Abstract
An electronic model of Duffing oscillator with a characteristic memristive nonlinear element is proposed instead of the classical cubic nonlinearity. The memristive Duffing oscillator circuit system is mathematically modeled, and the stability analysis presents the evolution of the proposed system. The dynamical behavior of this circuit is investigated through numerical simulations, statistical analysis, and real-time hardware experiments, which have been carried out under the external periodic force. The chaotic dynamics of the circuit is studied by means of phase diagram. It is found that the proposed circuit system shows complex behaviors, like bifurcations and chaos, three tori, transient chaos, and intermittency for a certain range of circuit parameters. The observed phenomena and scenario are illustrated in detail through experimental and numerical studies of memristive Duffing oscillator circuit. The existence of regular and chaotic behaviors is also verified by using 0–1 test measurements. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio. The numerically observed results are confirmed from the laboratory experiment.
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Ho, Y., Huang, G.M., Li, P.: “Nonvolatile memristor memory: device characteristics and design implications, In: Computer-Aided Design-Digest of Technical Papers, 2009. ICCAD 2009. IEEE/ACM International Conference on (IEEE, 2009) pp. 485–490
Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: “The missing memristor found”. Nature 453, 80–83 (2008)
Pershin, Y.V., La Fontaine, S., Di Ventra, M.: “Memristive model of amoeba learning”. Phys. Rev. E 80, 021926 (2009)
Pershin, Y.V., Di Ventra, M.: “Spin memristive systems: spin memory effects in semiconductor spintronics”. Phys. Rev. B 78, 113309 (2008)
Pershin, Y.V., Ventra, D.: “Frequency doubling and memory effects in the spin Hall effect”. Phys. Rev. B 79, 153307 (2009)
Tour, J.M., He, T.: “Electronics: the fourth element”. Nature 453, 42–43 (2008)
Mohanty, S.P.: Memristor: from basics to deployment. Potentials IEEE 32, 34–39 (2013)
Chua, L.O.: Memristor-the missing circuit element. Circuit Theory IEEE Trans. 18, 507–519 (1971)
Wang, D., Hu, Z., Yu, X., Yu, J.: “A pwl model of memristor and its application example”, In: Communications, Circuits and Systems, 2009. ICCCAS 2009. International Conference on (IEEE, 2009) pp. 932–934
Corinto, F., Ascoli, A., Gilli, M.: “Nonlinear dynamics of memristor oscillators”. Circuits Syst. I Regul. Pap. IEEE Trans. 58, 1323–1336 (2011)
Teng, L., Iu, H.H.C., Wang, X., Wang, X.: “Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial”. Nonlinear Dyn. 77, 231–241 (2014)
Wu, H., Bao, B., Liu, Z., Xu, Q., Jiang, P.: “Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator”. Nonlinear Dyn. 83, 893–903 (2016)
Cafagna, D., Grassi, G.: “On the simplest fractional-order memristor-based chaotic system”. Nonlinear Dyn. 70, 1185–1197 (2012)
Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20, 1335–1350 (2010)
Muthuswamy, B., Kokate, P.P.: Memristor-based chaotic circuits. IETE Tech. Rev. 26, 417–429 (2009)
Radwan, A.G., Zidan, M.A., Salama, K.N.: “Hp memristor mathematical model for periodic signals and dc”, In: Circuits and Systems (MWSCAS), 2010 53rd IEEE International Midwest Symposium on (IEEE, 2010) pp. 861–864
Prodromakis, T., Peh, B.P., Papavassiliou, C., Toumazou, C.: “A versatile memristor model with nonlinear dopant kinetics”. Electron Devices IEEE Trans. 58, 3099–3105 (2011)
Ahamed, A.I., Lakshmanan, M.: “Nonsmooth bifurcations, transient hyperchaos and hyperchaotic beats in a memristive Murali-Lakshmanan-Chua circuit”. Int. J. Bifurc. Chaos 23, 1350098 (2013)
Chen, M., Yu, J., Yu, Q., Li, C., Bao, B.: “A memristive diode bridge-based canonical Chua’s circuit”. Entropy 16, 6464–6476 (2014)
Bo-Cheng, B., Jian-Ping, X., Guo-Hua, Z., Zheng-Hua, M., Ling, Z.: “Chaotic memristive circuit: equivalent circuit realization and dynamical analysis”. Chin. Phys. B 20, 120502 (2011)
Talukdar, A.H.: Nonlinear dynamics of memristor based 2nd and 3rd order oscillators, Ph.D. thesis (2011)
Bo-Cheng, B., Jian-Ping, X., Zhong, L.: Initial state dependent dynamical behaviors in a memristor based chaotic circuit. Ł 27, 70504–070504 (2010)
Chua, L.: Resistance switching memories are memristors. Appl. Phys. A. 102, 765–783 (2011)
Borghetti, J., Li, Z., Straznicky, J., Li, X., Ohlberg, D.A.A., Wu, W., Stewart, D.R., Williams, R.S.: “A hybrid nanomemristor/transistor logic circuit capable of self-programming”. Proc. Natl. Acad. Sci. 106, 1699–1703 (2009)
Xu, C., Dong, X., Jouppi, N.P., Xie, Y.: “Design implications of memristor-based RRAM cross-point structures”, In: Design, Automation & Test in Europe Conference & Exhibition (DATE), 2011 (IEEE, 2011) pp. 1–6
