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Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme Multistability, and Bifurcations Without Parameters

  • Fernando Corinto
  • Mauro Forti
  • Leon O. Chua
Chapter
  • 110 Downloads

Abstract

Chapter  5 has developed the flux-charge analysis method (FCAM) for the analysis of a class \(\mathcal {L}\mathcal {M}\) of memristor circuits containing memristors, linear resistors, inductors, capacitors, and independent voltage and current sources. The formulation of circuit equations (DAEs and SEs) has been provided in the (φ, q)-domain and in the (v, i)-domain and relationships between the analysis in the two domains have been discussed. A fundamental property is that the SE description in the (φ, q)-domain of a circuit in \(\mathcal {L}\mathcal {M}\) has a lower order with respect to the corresponding description in the (v, i)-domain, so that it is expected that the dynamic analysis in the former domain is simpler. The goal of this chapter is to highlight some main advantages of FCAM by studying the dynamics of a number of low-order autonomous memristor circuits in the (φ, q)-domain, namely, circuits described by a first-order SE in the (φ, q)-domain, oscillatory circuits described by a second-order SE in the (φ, q)-domain, and chaotic circuits described by a third-order SE in the (φ, q)-domain. Such a study permits to show that memristor circuits display a number of fundamental peculiar dynamic features. First of all, the state space in the (v, i)-domain can be foliated in a continuum of invariant manifolds, thus implying the coexistence of infinitely many different reduced-order dynamics and attractors. Moreover, such circuits display bifurcations due to changing the initial conditions for a fixed set of circuit parameters that are named bifurcations without parameters.

