Autonomous Systems Based on Dynamics Close to Heteroclinic Cycle

  • Sergey P. Kuznetsov


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.


Lyapunov Exponent Autonomous System Large Lyapunov Exponent Heteroclinic Cycle Positive Lyapunov Exponent 
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© 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

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