Mouttet, B.: “Proposal for memristors in signal processing”, In: Nano-Net (Springer) pp. 11–13 (2009)
Thomas, A.: Memristor-based neural networks. J. Phys. D Appl. Phys. 46, 093001 (2013)
Rajendran, J., Manem, H., Karri, R., Rose, G.S.: “Memristor based programmable threshold logic array”, In: Proceedings of the 2010 IEEE/ACM International Symposium on Nanoscale Architectures (IEEE Press, 2010) pp. 5–10
Kim, K.-H., Gaba, S., Wheeler, D., Cruz-Albrecht, J.M., Hussain, T., Srinivasa, N., Lu, W.: A functional hybrid memristor crossbar-array/cmos system for data storage and neuromorphic applications. Nano Lett. 12, 389–395 (2011)
Itoh, M., Chua, L.O.: “Memristor oscillators”. Int. J. Bifurc. Chaos 18, 3183–3206 (2008)
Zhang, G., Hu, J., Shen, Y.: “New results on synchronization control of delayed memristive neural networks”. Nonlinear Dyn. 81, 1167–1178 (2015)
Zhang, G., Shen, Y.: “Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control”. Neural Netw. 55, 1–10 (2014)
Podhaisky, H., Marszalek, W.: “Bifurcations and synchronization of singularly perturbed oscillators: an application case study”. Nonlinear Dyn. 69, 949–959 (2012)
Marszalek, W., Podhaisky, H.: “2d bifurcations and Newtonian properties of memristive Chua’s circuits”. EPL 113, 10005 (2016)
Megam Ngouonkadi, E.B., Fotsin, H.B., Fotso, P.L.: “Implementing a memristive van der pol oscillator coupled to a linear oscillator: synchronization and application to secure communication”. Phys. Scr. 89, 035201 (2014)
George, D.: “Erzwungene schwingung bei vernderlicher eigenfrequenz und ihre technische bedeutung”, Vieweg (1918)
Kovacic, I., Brennan, M.J.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, UK (2011)
Linsay, P.S.: Period doubling and chaotic behavior in a driven anharmonic oscillator. Phys. Rev. Lett. 47, 1349 (1981)
Sabarathinam, S., Thamilmaran, K.: Transient chaos in a globally coupled system of nearly conservative Hamiltonian Duffing oscillators. Chaos Solitons Fractals 73, 129–140 (2015)
Bao, B., Jiang, P., Wu, H., Hu, F.: “Complex transient dynamics in periodically forced memristive chuas circuit”. Nonlinear Dyn. 79, 2333–2343 (2014)
Wu, W., Chen, Z., Yuan, Z.: “The evolution of a novel four-dimensional autonomous system: among 3-torus, limit cycle, 2-torus, chaos and hyperchaos”. Chaos Solitons Fractals 39, 2340–2356 (2009)
Prasad, A., Mehra, V., Ramaswamy, R.: “Intermittency route to strange nonchaotic attractors”. Phys. Rev. Lett. 79, 4127 (1997)
Lakshmanan, M., Rajaseekar, S.: Nonlinear Dynamics: Integrability, Chaos and Patterns. Springer, Berlin (2012)
Johnson, D.H.: “Signal-to-noise ratio”. 1,2088, revision 91770 (2006)
Gottwald, G.A., Melbourne, I.: “On the implementation of the 0–1 test for chaos”. SIAM J. Appl. Dyn. Syst. 8, 129–145 (2009)
Falconer, I., Gottwald, G.A., Melbourne, I., Wormnes, K.: “Application of the 0–1 test for chaos to experimental data”. SIAM J. Appl. Dyn. Syst. 6, 395–402 (2007)
Lai, Y.-C., Tél, T.: Transient Chaos: Complex Dynamics on Finite Time Scales, vol. 173. Springer, Berlin (2011)
Lai, Y.-C., Winslow, R.L.: “Geometric properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems”. Phys. Rev. Lett. 74, 5208 (1995)
Bleher, S., Grebogi, C., Ott, E.: “Bifurcation to chaotic scattering”. Phys. D Nonlinear Phenom. 46, 87–121 (1990)
Jung, C., Tél, T., Ziemniak, E.: “Application of scattering chaos to particle transport in a hydrodynamical flow”. Chaos Interdiscip. J. Nonlinear Sci. 3, 555–568 (1993)
Dhamala, M., Lai, Y.-C.: “Controlling transient chaos in deterministic flows with applications to electrical power systems and ecology”. Phys. Rev. E 59, 1646 (1999)
Tél, T., Lai, Y.-C.: “Chaotic transients in spatially extended systems”. Phys. Rep. 460, 245–275 (2008)
Bo-Cheng, B., Zhong, L., Jian-Ping, X.: “Transient chaos in smooth memristor oscillator”. Chin. Phys. B 19, 030510 (2010)
Mukouyama, Y., Kawasaki, H., Hara, D., Nakanishi, S.: “Transient chaotic behavior during simultaneous occurrence of two electrochemical oscillations”. J. Solid State Electrochem. 19, 3253–3263 (2015)
Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: “Phase synchronization of chaotic oscillators”. Phys. Rev. Lett. 76, 1804 (1996)
Manneville, P., Pomeau, Y.: “Intermittency and the Lorenz model”. Phys. Lett. A 75, 1–2 (1979)
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S.S. acknowledges University Grants Commission (UGC) for the financial assistance through RFSMS scheme. K.T. acknowledges DST, Govt. of India, for the financial support through the Grant No. SB/EMEQ-077/2013.
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Sabarathinam, S., Volos, C.K. & Thamilmaran, K. Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator. Nonlinear Dyn 87, 37–49 (2017). https://doi.org/10.1007/s11071-016-3022-8
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DOI: https://doi.org/10.1007/s11071-016-3022-8