Method for Finding Periodic Trajectories of Centrally Symmetric Dynamical Systems on the Plane

Abstract

The problem of finding the cycles of a dynamical system on the plane is considered under the assumption that the system is centrally symmetric. We suggest an iteration method where, at each step, the function describing an approximation of a periodic trajectory is determined as a trajectory of some Hamiltonian system. If the resulting function sequence converges, then the limit is a periodic trajectory of the exact system. The efficiency of the method is illustrated by examples of seeking the cycles in the classical problems on the van der Pol oscillator and the perturbed Duffing oscillator for the case in which the coefficient of nonconservative terms takes values of the order of unity.