Parametric Generators of Hyperbolic Chaos

  • Sergey P. Kuznetsov


Excitation transfer between alternately active oscillators with phases transformed in accordance with chaotic maps, like that in the previous two chapters, may be regarded as some general principle of design of systems with attractors of the Smale-Williams type. Appropriate and convenient for implementation of this principle, are parametric oscillatory systems (Mandelshtam, 1972; Louisell, 1960; Akhmanov and Khokhlov, 1966; Rabinovich and Trubetskov, 1989; Damgov, 2004), The term relates to a class of systems with excitation of oscillations caused by periodic variation of some parameter. Parametric oscillations occur in systems of mechanics, electronics, acoustics, nonlinear optics, etc. In mechanics, a commonly known example relates to a person on a swing, who gradually increases the amplitude of the oscillations due to variation of the body position that corresponds to periodic change of the effective length of the equivalent pendulum (Fig. 6.1 (a) and (b)). In electric LC-circuit one can increase amplitude of the oscillations step by step if vary periodically the capacity: increasing the distance between the plates of the capacitor at the instants of its maximal charge, and decreasing it back at the instants of maximal current in the inductance, when the charge of the capacitor is close to zero (Fig. 6.1 (c)). Clearly, the mechanical work against the force of attraction of the charged plates in the course of the process converts into the energy of electromagnetic oscillations in the circuit; so, the amplitude will grow gradually. The main part of this chapter starts with consideration of the three-frequency parametric oscillator (Louisell, 1960), in which the excitation takes place in two coupled oscillators of frequencies ω1 and ω2 due to periodic variation of the coupling strength with the pump frequency ω3 = ω1 + ω2. Saturation of the excitation may be ensured by nonlinear damping in the scheme, which manifests then regular sustained regime of parametric generation. After that, we turn to designs, in which phases of tire parametrically excited oscillators undergo transformations corresponding to the expanding circle map in some period of time. Accounting compression of the phase volume in the state space along other directions, these will be systems with attractors of Smale-Williams type in the stroboscope Poincaré maps. In the systems examined here, the phase manipulation is performed as energy is transferred between oscillators, whereas the key role in systems discussed in the previous two chapters is played by a non-oscillatory external source of energy that compensates for oscillator losses, rather than by the energy transfer.
Fig. 6.1

Simple examples of parametric excitation: (a) a child increases amplitude of the swing by variation of the body position that is equivalent to periodic variation of the effective length of a pendulum (b); in LC-circuit the electromagnetic oscillations grow in amplitude due to periodic variation of the capacity (c).


Lyapunov Exponent Chaotic Attractor Parametric Generator Couple Oscillator Parametric Excitation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akhmanov, S.A., Khokhlov, R.V.: Parametric amplifiers and generators of light. Soviet Physics Uspekhi 9(2), 210(1966).ADSCrossRefGoogle Scholar
  2. Arnold, V.I.: Ordinary Differential Equations. The MIT Press, Somerville. MA (1978).Google Scholar
  3. Damgov, V.: Nonlinear and parametric phenomena: theory and applications in radiophysical and mechanical systems. World Scientific Publ. (2004).Google Scholar
  4. Kuznetsov. S.P.: On feasibility of a parametric generator of hyperbolic chaos. JF.TP 106, 380–387 (2008).Google Scholar
  5. Kuznetsov, A.S., Kuznetsov, S.P., Sataev, I.R.: Parametric generator of hyperbolic chaos based on two coupled oscillators with nonlinear dissipation. Technical Physics 55, 1707–1715 (2010).ADSCrossRefGoogle Scholar
  6. Louisell, W.H.: Coupled Mode and Paramagnetic Electronics. Wiley, New York (1960).Google Scholar
  7. Mandelshtam.
    L.I.: Lectures on oscillation theory. Nauka, Moscow (1972) (In Russian).Google Scholar
  8. Marsden.
    J.E., Ratiu, T.S.: Introduction To Mechanics And Symmetry: A Basic Exposition Of Classical Mechanical Systems. Springer, New York (1999).Google Scholar
  9. Rabinovich.
    M.I.. Trubetskov, D.I.: Oscillations and Waves: In Linear and Nonlinear Systems. Kluwer Academic Publication, The Netherlands (1989).Google 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