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The Effects of Symmetry Breaking Perturbation on the Dynamics of a Novel Chaotic System with Cyclic Symmetry: Theoretical Analysis and Circuit Realization

    Symmetry is an important property shared by a large number of nonlinear dynamical systems. Although the study of nonlinear systems with a symmetry property is very well documented, the literature has no sufficient investigation on the important issues concerning the behavior of such systems when their symmetry is broken or altered. In this work, we introduce a novel autonomous 3D system with cyclic symmetry having a piecewise quadratic nonlinearity φk(x)=axk|x|x|x| where parameter a is fixed and parameter k controls the symmetry and the nonlinearity of the model. Obviously, for k=0 the system presents both cyclic and inversion symmetries while the inversion symmetry is explicitly broken for k0. We consider in detail the dynamics of the new system for both two regimes of operation by using classical nonlinear analysis tools (e.g. bifurcation diagrams, plots of largest Lyapunov exponents, phase space trajectory plots, etc.). Several nonlinear patterns are reported such as period doubling, periodic windows, parallel bifurcation branches, hysteresis, transient chaos, and the coexistence of multiple attractors of different topologies as well. One of the most gratifying features of the new system introduced in this work is the existence of several parameter ranges for which up to twelve disconnected periodic and chaotic attractors coexist. This latter feature is rarely reported, at least for a simple system like the one discussed in this work. An electronic analog device of the new cyclic system is designed and implemented in PSpice. A very good agreement is observed between PSpice simulation and the theoretical results.


    • Bao, B., Qian, H., Xu, Q., Chen, M., Wang, J. & Yu, Y. [2017] “ Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network,” Front. Comput. Neurosci. 11, 81. Crossref, ISIGoogle Scholar
    • Bao, H., Wang, N., Bao, B., Chen, M., Jin, P. & Wang, G. [2018] “ Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria,” Commun. Nonlin. Sci. Numer. Simulat. 57, 264–275. Crossref, ISIGoogle Scholar
    • Barati, K., Jafari, S., Sprott, J. C. & Pham, V.-T. [2016] “ Simple chaotic flows with a curve of equilibria,” Int. J. Bifurcation and Chaos 26, 1630034-1–11. Link, ISIGoogle Scholar
    • Bishop, S., Sofroniou, A. & Shi, P. [2005] “ Symmetry-breaking in the response of the parametrically excited pendulum model,” Chaos Solit. Fract. 25, 257–264. Crossref, ISIGoogle Scholar
    • Cao, H. & Jing, Z. [2001] “ Chaotic dynamics of Josephson equation driven by constant DC and AC forcings,” Chaos Solit. Fract. 12, 1887–1895. Crossref, ISIGoogle Scholar
    • Cao, H., Seoane, J. M. & Sanjuán, M. A. [2007] “ Symmetry-breaking analysis for the general Helmholtz–Duffing oscillator,” Chaos Solit. Fract. 34, 197–212. Crossref, ISIGoogle Scholar
    • Chedjou, J., Fotsin, H., Woafo, P. & Domngang, S. [2001] “ Analog simulation of the dynamics of a van der Pol oscillator coupled to a Duffing oscillator,” IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 48, 748–757. Crossref, ISIGoogle Scholar
    • de Paula, A. S., Savi, M. A. & Pereira-Pinto, F. H. I. [2006] “ Chaos and transient chaos in an experimental nonlinear pendulum,” J. Sound Vibr. 294, 585–595. Crossref, ISIGoogle Scholar
    • Gugapriya, G., Rajagopal, K., Karthikeyan, A. & Lakshmi, B. [2019] “ A family of conservative chaotic systems with cyclic symmetry,” Pramana 92, 48. Crossref, ISIGoogle Scholar
    • Hamill, D. C. [1993] “ Learning about chaotic circuits with SPICE,” IEEE Trans. Ed. 36, 28–35. Crossref, ISIGoogle Scholar