References

  1. 1.
    H.K. Khalil, Nonlinear Systems (Prentice Hall, Englewood Cliffs, 2002)Google Scholar
  2. 2.
    L.O. Chua, Dynamic nonlinear networks: state-of-the-art. IEEE Trans. Circuits Syst. 27(11), 1059–1087 (1980)MathSciNetCrossRefGoogle Scholar
  3. 3.
    T. Matsumoto, L. Chua, A. Makino, On the implications of capacitor-only cutsets and inductor-only loops in nonlinear networks. IEEE Trans. Circuits Syst. 26(10), 828–845 (1979)MathSciNetCrossRefGoogle Scholar
  4. 4.
    V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, 1988)CrossRefGoogle Scholar
  5. 5.
    O. Merino, A short history of complex numbers. University of Rhode Island, Kingston (2006)Google Scholar
  6. 6.
    B. Fiedler, S. Liebscher, J.C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters: I. Theory. J. Differ. Equ. 167(1), 16–35 (2000)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Itoh, L.O. Chua, Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.P. Aubin, A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory (Springer, Berlin, 1984)Google Scholar
  9. 9.
    A. Mees, L. Chua, The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems. IEEE Trans. Circuits Syst. 26(4), 235–254 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C.R. Hens, R. Banerjee, U. Feudel, S.K. Dana, How to obtain extreme multistability in coupled dynamical systems. Phys. Rev. E 85(3), 035202 (2012)Google Scholar
  11. 11.
    M. da Cruz Scarabello, M. Messias, Bifurcations leading to nonlinear oscillations in a 3D piecewise linear memristor oscillator. Int. J. Bifurc. Chaos 24(1), 1430001 (2014)Google Scholar
  12. 12.
    M. Messias, C. Nespoli, V.A. Botta, Hopf bifurcation from lines of equilibria without parameters in memristor oscillators. Int. J. Bifurcation Chaos 20(2), 437–450 (2010)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Amador, E. Freire, E. Ponce, J. Ros, On discontinuous piecewise linear models for memristor oscillators. Int. J. Bifurcation Chaos 27(6), 1730022 (2017)Google Scholar
  14. 14.
    F. Corinto, A. Ascoli, M. Gilli, Nonlinear dynamics of memristor oscillators. IEEE Trans. Circuits Syst. I: Regul. Pap. 58(6), 1323–1336 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Gilli, F. Corinto, P. Checco, Periodic oscillations and bifurcations in cellular nonlinear networks. IEEE Trans. Circ. Syst. I: Regul. Pap. 51(5), 948–962 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Di Marco, M. Forti, G. Innocenti, A. Tesi, Harmonic balance method to analyze bifurcations in memristor oscillatory circuits. Int. J. Circuit Theory Appl. 46(1), 66–83 (2018)CrossRefGoogle Scholar
  17. 17.
    G. Innocenti, M. Di Marco, M. Forti, A. Tesi, Prediction of period doubling bifurcations in harmonically forced memristor circuits. Nonlinear Dyn. 96(2), 1169–1190 (2019)CrossRefGoogle Scholar
  18. 18.
    B.-C. Bao, Q. Xu, H. Bao, M. Chen, Extreme multistability in a memristive circuit. Electron. Lett. 52(12), 1008–1010 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    A. Buscarino, C. Corradino, L. Fortuna, M. Frasca, L.O. Chua, Turing patterns in memristive cellular nonlinear networks. IEEE Trans. Circuits Syst. I: Regul. Pap. 63(8), 1222–1230 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A.I. Ahamed, M. Lakshmanan, Nonsmooth bifurcations, transient hyperchaos and hyperchaotic beats in a memristive Murali–Lakshmanan–Chua circuit. Int. J. Bifurcation Chaos 23(6), 1350098 (2013)Google Scholar
  21. 21.
    V.-T. Pham, S. Vaidyanathan, C.K. Volos, S. Jafari, N.V. Kuznetsov, T.M. Hoang, A novel memristive time-delay chaotic system without equilibrium points. Eur. Phys. J. Spec. Top. 225(1), 127–136 (2016)CrossRefGoogle Scholar
  22. 22.
    B. Bao, Z. Ma, J. Xu, Z. Liu, Q. Xu, A simple memristor chaotic circuit with complex dynamics. Int. J. Bifurc. Chaos 21(9), 2629–2645 (2011)CrossRefGoogle Scholar
  23. 23.
    Q. Li, S. Hu, S. Tang, G. Zeng, Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int. J. Circuit Theory Appl. 42(11), 1172–1188 (2014)CrossRefGoogle Scholar
  24. 24.
    R. Riaza, C. Tischendorf, Semistate models of electrical circuits including memristors. Int. J. Circuit Theory Appl. 39(6), 607–627 (2011)CrossRefGoogle Scholar
  25. 25.
    R. Riaza, Manifolds of equilibria and bifurcations without parameters in memristive circuits. SIAM J. Appl. Math. 72(3), 877–896 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    F. Corinto, A. Ascoli, M. Gilli, Analysis of current–voltage characteristics for memristive elements in pattern recognition systems. Int. J. Circuit Theory Appl. 40(12), 1277–1320 (2012)CrossRefGoogle Scholar
  27. 27.
    B.C. Bao, H. Bao, N. Wang, M. Chen, Q. Xu, Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94, 102–111 (2017)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    H. Bao, N. Wang, B. Bao, M. Chen, P. Jin, G. Wang, Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun. Nonlinear Sci. Numer. Simul. 57, 264–275 (2018)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    A. Ascoli, R. Tetzlaff, Z. Biolek, Z. Kolka, The art of finding accurate memristor model solutions. IEEE J. Emerg. Sel. Top. Circuits Syst. 5(2), 133–142 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2021

Authors and Affiliations

  • Fernando Corinto
    • 1
  • Mauro Forti
    • 2
  • Leon O. Chua
    • 3
  1. 1.Department of Electronics & TelecommunicationsPolitecnico di TorinoTorinoItaly
  2. 2.Department of Information Engineering and MathematicsUniversity of SienaSienaItaly
  3. 3.University of CaliforniaBerkeleyUSA

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