Residual diffusion in the autooscillatory system, demonstrating the stochastic web in the conservative limit, at small values of nonlinear dissipation

Abstract

We investigate dynamics of the weakly dissipative pulse-driven van der Pol
system, which transforms into the stochastic web generator in the conservative
limit due to the special form of the external pulses. The conservative system
demonstrates the unbounded diffusion in the phase space through the stochastic
layer, and average energy of an ensemble of systems with different initial
conditions grows linearly versus time. In the autooscillatory system dependence of
the average energy on time is approximately linear in the double logarithmic scale
in the certain time interval, before trajectories converge to the attractors — we call
this phenomenon the residual diffusion. Slope of this dependence is different for
different dissipation parameters and different values of initial energy of an
ensemble. We propose several characteristics to estimate the expansion of such
residual diffusion in the phase space. The saturation radius determines size of the
domain in the phase space, where ensemble of initial conditions should be chosen
in order to obtain diffusion; the minimal diffusion radius determines the size of the
region, where a trajectory, starting from the vicinity of the origin, remains after the
transient process; the maximal diffusion radius is determined by lower boundary of
the domain, where enegry decreases with time. We investigate how these radii
depend on the parameter of nonlinear dissipation and show that these dependencies
look similar for different values of the linear dissipation parameter.

Speaker

Golokolenov Alexander
Saratov State University
Russia

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