TWO-DIMENSIONAL DYNAMICAL SYSTEMS WITH PERIODIC COEFFICIENTS

Abstract

This paper is devoted to the properties conferred on phase space by a vector field of generic second order autonomous dynamical systems with periodic coefficients, called parametric autonomous dynamical systems (PADS). At first, an associated periodical parametric linear equation (APPLE) is defined at every point of the phase plane. The exact value of the Floquet–Liapunov exponent of the APPLE is computed using a fast algorithm, without integration. The role of Floquet–Liapunov exponents is known to establish the stability of periodic solutions. In this work, it is pointed out that, under certain conditions, they bring information on local characteristics of PADS solutions according to their location in the phase plane, such as sensitivity to initial conditions, oscillation frequency, period doubling, parametric resonance, funneling. Then, an invariant manifold of an associated constant coefficients equivalent system (ACCES) is defined. It is shown that this manifold is periodically crossed by solutions of the PADS. This manifold crossing property contributes to the structure of the PADS solutions in phase plane. The implementation of this method is shown on a Van der Pol equation with a periodic coefficient in order to illustrate all kinds of solution patterns near the manifold in the phase plane according to the Floquet–Liapunov exponent local value. The manifold crossing property can be observed in all cases. Then, a parametric Duffing equation is processed. A numerical study shows some chaos routes, their bifurcation diagram and the top Liapunov exponent variations. The ACCES of the Duffing equation does not have any slow manifold. However, the Floquet–Liapunov exponent computation allows to specify the locus in the phase plane where the curvature of the trajectories changes, giving rise to chaos.