Stability in non-autonomous periodic systems with grazing stationary impacts

Abstract

This paper examines impulsive non-autonomous periodic systems whose surfaces of discontinuity and impact functions are not depending on the time variable. The W−map which alters the system with variable moments of impulses to that with fixed moments and facilitates the investigations, is presented. A particular linearizion system with two compartments is utilized to analyze stability of a grazing periodic solution. A significant way to keep down a singularity in linearizion is demonstrated. A concise review on sufficient conditions for the linearizion and stability is presented. An example is given to actualize the theoretical results.

Periodic motions generated from non-autonomous grazing dynamics

“…Lemma 3.1. [32] Assume that the conditions (H1) and (A2) are valid. Then, the function τ i (x) is continuous on the set of points near a grazing point which satisfy condition (N 1).…”

“…For each of these systems, we find the matrix of monodromy, U j (T ) and denote corresponding Floquet multipliers by ρ Theorem 4.1. [32] Assume that the conditions (H1), (A1) − (A4) are valid. Then T − periodic solution Ψ(t) of (2.1) is asymptotically stable.…”

“…For our investigation, we utilized an approach which was summarized in [30]. For the analysis of grazing, the Bequivalence method recently started to be applied [31,32]. It is urgent to mention that diversity of methods determine the strength of mathematics.…”

“…The newly developed linearization for non-autonomous systems with stationary impacts was considered in our paper [32]. We investigate there equations, which are non-perturbed, while the present research is devoted to extension of the results to more complex problems, when grazing effects changes connected to a parameter variations.…”