciNet  Google Scholar 

  40. Favache, A., Dochain, D., Maschke, B.: An entropy-based formulation of irreversible processes based on contact structures. Chem. Eng. Sci. 65(18), 5204–5216 (2010)

    Article  Google Scholar 

  41. van der Schaft, A.J.: Port-controlled Hamiltonian systems: towards a theory for control and design of nonlinear physical systems. J. Soc. Instrum. Control Eng. 39(2), 91–98 (2000)

    Google Scholar 

  42. van der Schaft, A., Jeltsema, D.: Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends in Systems and Control. Now Publishers (2014)

  43. García-Sandoval, J.P., Hudon, N., Dochain, D.: Generalized Hamiltonian representation of thermo-mechanical systems based on an entropic formulation. J. Process Control 51, 18–26 (2017)

    Article  Google Scholar 

  44. Yu, H., Yu, J., Wu, H., Li, H.: Energy-shaping and integral control of the three-tank liquid level system. Nonlinear Dyn. 73(4), 2149–2156 (2013)

    Article  MathSciNet  Google Scholar 

  45. Kang, J., Zhu, Z.H.: A unified energy-based control framework for tethered spacecraft deployment. Nonlinear Dyn. 95(2), 1117–1131 (2019)

    Article  Google Scholar 

  46. García-Sandoval, J.P., Hudon, N., Dochain, D.: Conservative and dissipative phenomena in thermodynamical systems stability. IFAC-PapersOnLine 49(24), 28–33 (2016)

    Article  Google Scholar 

  47. Chua, L.O., Huynh, L.T.: Bifurcation analysis of Chua’s circuit. In: Proceedings of the 35th Midwest Symposium on Circuits and Systems, pp. 746–751. IEEE (1992)

  48. Komuro, M., Tokunaga, R., Matsumoto, T., Chua, L.O., Hotta, A.: Global bifurcation analysis of the double scroll circuit. Int. J. Bifurc. Chaos 1(01), 139–182 (1991)

    Article  MathSciNet  Google Scholar 

  49. Parlitz, U.: Lyapunov exponents from Chua’s circuit. J. Circuits Syst. Comput. 3(02), 507–523 (1993)

  50. Sira-Ramirez, H., Cruz-Hernández, C.: Synchronization of chaotic systems: a generalized Hamiltonian systems approach. Int. J. Bifur. Chaos 11(05), 1381–1395 (2001)

    Article  Google Scholar 

  51. Bar-Yam, Y.: Dynamics of Complex Systems. Addison-Wesley, Boston (1997)

    MATH  Google Scholar 

  52. Chen, L., Zhou, Y., Yang, F., Zhong, S., Zhang, J.: Complex dynamical behavior in memristor-capacitor systems. Nonlinear Dyn. 98(1), 517–537 (2019)

    Article  Google Scholar 

  53. Ramirez, H., Le Gorrec, Y., Maschke, B., Couenne, F.: Passivity based control of irreversible port Hamiltonian systems. IFAC Proc. Vol. 46(14), 84–89 (2013)

    Article  Google Scholar 

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Acknowledgements

Sergio Javier Munguía-Medina thanks Consejo Nacional de Ciencia y Tecnología (CONACyT) for the financial support under Grant Agreement No. 706022.

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Authors and Affiliations

  1. Departamento de Ingeniería Química, Universidad de Guadalajara, Blvd. Marcelino García Barragán #1451, Guadalajara, Jalisco, 44430, Mexico

    Sergio Javier Munguía-Medina, Juan Paulo García-Sandoval & Alejandro González-Álvarez

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  1. Sergio Javier Munguía-Medina
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  2. Juan Paulo García-Sandoval
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  3. Alejandro González-Álvarez
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Contributions

Sergio Javier Munguía-Medina involved in formal analysis, investigation, resources, software, visualization, writing—original draft. Juan Paulo García-Sandoval involved in conceptualization, formal analysis, methodology, project administration, resources, supervision, validation, writing—review & editing. Alejandro González-Álvarez involved in resources, supervision, writing—review & editing.

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Correspondence to Juan Paulo García-Sandoval.

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Munguía-Medina, S.J., García-Sandoval, J.P. & González-Álvarez, A. Stability analysis of a class of electronic circuits based on thermodynamic principles part II: analysis of chaos in Chua’s circuit. Nonlinear Dyn 105, 3637–3658 (2021). https://doi.org/10.1007/s11071-021-06753-0

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