Stable Manifolds for Periodically Perturbed Maps and Application to Bipedal Locomotion

Abstract.

We prove that if a certain entry in the map of the Hadamard–Perron theorem is [Formula: see text]-periodic in one of the variables, then the stable manifold guaranteed by the theorem is a graph of a [Formula: see text]-periodic function. As an application, we first extend the classical Levinson’s result about the occurrence of an attracting closed invariant curve near a stable cycle of a system of autonomous equations under periodic perturbations to hybrid differential equations. Second, we validate the theory via an example of a compass-gait passive walker (system of differential equations with impacts) where we apply a [Formula: see text]-periodic perturbation to various parameters of the walker and, as predicted by the theory, document the occurrence of an attractive invariant curve for a suitable Poincare map in simulations.