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.