  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.
