Complex Dynamics and Synchronization Phenomena in Arrays of Memristor Oscillators

  • Fernando Corinto
  • Mauro Forti
  • Leon O. Chua


A great deal of efforts have been traditionally devoted in circuit theory to analyze nonstationary steady-state behaviors in networks obtained by locally coupled arrays of simple dynamic circuits (also named cells, oscillators, units, etc.). These arrays can be thought of as a bio-inspired circuit model of complex nonlinear phenomena observable in nature with potential applications in signal processing and computing systems. In fact, on one hand, they are a mean for reproducing, analyzing, and understanding spatiotemporal nonlinear phenomena displayed by spatially extended networks found in such diverse fields as electrical engineering, computer science, biology, and physics. On the other hand, complex spatiotemporal dynamics including chaos are potentially useful for developing future analogue computing systems. Recent studies have shown for instance that chaos can play a crucial role in searching for the global solution of combinatorial optimization problems and chaotic relaxation oscillators with memristors have been used to boot efficiency and accuracy of Hopfield-like computing networks.


  1. 1.
    L.O. Chua (Ed.), Special issue on nonlinear waves, patterns and spatio-temporal chaos in dynamic arrays. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 42(10), 557–823 (1995)Google Scholar
  2. 2.
    S. Kumar, J.P. Strachan, R. Stanley Williams, Chaotic dynamics in nanoscale NbO 2 Mott memristors for analogue computing. Nature 548(7667), 318 (2017)Google Scholar
  3. 3.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, vol. 12 (Cambridge University Press, Cambridge, 2003)CrossRefGoogle Scholar
  4. 4.
    A.-L. Barabási, Linked: The New Science of Networks (American Association of Physics Teachers, College Park, 2003)Google Scholar
  5. 5.
    G.V. Osipov, V.D. Shalfeev, The evolution of spatio-temporal disorder in a chain of unidirectionally-coupled Chua’s circuits. IEEE Trans. Circ. Syst. I Fund. Theory Appl. 42(10), 687–692 (1995)CrossRefGoogle Scholar
  6. 6.
    M.J. Ogorzalek, Z. Galias, A.M. Dabrowski, W.R. Dabrowski, Chaotic waves and spatio-temporal patterns in large arrays of doubly-coupled Chua’s circuits. IEEE Trans. Circ. Syst. I Fund. Theory Appl. 42(10), 706–714 (1995)CrossRefGoogle Scholar
  7. 7.
    F. Kavaslar, C. Guzelis, A computer-assisted investigation of a 2-D array of Chua’s circuits. IEEE Trans. Circ. Syst. I Fund. Theory Appl. 42(10), 721–735 (1995)CrossRefGoogle Scholar
  8. 8.
    E. Sánchez, M.A. Matías, V. Pérez-Muñuzuri, Chaotic synchronization in small assemblies of driven Chua’s circuits. IEEE Trans. Circ. Syst. I Fund. Theory Appl. 47(5), 644–654 (2000)CrossRefGoogle Scholar
  9. 9.
    M. de Magistris, M. di Bernardo, E. Di Tucci, S. Manfredi, Synchronization of networks of non-identical Chua’s circuits: analysis and experiments. IEEE Trans. Circ. Syst. I Regul. Pap. 59(5), 1029–1041 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Bogojeska, M. Mirchev, I. Mishkovski, L. Kocarev, Synchronization and consensus in state-dependent networks. IEEE Trans. Circ. Syst. I Regul. Pap. 61(2), 522–529 (2014)CrossRefGoogle Scholar
  11. 11.
    H. Liu, M. Cao, C.W. Wu, Coupling strength allocation for synchronization in complex networks using spectral graph theory. IEEE Trans. Circ. Syst. I Regul. Pap. 61(5), 1520–1530 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    W.K. Wong, W. Zhang, Y. Tang, X. Wu, Stochastic synchronization of complex networks with mixed impulses. IEEE Trans. Circ. Syst. I Regul. Pap. 60(10), 2657–2667 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Kim, M. Jin, P.H. Chang, Control and synchronization of the generalized Lorenz system with mismatched uncertainties using backstepping technique and time-delay estimation. Int. J. Circuit Theory Appl. (2017).
