Abstract
Memristive circuit with infinitely many equilibrium points can exhibit the special phenomenon of extreme multistability, whose dynamics mechanism and physical control are significant issues deserving in-depth investigations. In this paper, a control strategy for extreme multistability exhibited in an active band pass filter-based memristive circuit is explored in flux–charge domain. To this end, an incremental flux–charge model is established with four additional constant parameters reflecting the initial conditions of all dynamic elements. Thus, the line equilibrium point only related to memristor initial condition in the voltage–current domain is transformed into some determined equilibrium points, whose locations and stabilities are explicitly related to all four initial conditions. Consequently, the initial condition-dependent extreme multistability phenomenon, which has not been quantitatively analyzed in the voltage–current domain, can readily be investigated through evaluating these determined equilibrium points. Most important of all, the initial condition-dependent dynamical behaviors are formulated as the system parameter-dependent behaviors in the newly constructed flux–charge model and thus can be rigorously captured in a hardware equivalent realization circuit. Numerical simulations and experimental measurements reveal that the control of extreme multistability is successfully achieved in flux–charge domain, which is significant for seeking potential engineering applications of multistable memristive circuits.
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Fortuna, L., Frasca, M., Xibilia, M.G.: Chua’s Circuit Implementations: Yesterday, Today and Tomorrow. World Scientific, Singapore (2009)
Wang, X., Vaidyanathan, S., Volos, C., Pham, V.T., Kapitaniak, T.: Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors. Nonlinear Dyn. 89(3), 1673–1687 (2017)
Pham, V.T., Volos, C., Jafari, S., Kapitaniak, T.: Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. 87(3), 2001–2010 (2017)
Wang, Z., Akgul, A., Pham, V.T., Jafari, S.: Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors. Nonlinear Dyn. 89(3), 1877–1887 (2017)
Sprott, J.C., Jafari, S., Khalaf, A.J.M., Kapitaniak, T.: Megastability: coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. Eur. Phys. J. Spec. Top. 226(9), 1979–1985 (2017)
Ngouonkadi, E.B.M., Fotsin, H.B., Fotso, P.L., Tamba, V.K., Cerdeira, H.A.: Bifurcations and multistability in the extended hindmarsh-rose neuronal oscillator. Chaos Solitons Fractals 85, 151–163 (2016)
Li, C.B., Sprott, J.C.: Multistability in the Lorenz system: a broken butterfly. Int. J. Bifurc. Chaos 24(10), 1450131 (2014)
Xu, Q., Lin, Y., Bao, B.C., Chen, M.: Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solitons Fractals 83, 186–200 (2016)
Ojoniyi, O.S., Njah, A.N.: A 5D hyperchaotic Sprott B system with coexisting hidden attractors. Chaos Solitons Fractals 87, 172–181 (2016)
Jaros, P., Perlikowski, P., Kapitaniak, T.: Synchronization and multistability in the ring of modified Rössler oscillators. Eur. Phys. J. Spec. Top. 224(8), 1541–1552 (2015)
Njitacke, Z.T., Fotsin, H.B., Negou, A.N., Tchiotsop, D.: Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit. Chaos Solitons Fractals 91, 180–197 (2016)
Kengne, J., Tabekoueng, Z.N., Tamba, V.K., Negou, A.N.: Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos 25(10), 103126 (2015)
Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83(1–2), 751–765 (2016)
Bao, B.C., Li, Q.D., Wang, N., Xu, Q.: Multistability in Chua’s circuit with two stable node-foci. Chaos 26(4), 043111 (2016)
Chen, M., Xu, Q., Lin, Y., Bao, B.C.: Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit. Nonlinear Dyn. 87(2), 789–802 (2017)
Bao, B.C., Jiang, T., Wang, G.Y., Jin, P.P., Bao, H., Chen, M.: Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 89(2), 1157–1171 (2017)
