Invariance and Contraction in Geometrically Periodic Systems with Differential Inclusions

The objective of this paper is to derive the essential invariance and contraction properties for the geometric periodic systems, which can be formulated as a category of differential inclusions, and primarily rendered in the phase coordinate, or the cycle coordinate. First, we introduce the geometric averaging method for this category of systems, and also analyze the accuracy of its averaging approximation. Specifically, we delve into the details of the geometrically periodic system through the tunnel of considering the convergence between the system and its geometrically averaging approximation. Under different corresponding conditions, the approximation on infinite time intervals can achieve certain accuracies, such that one can use the stability result of either the original system or the averaging system to deduce the stability of the other. After that, we employ the graphical stability to investigate the "pattern stability" with respect to the phase-based system. Finally, by virtue of the contraction analysis on the Finsler manifold, the idea of accentuating the periodic pattern incremental stability and convergence is nurtured in the phase-based differential inclusion system, and comes to its preliminary fruition in application to biomimetic mechanic robot control problem.