Diffusion in the Phase Space of the Autooscillatory System, Demonstrating the Stochastic Web in the Conservative Limit: Numerical Investigation

Abstract

The paper investigates diffusion in the phase space of the weakly dissipative version of the 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. Whilst the conservative system demonstrates the unbounded diffusion in the phase space through the stochastic layer, trajectories of the autooscillatory system converge to several attractors, and diffusion can be obtained only in some limited time interval. The trajectories demonstrating diffusion properties were detected 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 grows linearly versus time, in the autooscillatory system this dependence appears to be rather complex. In the time interval associated with existence of diffusion it can be, however, approximated with the power law. The dependence of it’s exponent on the dissipation parameter value and on the initial energy of the ensemble was investigated. The exponent increases with the decrease of dissipation and decreases up to 0 with the increase of the initial ensemble energy. Dependence on the initial energy have the same shape in wide interval of dissipation parameter values.