Bifurcations from nondegenerate families of periodic solutions

TL;DR

This paper investigates bifurcations from nondegenerate families of periodic solutions, providing new insights into solution uniqueness and stability without requiring differentiability, under Lipschitz conditions.

Contribution

It extends bifurcation theory by analyzing cases where algebraic multiplicity is unknown, demonstrating solution uniqueness without differentiability assumptions.

Findings

Solution uniqueness can be established without asymptotic stability.

Differentiability of the system is not necessary; Lipschitz continuity suffices.

Theoretical results apply to broader classes of systems with Lipschitz conditions.

Abstract

By a nondegenerate $k$-parameterized family $K$ of periodic solutions we understand the situation when the geometric multiplicity of the multiplier +1 of the linearized on $K$ system equals to $k.$ Bifurcation of asymptotically stable periodic solutions from $K$ is well studied in the literature and different conditions have been proposeddepending on whether the algebraic multiplicity of +1 is $k$ or not (by Malkin, Loud, Melnikov, Yagasaki). In this paper we assume that the later is unknown. Asymptotic stability can not be understood in this case, but we demonstrate that the information about uniqueness of periodic solutions is still available. Moreover, we show that differentiability of the right hand sides is not necessary for the results of this kind and our theorems are proven under a kind of Lipschitz continuity.