Vector Fields and Maps

Abstract

This paper looks at some relationships between continuous vector field descriptions of nonlinear systems and associated return maps defined on suitable Poincaré sections. Limit cycles of the vector field become fixed points of the return map. We find descriptions of the return map in terms of perturbation expansions about the fixed points. The first order coefficients, of course, are just the Floquet multipliers; so our approach is an extension of Floquet ideas to higher order. A time correction is used to ensure that a perturbed limit cycle trajectory is evaluated after almost one period on the Poincaré section.