Impact Factor

    Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators

    Received 31 August 2018; accepted 19 October 2018

    2018, Vol. 14, no. 4, pp.  435-451

    Author(s): Doroshenko V. M., Kruglov V. P., Kuznetsov S. P.

    The principle of constructing a new class of systems demonstrating hyperbolic chaotic attractors is proposed. It is based on using subsystems, the transfer of oscillatory excitation between which is provided resonantly due to the difference in the frequencies of small and large (relaxation) oscillations by an integer number of times, accompanied by phase transformation according to an expanding circle map. As an example, we consider a system where a Smale – Williams attractor is realized, which is based on two coupled Bonhoeffer – van der Pol oscillators. Due to the applied modulation of parameter controlling the Andronov – Hopf bifurcation, the oscillators manifest activity and suppression turn by turn. With appropriate selection of the modulation form, relaxation oscillations occur at the end of each activity stage, the fundamental frequency of which is by an integer factor $M = 2, 3, 4, \ldots$ smaller than that of small oscillations. When the partner oscillator enters the activity stage, the oscillations start being stimulated by the $M$-th harmonic of the relaxation oscillations, so that the transformation of the oscillation phase during the modulation period corresponds to the $M$-fold expanding circle map. In the state space of the Poincaré map this corresponds to an attractor of Smale – Williams type, constructed with $M$-fold increase in the number of turns of the winding at each step of the mapping. The results of numerical studies confirming the occurrence of the hyperbolic attractors in certain parameter domains are presented, including the waveforms of the oscillations, portraits of attractors, diagrams illustrating the phase transformation according to the expanding circle map, Lyapunov exponents, and charts of dynamic regimes in parameter planes. The hyperbolic nature of the attractors is verified by numerical calculations that confirm the absence of tangencies of stable and unstable manifolds for trajectories on the attractor (“criterion of angles”). An electronic circuit is proposed that implements this principle of obtaining the hyperbolic chaos and its functioning is demonstrated using the software package Multisim.
    Keywords: uniformly hyperbolic attractor, Smale – Williams solenoids, Bernoulli mapping, Lyapunov exponents, Bonhoeffer – van der Pol oscillators
    Citation: Doroshenko V. M., Kruglov V. P., Kuznetsov S. P., Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  435-451

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    [1] Smale, S., “Differentiable Dynamical Systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817  crossref  mathscinet  zmath
    [2] Williams, R., “Expanding Attractors”, Inst. Hautes Études Sci. Publ. Math., 1974, no. 43, 169–203  crossref  mathscinet
    [3] Shilnikov, L., “Mathematical Problems of Nonlinear Dynamics: A Tutorial”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7:9 (1997), 1953–2001  crossref  mathscinet  zmath
    [4] Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995, 802 pp.  mathscinet  zmath
    [5] Dynamical Systems 9: Dynamical Systems with Hyperbolic Behaviour, Encyclopaedia Math. Sci., 66, ed. D. V. Anosov, Springer, Berlin, 1995, VIII+236 pp.
    [6] Afraimovich, V. S. and Hsu, S.-B., Lectures on Chaotic Dynamical Systems, AMS/IP Stud. Adv. Math., 28, AMS, Providence, R.I., 2003, 353 pp.  crossref  mathscinet  zmath
    [7] Kuznetsov, S. P., Hyperbolic Chaos: A Physicist's View, Springer, Berlin, 2012, 336 pp.  zmath  adsnasa
    [8] Kuznetsov, S. P., “Example of a Physical System with a Hyperbolic Attractor of the Smale – Williams Type”, Phys. Rev. Lett., 95:14 (2005), 144101, 4 pp.  crossref  adsnasa
    [9] Kuznetsov, S. P. and Seleznev, E. P., “Strange Attractor of Smale – Williams Type in the Chaotic Dynamics of a Physical System”, J. Exp. Theor. Phys., 102:2 (2006), 355–364  crossref  mathscinet  adsnasa; Zh. Èksper. Teoret. Fiz., 129:2 (2006), 400–412 (Russian)
    [10] Kuznetsov, S. P. and Sataev, I. R., “Hyperbolic Attractor in a System of Coupled Non-Autonomous van der Pol Oscillators: Numerical Test for Expanding and Contracting Cones”, Phys. Lett. A, 365:1–2 (2007), 97–104  crossref  zmath  adsnasa
    [11] Kuznetsov, S. P. and Kruglov, V. P., “Hyperbolic Chaos in a System of Two Froude Pendulums with Alternating Periodic Braking”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 152–161  crossref  mathscinet  adsnasa
    [12] Isaeva, O. B., Jalnine, A. Yu., and Kuznetsov, S. P., “Arnold's Cat Map Dynamics in a System of Coupled Nonautonomous van der Pol Oscillators”, Phys. Rev. E, 74:4 (2006), 046207, 5 pp.  crossref  adsnasa
    [13] Kuznetsov, S. P. and Pikovsky, A., “Autonomous Coupled Oscillators with Hyperbolic Strange Attractors”, Phys. D, 232:2 (2007), 87–102  crossref  mathscinet  zmath
    [14] Kuznetsov, S. P., “Example of Blue Sky Catastrophe Accompanied by a Birth of Smale – Williams Attractor”, Regul. Chaotic Dyn., 15:2–3 (2010), 348–353  crossref  mathscinet  zmath  adsnasa
    [15] FitzHugh, R., “Impulses and Physiological States in Theoretical Models of Nerve Membrane”, Biophys. J., 1:6 (1961), 445–466  crossref
    [16] Nagumo, J., Arimoto, S., and Yoshizawa, S., “An Active Pulse Transmission Line Simulating Nerve Axon”, Proc. of the IRE, 50:10 (1962), 2061–2070  crossref
    [17] Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory”, Meccanica, 15 (1980), 9–20  crossref  zmath  adsnasa
    [18] Shimada, I. and Nagashima, T., “A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems”, Progr. Theoret. Phys., 61:6 (1979), 1605–1616  crossref  mathscinet  zmath  adsnasa
    [19] Lai, Y.-Ch., Grebogi, C., Yorke, J. A., and Kan, I., “How Often Are Chaotic Saddles Nonhyperbolic?”, Nonlinearity, 6:5 (1993), 779–797  crossref  mathscinet  zmath  adsnasa
    [20] Anishchenko, V. S., Kopeikin, A. S., Kurths, J., Vadivasova, T. E., and Strelkova, G. I., “Studying Hyperbolicity in Chaotic Systems”, Phys. Lett. A, 270:6 (2000), 301–307  crossref  mathscinet  zmath  adsnasa
    [21] Kuptsov, P. V., “Fast Numerical Test of Hyperbolic Chaos”, Phys. Rev. E, 85:1 (2012), 015203(R), 4 pp.  crossref  adsnasa
    [22] Kuznetsov, S. P. and Kruglov, V. P., “Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 160–174  mathnet  crossref  mathscinet  zmath  adsnasa

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