    • Heinrich, M., Dahms, T., Flunkert, V., Teitsworth, S. W. & Schöll, E. [2010] “ Symmetry-breaking transitions in networks of nonlinear circuit elements,” New J. Phys. 12, 113030. Crossref, ISIGoogle Scholar
    • Itoh, M. [2001] “ Synthesis of electronic circuits for simulating nonlinear dynamics,” Int. J. Bifurcation and Chaos 11, 605–653. Link, ISIGoogle Scholar
    • Izrailev, F., Timmermann, B. & Timmermann, W. [1988] “ Transient chaos in a generalized Hénon map on the torus,” Phys. Lett. A 126, 405–410. Crossref, ISIGoogle Scholar
    • Jafari, S., Sprott, J. & Golpayegani, S. M. R. H. [2013] “ Elementary quadratic chaotic flows with no equilibria,” Phys. Lett. A 377, 699–702. Crossref, ISIGoogle Scholar
    • Jafari, S., Sprott, J. & Nazarimehr, F. [2015] “ Recent new examples of hidden attractors,” The Europ. Phys. J. Special Topics 224, 1469–1476. Crossref, ISIGoogle Scholar
    • Jafari, S., Sprott, J., Pham, V.-T., Volos, C. & Li, C. [2016a] “ Simple chaotic 3D flows with surfaces of equilibria,” Nonlin. Dyn. 86, 1349–1358. Crossref, ISIGoogle Scholar
    • Jafari, S., Sprott, J. C. & Molaie, M. [2016b] “ A simple chaotic flow with a plane of equilibria,” Int. J. Bifurcation and Chaos 26, 1650098-1–6. Link, ISIGoogle Scholar
    • Jafari, M. A., Mliki, E., Akgul, A., Pham, V.-T., Kingni, S. T., Wang, X. & Jafari, S. [2017] “ Chameleon: The most hidden chaotic flow,” Nonlin. Dyn. 88, 2303–2317. Crossref, ISIGoogle Scholar
    • Kengne, J. & Mogue, R. L. T. [2019] “ Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: Chaos, multi-scroll, and multiple coexisting attractors,” Int. J. Dyn. Contr. 7, 112–133. CrossrefGoogle Scholar
    • Kengne, J., Abdolmohammadi, H., Signing, V. F., Jafari, S. & Kom, G. [2020] “ Chaos and coexisting bifurcations in a novel 3D autonomous system with a non-hyperbolic fixed point: Theoretical analysis and electronic circuit implementation,” Brazilian J. Phys. 50, 442–453. Crossref, ISIGoogle Scholar
    • Kingni, S. T., Rajagopal, K., Çiçek, S., Srinivasan, A. & Karthikeyan, A. [2020] “ Dynamic analysis, FPGA implementation, and cryptographic application of an autonomous 5D chaotic system with offset boosting,” Front. Inform. Technol. Electron. Engin. 21, 950–961. Crossref, ISIGoogle Scholar
    • Kuate, P. D. K., Lai, Q. & Fotsin, H. [2019] “ Complex behaviors in a new 4D memristive hyperchaotic system without equilibrium and its microcontroller-based implementation,” The Europ. Phys. J. Special Topics 228, 2171–2184. Crossref, ISIGoogle Scholar
    • Kunkel, P. [1997] Kuznetsov, YA: Elements of Applied Bifurcation Theory (Springer-Verlag, NY) 1995. XV, 515 pp., 232 figs., DM 98.-ISBN 0-387-94418-4 (Applied Mathematical Sciences 112). ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f’́ur Angewandte Mathematik und Mechanik 77(5): 392–392. Google Scholar
    • Lai, Q. & Chen, S. [2016] “ Generating multiple chaotic attractors from Sprott B system,” Int. J. Bifurcation and Chaos 26, 1650177-1–13. Link, ISIGoogle Scholar
    • Lai, Q., Wan, Z., Kuate, P. D. K. & Fotsin, H. [2020] “ Coexisting attractors, circuit implementation and synchronization control of a new chaotic system evolved from the simplest memristor chaotic circuit,” Commun. Nonlin. Sci. Numer. Simulat., 105341. Crossref, ISIGoogle Scholar
    • Leonov, G. A. & Kuznetsov, N. V. [2013] “ Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits,” Int. J. Bifurcation and Chaos 23, 1330002-1–16. Link, ISIGoogle Scholar
    • Letellier, C. & Gilmore, R. [2007] “ Symmetry groups for 3D dynamical systems,” J. Phys. A: Math. Theoret. 40, 5597. Crossref, ISIGoogle Scholar
    • Li, C., Wang, J. & Hu, W. [2012] “ Absolute term introduced to rebuild the chaotic attractor with constant Lyapunov exponent spectrum,” Nonlin. Dyn. 68, 575–587. Crossref, ISIGoogle Scholar
    • Li, C. & Sprott, J. [2013] “ Amplitude control approach for chaotic signals,” Nonlin. Dyn. 73, 1335–1341. Crossref, ISIGoogle Scholar