  14. 14.
    L.O. Chua, Memristor-The missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  15. 15.
    A. Ascoli, V. Lanza, F. Corinto, R. Tetzlaff, Synchronization conditions in simple memristor neural networks. J. Franklin Inst. 352(8), 3196–3220 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    V. Erokhin, T. Berzina, P. Camorani, A. Smerieri, D. Vavoulis, J. Feng, M.P. Fontana, Material memristive device circuits with synaptic plasticity: learning and memory. BioNanoScience 1(1–2), 24–30 (2011)CrossRefGoogle Scholar
  17. 17.
    Z. Wang, S. Ambrogio, S. Balatti, D. Ielmini, A 2-transistor/1-resistor artificial synapse capable of communication and stochastic learning in neuromorphic systems. Front. Neurosci. 8, (2014)Google Scholar
  18. 18.
    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
  19. 19.
    L.V. Gambuzza, A. Buscarino, L. Fortuna, M. Frasca, Memristor-based adaptive coupling for consensus and synchronization. IEEE Trans. Circ. Syst. I Regul. Pap. 62(4), 1175–1184 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    E. Bilotta, F. Chiaravalloti, P. Pantano, Spontaneous synchronization in two mutually coupled memristor-based Chua’s circuits: numerical investigations. Math. Problems Eng. 2014, 594962 (2014)CrossRefGoogle Scholar
  21. 21.
    E. Bilotta, F. Chiaravalloti, P. Pantano, Synchronization and waves in a ring of diffusively coupled memristor-based Chua’s circuits. Acta Appl. Math. 132(1), 83–94 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    J. Ma, F. Wu, G. Ren, J. Tang, A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Q. Xu, Y. Lin, B. Bao, M. Chen, Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solitons Fractals 83, 186–200 (2016)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    B. Bao, T. Jiang, G. Wang, P. Jin, H. Bao, M. Chen, Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 1–15 (2017)Google Scholar
  26. 26.
    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
  27. 27.
    W. Lu, T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems. Phys. D Nonlinear Phenom. 213(2), 214–230 (2006)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    C.W. Wu, L.O. Chua, Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circuits Syst. I Fund. Theory Appl. 42(8), 430–447 (1995)MathSciNetCrossRefGoogle Scholar
  29. 29.
    M. Itoh, L.O. Chua, Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    B.-C. Bao, Q. Xu, H. Bao, M. Chen, Extreme multistability in a memristive circuit. Electron. Lett. 52(12), 1008–1010 (2016)ADSCrossRefGoogle Scholar
  31. 31.
    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
  32. 32.
    M.E. Yalcin, Dynamic behavior of 1-D array of the memristively-coupled Chua’s circuits, in 2013 8th International Conference on Electrical and Electronics Engineering (ELECO) (IEEE, Piscataway, 2013), pp. 13–16CrossRefGoogle Scholar
  33. 33.
    L. Ponta, V. Lanza, M. Bonnin, F. Corinto, Emerging dynamics in neuronal networks of diffusively coupled hard oscillators. Neural Netw. 24(5), 466–475 (2011)CrossRefGoogle Scholar
  34. 34.
    V. Lanza, L. Ponta, M. Bonnin, F. Corinto, Multiple attractors and bifurcations in hard oscillators driven by constant inputs. Int. J. Bifurc. Chaos 22(11), 1250267 (2012)Google Scholar
  35. 35.
    R. Genesio, A. Tesi, Distortion control of chaotic systems: the Chua’s circuit. J. Circuits Syst. Comput. 3(1), 151–171 (1993)MathSciNetCrossRefGoogle Scholar
  36. 36.
    F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks, vol. 126 (Springer, Berlin, 2012)Google Scholar

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© 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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