Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium

TL;DR

This paper investigates the existence and multiplicity of periodic solutions in nonlinear delayed systems under small perturbations, linking topological invariants to solution counts.

Contribution

It establishes a generic lower bound on the number of periodic solutions based on the Euler characteristic, connecting topological methods with delayed system analysis.

Findings

At least |hi |+1 periodic solutions exist under certain conditions.

Connections between fixed point and Poincare9 operators are elucidated.

Results apply to nonlinear delayed systems near equilibrium.

Abstract

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset \R^N$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the Euler characteristic of $\Omega$. Moreover, some connections between the associated fixed point operator and the Poincar\'e operator are explored.