A note on periodic differential equations

TL;DR

This paper investigates the asymptotic behavior of solutions to periodic linear differential equations in Banach spaces, linking continuous-time dynamics to discrete-time operator flows.

Contribution

It provides new insights into the recurrence and chain recurrence properties of solutions to periodic differential equations in Banach spaces.

Findings

Characterization of asymptotic behavior of solutions

Connection between continuous and discrete dynamical systems

Conditions for recurrence and chain recurrence

Abstract

Let $F$ be a Banach space and $L(F)$ be the set of all its bounded linear operators. In this note, we are interested in the asymptotic behavior (recurrence and chain recurrence) of the solution of the following initial value problem \label{eqlinear} x'(t) = X(t)x(t), \qquad x(0) = x, where $x \in F$ and the map $t \mapsto X(t) \in L(F)$ is a $T$-periodic continuous curve. This asymptotic behavior is related to the asymptotic behavior of the discrete-time flow on $F$ generated by the invertible operator $g \in L(F)$ given by the associated fundamental solution at time $T$.