    • Li, C., Sprott, J. C., Yuan, Z. & Li, H. [2015] “ Constructing chaotic systems with total amplitude control,” Int. J. Bifurcation and Chaos 25, 1530025-1–33. Link, ISIGoogle Scholar
    • Li, C. & Sprott, J. C. [2018] “ An infinite 3D quasiperiodic lattice of chaotic attractors,” Phys. Lett. A 382, 581–587. Crossref, ISIGoogle Scholar
    • Molaie, M., Jafari, S., Sprott, J. C. & Golpayegani, S. M. R. H. [2013] “ Simple chaotic flows with one stable equilibrium,” Int. J. Bifurcation and Chaos 23, 1350188-1–7. Link, ISIGoogle Scholar
    • Nayfeh, A. H. & Balachandran, B. [2008] Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods (John Wiley & Sons). Google Scholar
    • Rajagopal, K., Karthikeyan, A. & Srinivasan, A. K. [2017] “ FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization,” Nonlin. Dyn. 87, 2281–2304. Crossref, ISIGoogle Scholar
    • Rajagopal, K., Panahi, S., Karthikeyan, A., Alsaedi, A., Pham, V.-T. & Hayat, T. [2018a] “ Some new dissipative chaotic systems with cyclic symmetry,” Int. J. Bifurcation and Chaos 28, 1850164-1–10. Link, ISIGoogle Scholar
    • Rajagopal, K., Jafari, S., Karthikeyan, A., Srinivasan, A. & Ayele, B. [2018b] “ Hyperchaotic memcapacitor oscillator with infinite equilibria and coexisting attractors,” Circuits Syst. Sign. Process. 37, 3702–3724. Crossref, ISIGoogle Scholar
    • Rajagopal, K., Akgul, A., Pham, V.-T., Alsaadi, F. E., Nazarimehr, F., Alsaadi, F. E. & Jafari, S. [2019] “ Multistability and coexisting attractors in a new circulant chaotic system,” Int. J. Bifurcation and Chaos 29, 1950174-1–18. Link, ISIGoogle Scholar
    • Rynio, R. & Okniński, A. [1998] “ Symmetry breaking and fractal dependence on initial conditions in dynamical systems: Ordinary differential equations of thermal convection,” Chaos Solit. Fract. 9, 1723–1732. Crossref, ISIGoogle Scholar
    • Signing, V. F., Kengne, J. & Kana, L. K. [2018] “ Dynamic analysis and multistability of a novel four-wing chaotic system with smooth piecewise quadratic nonlinearity,” Chaos Solit. Fract. 113, 263–274. Crossref, ISIGoogle Scholar
    • Sprott, J. C. & Chlouverakis, K. E. [2007] “ Labyrinth chaos,” Int. J. Bifurcation and Chaos 17, 2097–2108. Link, ISIGoogle Scholar
    • Sprott, J. C. [2014] “ Simplest chaotic flows with involutional symmetries,” Int. J. Bifurcation and Chaos 24, 1450009-1–9. Link, ISIGoogle Scholar
    • Sprott, J. C. [2015] “ Symmetric time-reversible flows with a strange attractor,” Int. J. Bifurcation and Chaos 25, 1550078-1–7. Link, ISIGoogle Scholar
    • Strogatz, S. H. [1996] Nonlinear Dynamics and Chaos (Westwiew Press, Nashville, TN, USA). Google Scholar
    • Thomas, R. [1999] “ Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, labyrinth chaos,” Int. J. Bifurcation and Chaos 9, 1889–1905. Link, ISIGoogle Scholar
    • Wang, X. & Chen, G. [2012] “ A chaotic system with only one stable equilibrium,” Commun. Nonlin. Sci. Numer. Simulat. 17, 1264–1272. Crossref, ISIGoogle Scholar
    • Wei, Z. [2011] “ Dynamical behaviors of a chaotic system with no equilibria,” Phys. Lett. A 376, 102–108. Crossref, ISIGoogle Scholar
    • Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] “ Determining Lyapunov exponents from a time series,” Physica D 16, 285–317. Crossref, ISIGoogle Scholar
    • Yang, X.-S. & Yuan, Q. [2005] “ Chaos and transient chaos in simple Hopfield neural networks,” Neurocomputing 69, 232–241. Crossref, ISIGoogle Scholar
    • Yorke, J. A. & Yorke, E. D. [1979] “ Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model,” J. Statist. Phys. 21, 263–277. Crossref, ISIGoogle Scholar
    • Zhou, P. & Cao, H. [2008] “ The effect of symmetry-breaking on the parameterically excited pendulum,” Chaos Solit. Fract. 38, 590–597. Crossref, ISIGoogle Scholar
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