Bao, B.C., Jiang, T., Xu, Q., Chen, M., Wu, H.G., Hu, Y.H.: Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn. 86(3), 1711–1723 (2016)
Yuan, F., Wang, G.Y., Wang, X.W.: Extreme multistability in a memristor-based multi-scroll hyper-chaotic system. Chaos 26(7), 073107 (2016)
Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractor manipulation and robustness. Chaos 25(5), 053112 (2015)
Patel, M.S., Patel, U., Sen, A., Sethia, G.C., Hens, C., Dana, S.K., Feudel, U., Showalter, K., Ngonghala, C.N., Amritkar, R.E.: Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators. Phys. Rev. E 89(2), 022918 (2014)
Hens, C.R., Banerjee, R., Feudel, U., Dana, S.K.: How to obtain extreme multistability in coupled dynamical systems. Phys. Rev. E 85(3), 035202 (2012)
Ngonghala, C.N., Feudel, U., Showalter, K.: Extreme multistability in a chemical model system. Phys. Rev. E 83(5), 056206 (2011)
Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540, 167–218 (2014)
Li, C.B., Pehlivan, I., Sprott, J.C.: Amplitude-phase control of a novel chaotic attractor. Turk. J. Electr. Eng. Comput. Sci. 24, 1–11 (2016)
Sharma, P.R., Shrimali, M.D., Prasad, A., Kuznetsov, N.V., Leonov, G.A.: Control of multistability in hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1485–1491 (2015)
Dudkowski, D., Prasad, A., Kapitaniak, T.: Perpetual points: new tool for localization of coexisting attractors in dynamical systems. Int. J. Bifurc. Chaos 27(4), 1750063 (2017)
Gotthans, T., Petrzela, J.: New class of chaotic systems with circular equilibrium. Nonlinear Dyn. 81, 1143–1149 (2015)
Jafari, S., Sprott, J.C., Molaie, M.: A simple chaotic flow with a plane of equilibria. Int. J. Bifurc. Chaos 26(6), 1650098 (2016)
Li, Q.D., Hu, S.Y., Tang, S., Zeng, G.: 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)
Bao, B.C., Hu, F.W., Liu, Z., Xu, J.P.: Mapping equivalent approach to analysis and realization of memristor based dynamical circuit. Chin. Phys. B 23(7), 070503 (2014)
Fitch, A.L., Yu, D.S., Iu, H.H.C., Sreeram, V.: Hyperchaos in a memristor-based modified canonical Chua’s circuit. Int. J. Bifurc. Chaos 22(6), 1250133 (2012)
Yuan, F., Wang, G.Y., Wang, X.W.: Chaotic oscillator containing memcapacitor and meminductor and its dimensionality reduction analysis. Chaos 27(3), 033103 (2017)
Bao, B.C.: Reply: Comment on ’Is memristor a dynamic element?’. Electron. Lett. 50(19), 1344–1345 (2014)
Corinto, F., Forti, M.: Memristor circuits: flux–charge analysis method. IEEE Trans. Circuits Syst. I Reg. Pap. 63(11), 1997–2009 (2016)
Corinto, F., Forti, M.: Memristor circuits: bifurcations without parameters. IEEE Trans. Circuits Syst. I Reg. Pap. 64(6), 1540–1551 (2017)
Yang, Q.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos 18(5), 1393–1414 (2008)
Qi, G.Y., Chen, G.R.: A spherical chaotic system. Nonlinear Dyn. 81(3), 1381–1392 (2015)
Tahir, F.R., Jafari, S., Pham, V.T., Volos, C., Wang, X.: A novel no-equilibrium chaotic system with multiwing butterfly attractors. Int. J. Bifurc. Chaos 25(4), 1550056 (2015)
Li, H.F., Wang, L.D., Duan, S.K.: A memristor-based scroll chaotic system—design, analysis and circuit implementation. Int. J. Bifurc. Chaos 24(7), 1450099 (2014)
Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20(5), 1335–1350 (2010)
Bao, B.C., Bao, H., Wang, N., Chen, M., Xu, Q.: Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94, 102–111 (2017)
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This work was supported by the National Natural Science Foundation of China under Grant Nos. 61601062, 51607013, and 51277017 and the Natural Science Foundation of Jiangsu Province, China, under Grant No. BK20160282.
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Chen, M., Sun, M., Bao, B. et al. Controlling extreme multistability of memristor emulator-based dynamical circuit in flux–charge domain. Nonlinear Dyn 91, 1395–1412 (2018). https://doi.org/10.1007/s11071-017-3952-9
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DOI: https://doi.org/10.1007/s11071-017-3952-9