Topological orbital equivalence with asymptotic phase for a two time-scales discrete-time system

Abstract

Existence, smoothness, approximation, and attractiveness of a locally integral manifold are established for a two time-scales discrete-time system. This manifold contains all the solutions remaining in a specific compact subset. It allows us to define locally a triangular system which is topologically orbitally equivalent with asymptotic phase. It follows that (in)stability properties and existence of solutions of the original system, remaining after some time instant in the above-mentioned compact subset, can be established from the study of a reduced-order system. We study this reduced-order system for a weakly nonstationary case, applying the stroboscopic method to approximate it by a practically meaningful, slowly time-varying system.