Multistability and its Annihilation in the Chua’s Oscillator with Piecewise-Linear Nonlinearity

Year 2020, Volume: 2 Issue: 2, 77 - 89, 30.11.2020

Zeric Njıtacke Théophile Fozin Léandre Kamdjeu Kengne Gervais Leutcho, Edwige Mache Kengne Jacques Kengne

Abstract

This contribution uncovers numerical evidence of hysteric dynamical behaviors for the same set of the circuit parameters of the Chua’s circuit with traditional piecewise-linear nonlinearity. Stationary points
and the symmetry property of the model first forecast the possible evidence of coexisting attractors. Then,
well known nonlinear analysis approach based on the bifurcation diagrams, two-parameter diagrams, phase
portraits, two parameter Lyapunov exponent diagrams, graph of maximum Lyapunov exponents, and attraction basins are exploited to characterize the dynamical behavior of the oscillator including coexisting orbits. Finally, the simultaneous existence of both periodic and chaotic orbits highlighted in the Chua’s oscillator is also annihilated based on linear controller. Numerical findings indicate control method ’s efficacy by combining two periodic routes and one chaotic route with another chaotic route.

Keywords

Chua’s oscillator, Chaotic systems, piecewise-linear nonlinearity, Multistability control, merging crisis

References

• Adiyaman, Y., S. Emiroglu, M. K. Ucar, and M. Yildiz, 2020 Dynamical analysis, electronic circuit design and control application of a different chaotic system. Chaos Theory and Applications 2: 10–16.
• Bao, B., F. Hu, M. Chen, Q. Xu, and Y. Yu, 2015a Self-excited and hidden attractors found simultaneously in a modified chua’s circuit. International Journal of Bifurcation and Chaos 25: 1550075.
• Bao, B., P. Jiang, H.Wu, and F. Hu, 2015b Complex transient dynamics in periodically forced memristive chua’s circuit. Nonlinear Dynamics 79: 2333–2343.
• Bao, B.-C., Q. Xu, H. Bao, and M. Chen, 2016 Extreme multistability in a memristive circuit. Electronics Letters 52: 1008–1010.
• Chen, M., J. Yu, and B.-C. Bao, 2015 Finding hidden attractors in improved memristor-based chua’s circuit. Electronics Letters 51: 462–464.
• Chua, L., M. Komuro, and T. Matsumoto, 1986 The double scroll family. IEEE transactions on circuits and systems 33: 1072–1118.
• Chua, L. O., 1994 Chua’s circuit 10 years later. International Journal of Circuit Theory and Applications 22: 279–305. Chua, L. O., 1998 CNN: A paradigm for complexity, volume 31. World Scientific.
• Duan, Z., J.Wang, R. Li, and L. Huang, 2007 A generalization of smooth chua’s equations under lagrange stability. International Journal of Bifurcation and Chaos 17: 3047– 3059.
• Fonzin Fozin, T., R. Kengne, J. Kengne, K. Srinivasan, M. Souffo Tagueu, et al., 2019 Control of multistability in a self-excited memristive hyperchaotic oscillator. International Journal of Bifurcation and Chaos 29: 1950119.
• Fozin, T. F., G. Leutcho, A. T. Kouanou, G. Tanekou, R. Kengne, et al., 2019 Multistability control of hysteresis and parallel bifurcation branches through a linear augmentation scheme. Zeitschrift für Naturforschung A 75: 11–21.
• Gotthans, T. and J. Petržela, 2015 New class of chaotic systems with circular equilibrium. Nonlinear Dynamics 81: 1143–1149.
• Gotthans, T., J. C. Sprott, and J. Petrzela, 2016 Simple chaotic flow with circle and square equilibrium. International Journal of Bifurcation and Chaos 26: 1650137.
• Hartley, T. T., 1989 The duffing double scroll. In 1989 American Control Conference, pp. 419–424, IEEE.
• Huang, A., L. Pivka, C.W.Wu, and M. Franz, 1996 Chua’s equation with cubic nonlinearity. International Journal of Bifurcation and Chaos 6: 2175–2222.
• Jafari, S., J. Sprott, and S. M. R. H. Golpayegani, 2013 Elementary quadratic chaotic flows with no equilibria. Physics Letters A 377: 699–702.
• Kengne, J., 2017 On the dynamics of chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors. Nonlinear Dynamics 87: 363–375.
• Kengne, J., Z. N. Tabekoueng, and H. Fotsin, 2016 Coexistence of multiple attractors and crisis route to chaos in autonomous third order duffing–holmes type chaotic oscillators. Communications in Nonlinear Science and Numerical Simulation 36: 29–44.
• Kingni, S. T., C. Ainamon, V. K. Tamba, and J. C. OROU, 2020 Directly modulated semiconductor ring lasers: Chaos synchronization and applications to cryptography communications. Chaos Theory and Applications 2: 31–39.
