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Generally, the occurrence of multiple attractors indicates that the multi-stability existing in a nonlinear dynamical system and the long-time motion behavior are essentially different, depending on which basin of attraction the initial condition belongs to. Up to now, due to the emergence of multi-stability, some particular memristor-based nonlinear circuits whose dynamical behaviors are extremely related to memristor initial conditions or other initial conditions have attracted considerable attention. By replacing linear or nonlinear resistors with memristor emulators in some already-existing oscillating circuits or introducing memristor emulators with different nonlinearities into these oscillating circuits, various memristor-based nonlinear dynamical circuits have been constructed and broadly investigated. Motivated by these considerations, we present a novel fifth-order voltage-controlled memristor-based Chua's chaotic circuit in this paper, from which a wonderful phenomenon of bi-stability is well demonstrated by numerical simulations and PSIM circuit simulations. Note that the bi-stability is a special kind of multi-stability, which is rarely reported in the memristor-based chaotic circuits.
The proposed memristor-based Chua's chaotic circuit is constructed by inserting an inductor into the coupled resistor branch in series and substituting the Chua's diode with a voltage-controlled memristor in the classical Chua's circuit. Five-dimensional system model is established, of which the equilibrium point and its stability are investigated. Theoretical derivation results indicate that the proposed circuit owns one or three equilibrium points related to the circuit parameters. Especially, unlike the newly reported memristive circuit with bi-stability, the proposed memristor-based Chua's chaotic circuit has only one zero equilibrium point under the given parameters, but it can generate coexistent chaotic and periodic behaviors, and the bi-stability occurs in such a memristive Chua's circuit. By theoretical analyses, numerical simulations and PSIM circuit simulations, the bi-stability phenomenon of coexistent chaotic attractors and periodic limit cycles with different initial conditions and their formation mechanism are revealed and expounded. Besides, with the dimensionless system equations, the corresponding initial condition-dependent dynamical behaviors are further numerically explored through bifurcation diagram, Lyapunov exponents, phased portraits and attraction basin. Numerical simulation results demonstrate that the proposed memristive Chua's system can generate bi-stability under different initial conditions. The PSIM circuit simulations and numerical simulations are consistent well with each other, which perfectly verifies the theoretical analyses.-
Keywords:
- bi-stability /
- dynamical behaviors /
- voltage-controlled memristor /
- Chua's circuit
[1] Chua L O 1971 IEEE Trans. Circuit Theory 18 507
[2] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[3] Kim H, Sah M P, Yang C, Cho S, Chua L O 2012 IEEE Trans. Circuits Syst. I: Regular Papers 59 2422
[4] Yu D S, Iu H H C, Fitch A L, Liang Y 2014 IEEE Trans. Circuits Syst. I: Regular Papers 61 2888
[5] Sánchez-López C, Mendoza-López J, Carrasco-Aguilar M A, Muñiz-Montero C 2014 IEEE Trans. Circuits Syst. Ⅱ: Express Briefs 61 309
[6] Yang C, Choi H, Park S, Sah M P, Kim H, Chua L O 2015 Semicond. Sci. Technol. 30 015007
[7] Yu D S, Zheng C Y, Iu H H C, Fernando T, Chua L O 2017 IEEE Access 5 1284
[8] Corinto F, Ascoli A 2012 Electron. Lett. 48 824
[9] Bao B C, Yu J J, Hu F W, Liu Z 2014 Int. J. Bifurcation Chaos 24 1450143
[10] Wu H G, Bao B C, Liu Z, Xu Q, Jiang P 2016 Nonlinear Dyn. 83 893
[11] Xu Q, Lin Y, Bao B C, Chen M 2016 Chaos, Solitons Fractals 83 186
[12] Bao B C, Wang N, Xu Q, Wu H G, Hu Y H 2017 IEEE Trans. Circuits Syst. Ⅱ: Express Briefs 64 977
[13] Xu Q, Zhang Q L, Bao B C, Hu Y H 2017 IEEE Access 5 21039
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[15] Chen M, Li M Y, Yu Q, Bao B C, Xu Q, Wang J 2015 Nonlinear Dyn. 81 215
[16] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B 23 070503
