Autonomous Systems Based on Dynamics Close to Heteroclinic Cycle

  • Sergey P. Kuznetsov

Abstract

This chapter is devoted to autonomous models generating successive trains of oscillations with chaotic phase, which are designed on a base of systems with heteroclinic cycles in the phase space as suggested in our joint work (Kuznctsov and Pikovsky, 2007), The heteroclinic cycle, or the heteroclinic loop consists of several saddle fixed points and of trajectories joining them. The paradigm example for a. heteroclinic cycle (Guckenheimer and Holmes, 1988) is discussed in Sect. 9.1. In a frame of the present research, it is interpreted as a set of equations for real amplitudes of interacting oscillators. The equations are supplemented with appropriate additional coupling terms in such a way that the alternating cyclic excitation of the oscillators occurs accompanied by transformation of the phases in accordance with some chaotic map. The number of oscillators may be three or more; so, in the models constructed in this way, the minimal phase space dimension is six. In Sect. 9.2–9.4 we discuss three models, one with attractor of Smale-Williams type, one possessing attractor with dynamics governed approximately by the Arnold cat map, and an example of hyperchaos that corresponds to attractor with two positive Lyapunov exponents. All these examples are designed on a base of equations written for complex amplitudes, and they implement non-resonance mechanism of the excitation transfer between the alternately exciting oscillators. In Sect. 9.5 we advance a model composed of a large number of van der Pol oscillators. Because of slow variation of the natural frequencies of the oscillators around the ring structure of the device, it appears possible to use resonance mechanism for the excitation transfer; so, the system may have prospects for implementing high-frequency chaos generators.

Keywords

Lyapunov Exponent Autonomous System Large Lyapunov Exponent Heteroclinic Cycle Positive Lyapunov Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Guckenheimer, J., Holmes, P.: Structurally stable heteroclinic cycles. Math. Proc. Cambridge Philos. Soc. 103, 189–192 (1988).MathSciNetADSMATHCrossRefGoogle Scholar
  2. Huisman, I., Weissing, F.J.: Biological conditions for oscillations and chaos generated by multispecies competition. Ecology 82, 2682–2695 (2001).CrossRefGoogle Scholar
  3. Kruglov, V.P., Kuznelsov, S.P.: An autonomous system with allractor of Smale-Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators. CNSNS 16, 3219–3223 (2011).ADSGoogle Scholar
  4. Kuznetsov, S.P., Pikovsky, A.: Autonomous coupled oscillators with hyperbolic strange attractors. Physica D 232, 87–102 (2007).MathSciNetADSMATHCrossRefGoogle Scholar
  5. Letellier C., Rössler, O.E.: Hyperchaos. Scholarpedia 2, 1936 (2007).CrossRefGoogle Scholar
  6. Matsumoto. T., Chua. L.O., Kobayashi. K.: Hyperchaos laboratory experiment and numerical confirmation. IEEE Trans, Circuits & Syst. 33, 1143–1147 (1986).MathSciNetADSCrossRefGoogle Scholar
  7. Mercader, I., Prat, J., Knobloch, E.: Robust heteroclinic cycles in two-dimensional Rayleigh-Benard convection without Boussinesq symmetry. Int. J. Bifurcation and Chaos 12, 2501–2522 (2002).MathSciNetMATHCrossRefGoogle Scholar
  8. Rabinovich, M.I., Varona, P., Seiverston, A.I., Abarbanel, H.D.I.: Dynamical principles in neuro-science. Rev. Mod. Phys. 78, 1213–1265 (2006).ADSCrossRefGoogle Scholar
  9. Reiterer, P., Lainscsek, C., Schürrer, F., Lctellier C., Maquet, J.: A nine-dimcnsional Lorenz system to study high-dimensional chaos. Journal of Physics A 31, 7121–7139 (1998).MATHCrossRefGoogle Scholar
  10. Rössler, O.E.: An equation for hyperchaos. Physics Letters A 71, 155–157 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  11. Stoop, R., Peinke, J., Parisi, J., Röhrichl B., Hübener, R. P.: A p-Ge semiconductor experiment showing chaos and hyperchaos. Physica D 35, 425–435 (1989).ADSCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sergey P. Kuznetsov
    • 1
  1. 1.Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratov BranchSaratovRussian Federation

Personalised recommendations