Complex behavior of trajectories of the autooscillatory system, demonstrating the stochastic web in the conservative limit

Abstract

We investigate dynamics of the weakly dissipative pulse-driven van der Pol system. Amplitude of external pulses depends on the dynamical variable in the same way as in the Zaslavsky generator of the stochastic web, and the system under investigation transforms into the stochastic web generator in the conservative limit. The conservative system demonstrates the unbounded diffusion in the phase space through the stochastic layer, and in the autooscillatory system all trajectories converge to several attractors. Some of these trajectories demonstrate in a limited time interval behavior similar to diffusion in the conservative system. The trajectories demonstrating diffusion properties were deteсted using the finite-time Lyapunov exponents, and for an ensemble of such trajectories dependence of average energy on time was analyzed. Whilst in the conservative system average energy depends linearly on time, in the autooscillatory system this dependence appears to be rather complex. In the time interval associated with existence of duffusion it can be, however, approximated with the power law. Properties of this dependence at different values of the dissipation parameter were investigated. It was shown that the rate of diffusion increases with the decrease of dissipation and decreases down to zero with the increase of the initial ensemble energy. In the wide interval of dissipation parameter depemdence of the diffusion rate on the initial ensemble energy has the same shape.

Speaker

Alexandr Golokolenov
Saratov State University
Russia

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