The comparative discription of the kicked Van-der-Pol system
in terms of differential equations and maps
Ludmila Turukina
Department of Nonlinear Processes, Saratov state University,
Saratov, Russia
Abstract:
The kicked Van-der-Pol system is studied on three levelsof precision.
Fist, we use a corresponding differential equations. Second, the
reversible 2D map is considered, to derive it we use the approximate
analytical solution between the kicks for the autonomous system. Third,
we turn to the irreversible 1D map, which is appropriate for the
strongly dissipative case. The main purpose of the present work is a
iscription of the original system in terms of these third
models. As a result, the domains in the parameter plane, where
approximate discription is effective, are determined. Particular
attention has been given to the phenomena, which do not survive the
passage from invertible 1D maps to invertible 2D maps. The
synchronization tongues structure in parameter plane is found for each
model, and their evolution is examined as the parameter of dissipation
increases. The typical bifurcation trees and phase portraits are
presented. The scenarios of transition to chaos, which are predicted
based on each model, are discussed.