• Leonov, G., N. Kuznetsov, and V. Vagaitsev, 2011 Localization of hidden chua’s attractors. Physics Letters A 375: 2230–2233.
• Leutcho, G. D., S. Jafari, I. I. Hamarash, J. Kengne, Z. T. Njitacke, et al., 2020 A new megastable nonlinear oscillator with infinite attractors. Chaos, Solitons & Fractals 134: 109703.
• Lian, K.-Y., P. Liu, T.-S. Chiang, and C.-S. Chiu, 2002 Adaptive synchronization design for chaotic systems via a scalar driving signal. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49: 17– 27.
• Matsumoto, T., 1984 A chaotic attractor from chua’s circuit. IEEE Transactions on Circuits and Systems 31: 1055–1058. Negou, A. N. and J. Kengne, 2018 Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: Reversals of period doubling, offset boosting and coexisting bifurcations. AEU-International Journal of Electronics and Communications 90: 1–19.
• Njitacke, Z., J. Kengne, T. F. Fozin, B. Leutcha, and H. Fotsin, 2019 Dynamical analysis of a novel 4-neurons based hopfield neural network: emergences of antimonotonicity and coexistence of multiple stable states. International Journal of Dynamics and Control 7: 823–841.
• Njitacke, Z., J. Kengne, and A. N. Negou, 2017 Dynamical analysis and electronic circuit realization of an equilibrium free 3d chaotic system with a large number of coexisting attractors. Optik 130: 356–364.
• Njitacke, Z., J. Kengne, R. W. Tapche, and F. Pelap, 2018 Uncertain destination dynamics of a novel memristive 4d autonomous system. Chaos, Solitons & Fractals 107: 177–185.
• Peng, J., E. Ding, M. Ding, and W. Yang, 1996 Synchronizing hyperchaos with a scalar transmitted signal. Physical Review Letters 76: 904.
• Pham, V.-T., S. Jafari, C. Volos, and L. Fortuna, 2019 Simulation and experimental implementation of a line– equilibrium system without linear term. Chaos, Solitons & Fractals 120: 213–221.
• Pham, V.-T., S. Jafari, X. Wang, and J. Ma, 2016 A chaotic system with different shapes of equilibria. International Journal of Bifurcation and Chaos 26: 1650069.
• Ramírez-Ávila, G. M. and J. A. Gallas, 2010 How similar is the performance of the cubic and the piecewise-linear circuits of chua? Physics Letters A 375: 143–148.
• Sharma, P. R., A. Sharma, M. D. Shrimali, and A. Prasad, 2011 Targeting fixed-point solutions in nonlinear oscillators through linear augmentation. Physical Review E 83: 067201.
• Sharma, P. R., M. D. Shrimali, A. Prasad, and U. Feudel, 2013 Controlling bistability by linear augmentation. Physics Letters A 377: 2329–2332.
• Sharma, P. R., M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, 2015 Controlling dynamics of hidden attractors. International Journal of Bifurcation and Chaos 25: 1550061.
• Sprott, J. C., S. Jafari, A. J. M. Khalaf, and T. Kapitaniak, 2017 Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. The European Physical Journal Special Topics 226: 1979–1985.
• Tagne, R. M., J. Kengne, and A. N. Negou, 2019 Multistability and chaotic dynamics of a simple jerk system with a smoothly tuneable symmetry and nonlinearity. International Journal of Dynamics and Control 7: 476–495.
• Tsuneda, A., 2005 A gallery of attractors from smooth chua’s equation. International Journal of Bifurcation and Chaos 15: 1–49.
• Tuna, M., A. Karthikeyan, K. Rajagopal, M. Alcin, and ˙I. Koyuncu, 2019 Hyperjerk multiscroll oscillators with megastability: Analysis, fpga implementation and a novel ann-ring-based true random number generator. AEUInternational Journal of Electronics and Communications 112: 152941.
• Wang, X. and G. Chen, 2012 A chaotic system with only one stable equilibrium. Communications in Nonlinear Science and Numerical Simulation 17: 1264–1272.
• Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985 Determining lyapunov exponents from a time series. Physica D: Nonlinear Phenomena 16: 285–317.
• Xu, Q., Y. Lin, B. Bao, and M. Chen, 2016 Multiple attractors in a non-ideal active voltage-controlled memristor based chua’s circuit. Chaos, Solitons & Fractals 83: 186–200.
• Zhong, G.-Q., 1994 Implementation of chua’s circuit with a cubic nonlinearity. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 41: 934– 941.
• Zhong, G.-Q. and F. Ayrom, 1985 Experimental confirmation of chaos from chua’s circuit. International journal of circuit theory and applications 13: 93–98.