Geodesic
Chaos generator with the Smale–Williams attractor based on oscillation death
Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 303-315.

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A nonautonomous system with a uniformly hyperbolic attractor of Smale–Williams type in a Poincaré cross-section is proposed with generation implemented on the basis of the effect of oscillation death. The results of a numerical study of the system are presented: iteration diagrams for phases and portraits of the attractor in the stroboscopic Poincaré cross-section, power density spectra, Lyapunov exponents and their dependence on parameters, and the atlas of regimes. The hyperbolicity of the attractor is verified using the criterion of angles.
Keywords: uniformly hyperbolic attractor, Smale–Williams solenoid, oscillation death, Lyapunov exponents.
Mots-clés : Bernoulli map 
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V. M. Doroshenko; V. P. Kruglov; S. P. Kuznetsov. Chaos generator with the Smale–Williams attractor based on oscillation death. Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 303-315. http://geodesic.mathdoc.fr/item/ND_2017_13_3_a0/

[1] Smale S., “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 | DOI | MR | MR | Zbl

[2] Williams R., “Expanding attractors”, Inst. Hautes Études Sci. Publ. Math., 1974, no. 43, 169–203 | DOI | MR

[3] Shilnikov L., “Mathematical problems of nonlinear dynamics: A tutorial”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7:9 (1997), 1953–2001 | DOI | MR | Zbl

[4] Katok A., Hasselblatt B., Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995, 802 pp. | MR | Zbl

[5] Dynamical Systems 9: Dynamical Systems with Hyperbolic Behaviour, Encyclopaedia Math. Sci., 66, ed. D. V. Anosov, Springer, Berlin, 1995, VIII+236 pp. | DOI | MR | MR

[6] Afraimovich V., Hsu S.-B., Lectures on chaotic dynamical systems, AMS/IP Stud. Adv. Math., 28, AMS, Providence, R.I., 2003, 353 pp. | DOI | MR | Zbl

[7] Kuznetsov S. P., Dynamical chaos and hyperbolic attractors: From mathematics to physics, R Dynamics, Institute of Computer Science, Izhevsk, 2013 (Russian)

[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. | DOI

[9] Kuznetsov S. P., Seleznev E. P., “A strange attractor of the Smale – Williams type in the chaotic dynamics of a physical system”, JETP, 102:2 (2006), 355–364 | DOI | MR

[10] Isaeva O. B., Jalnine A. Yu., 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. | DOI

[11] Kuznetsov S. P., Pikovsky A., “Autonomous coupled oscillators with hyperbolic strange attractors”, Phys. D, 232:2 (2007), 87–102 | DOI | MR | Zbl

[12] 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 | DOI | MR | Zbl

[13] Kuznetsov A. P., Kuznetsov S. P., Pikovski A. S., Turukina L. V., “Chaotic dynamics in the systems of coupling nonautonomous oscillators with resonance and nonresonance communicator of the signal”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 15:6 (2007), 75–85 (Russian) | Zbl

[14] Kruglov V. P., Kuznetsov S. P., “An autonomous system with attractor of Smale – Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators”, Commun. Nonlinear Sci. Numer. Simul., 16:8 (2011), 3219–3223 | DOI | MR | Zbl

[15] Kuznetsov A. S., Kuznetsov S. P., Sataev I. R., “Parametric generator of hyperbolic chaos based on two coupled oscillators with nonlinear dissipation”, Tech. Phys., 55:12 (2010), 1707–1715 | DOI | MR

[16] Kuptsov P. V., Kuznetsov S. P., Pikovsky A., “Hyperbolic chaos of Turing patterns”, Phys. Rev. Lett., 108:19 (2012), 194101, 4 pp. | DOI

[17] Kuznetsov S. P., Pikovsky A., Rosenblum M., “Collective phase chaos in the dynamics of interacting oscillator ensembles”, Chaos, 20:4 (2010), 043134, 8 pp. | DOI | MR | Zbl

[18] Aronson D. G., Ermentrout G. B., Kopell N., “Amplitude response of coupled oscillators”, Phys. D, 41:3 (1990), 403–449 | DOI | MR | Zbl

[19] Ermentrout G. B., Kopell N., “Oscillator death in systems of coupled neural oscillators”, SIAM J. Appl. Math., 50:1 (1990), 125–146 | DOI | MR | Zbl

[20] Matthews P. C., Strogatz S. H., “Phase diagram for the collective behavior of limit-cycle oscillators”, Phys. Rev. Lett., 65:14 (1990), 1701–1704 | DOI | MR | Zbl

[21] Matthews P. C., Mirollo R. E., Strogatz S. H., “Dynamics of a large system of coupled nonlinear oscillators”, Phys. D, 52:2–3 (1991), 293–331 | DOI | MR | Zbl

[22] Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M., “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them: P. 1: Theory; P. 2: Numerical application”, Meccanica, 15 (1980), 9–30 | DOI

[23] Shimada I., Nagashima T., “A numerical approach to ergodic problem of dissipative dynamical systems”, Progr. Theoret. Phys., 61:6 (1979), 1605–1616 | DOI | MR | Zbl

[24] Lai Y.-C., Grebogi C., Yorke J. A., Kan I., “How often are chaotic saddles nonhyperbolic?”, Nonlinearity, 6:5 (1993), 779–797 | DOI | MR | Zbl

[25] Anishchenko V. S., Kopeikin A. S., Kurths J., Vadivasova T. E., Strelkova G. I., “Studying hyperbolicity in chaotic systems”, Phys. Lett. A, 270:6 (2000), 301–307 | DOI | MR | Zbl

[26] Kuptsov P. V., “Fast numerical test for hyperbolic chaos”, Phys. Rev. E, 85:1 (2012), 015203, 4 pp. | DOI | MR

[27] Kuznetsov S. P., Kruglov V. P., “Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 160–174 | DOI | MR | Zbl