[17] Bao B C, Jiang P, Wu H G, Hu F W 2015 Nonlinear Dyn. 79 2333
[18] Fitch A L, Yu D S, Iu H H C, Sreeram V 2012 Int. J. Bifurcation Chaos 22 1250133
[19] Bao B C, Ma Z H, Xu J P, Liu Z, Xu Q 2011 Int. J. Bifurcation Chaos 21 2629
[20] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[21] Zhao Y B, Zhang X Z, Xu J, Guo Y C 2015 Chaos, Solitons Fractals 81 315
[22] Bao B C, Hu F W, Chen M, Xu Q, Yu Y J 2015 Int. J. Bifurcation Chaos 25 1550075
[23] Li C B, Sprott J C 2014 Int. J. Bifurcation Chaos 24 1450034
[24] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255
[25] Wei Z C, Wang R R, Liu A P 2014 Math. Comput. Simulat. 100 13
[26] Bao B C, Wu H G, Xu L, Chen M, Hu W 2018 Int. J. Bifurcation Chaos 28 1850019
[27] Richter H 2008 Chaos, Solitons Fractals 36 559
[28] Kengne J, Tabekoueng Z N, Tamba V K, Negou A N 2015 Chaos 25 103126
[29] Bao H, Wang N, Wu H G, Song Z, Bao B C 2018 IETE Tech. Rev. 6 1
[30] Feudel U 2008 Int. J. Bifurcation Chaos 18 1607
[31] Chen M, Sun M X, Bao B C, Wu H G, Xu Q, Wang J 2018 Nonlinear Dyn. 91 1395
[32] Morfu S, Nofiele B, Marquié P 2007 Phys. Lett. A 367 192
[33] Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin. Phys. B 20 120502
[34] Chua L O 2012 Proc. IEEE 100 1920
[35] Ma J, Chen Z Q, Wang Z L, Zhang Q 2015 Nonlinear Dyn. 81 1275
[36] Bao B C, Jiang T, Xu Q, Chen M, Wu H G, Hu Y H 2016 Nonlinear Dyn. 86 1711
[37] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
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[1] Chua L O 1971 IEEE Trans. Circuit Theory 18 507
[2] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[3] Kim H, Sah M P, Yang C, Cho S, Chua L O 2012 IEEE Trans. Circuits Syst. I: Regular Papers 59 2422
[4] Yu D S, Iu H H C, Fitch A L, Liang Y 2014 IEEE Trans. Circuits Syst. I: Regular Papers 61 2888
[5] Sánchez-López C, Mendoza-López J, Carrasco-Aguilar M A, Muñiz-Montero C 2014 IEEE Trans. Circuits Syst. Ⅱ: Express Briefs 61 309
[6] Yang C, Choi H, Park S, Sah M P, Kim H, Chua L O 2015 Semicond. Sci. Technol. 30 015007
[7] Yu D S, Zheng C Y, Iu H H C, Fernando T, Chua L O 2017 IEEE Access 5 1284
[8] Corinto F, Ascoli A 2012 Electron. Lett. 48 824
[9] Bao B C, Yu J J, Hu F W, Liu Z 2014 Int. J. Bifurcation Chaos 24 1450143
[10] Wu H G, Bao B C, Liu Z, Xu Q, Jiang P 2016 Nonlinear Dyn. 83 893
[11] Xu Q, Lin Y, Bao B C, Chen M 2016 Chaos, Solitons Fractals 83 186
[12] Bao B C, Wang N, Xu Q, Wu H G, Hu Y H 2017 IEEE Trans. Circuits Syst. Ⅱ: Express Briefs 64 977
[13] Xu Q, Zhang Q L, Bao B C, Hu Y H 2017 IEEE Access 5 21039
[14] Chen M, Yu J J, Bao B C 2015 Electron. Lett. 51 462
[15] Chen M, Li M Y, Yu Q, Bao B C, Xu Q, Wang J 2015 Nonlinear Dyn. 81 215
[16] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B 23 070503
[17] Bao B C, Jiang P, Wu H G, Hu F W 2015 Nonlinear Dyn. 79 2333
[18] Fitch A L, Yu D S, Iu H H C, Sreeram V 2012 Int. J. Bifurcation Chaos 22 1250133
[19] Bao B C, Ma Z H, Xu J P, Liu Z, Xu Q 2011 Int. J. Bifurcation Chaos 21 2629
[20] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[21] Zhao Y B, Zhang X Z, Xu J, Guo Y C 2015 Chaos, Solitons Fractals 81 315
[22] Bao B C, Hu F W, Chen M, Xu Q, Yu Y J 2015 Int. J. Bifurcation Chaos 25 1550075
[23] Li C B, Sprott J C 2014 Int. J. Bifurcation Chaos 24 1450034
[24] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255
[25] Wei Z C, Wang R R, Liu A P 2014 Math. Comput. Simulat. 100 13
[26] Bao B C, Wu H G, Xu L, Chen M, Hu W 2018 Int. J. Bifurcation Chaos 28 1850019
[27] Richter H 2008 Chaos, Solitons Fractals 36 559
[28] Kengne J, Tabekoueng Z N, Tamba V K, Negou A N 2015 Chaos 25 103126
[29] Bao H, Wang N, Wu H G, Song Z, Bao B C 2018 IETE Tech. Rev. 6 1
[30] Feudel U 2008 Int. J. Bifurcation Chaos 18 1607
[31] Chen M, Sun M X, Bao B C, Wu H G, Xu Q, Wang J 2018 Nonlinear Dyn. 91 1395
[32] Morfu S, Nofiele B, Marquié P 2007 Phys. Lett. A 367 192
[33] Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin. Phys. B 20 120502
[34] Chua L O 2012 Proc. IEEE 100 1920
[35] Ma J, Chen Z Q, Wang Z L, Zhang Q 2015 Nonlinear Dyn. 81 1275
[36] Bao B C, Jiang T, Xu Q, Chen M, Wu H G, Hu Y H 2016 Nonlinear Dyn. 86 1711
[37] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
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