Hyers-Ulam stability and exponential dichotomy of linear differential periodic systems are equivalent

Abstract

Let $m$ be a positive integer and $q$ be a positive real number. We prove that the $m$-dimensional and $q$-periodic system \begin{equation}\tag{$\ast$} \dot x(t)=A(t)x(t),\qquad t\in\mathbb{R}_+, \qquad x(t)\in\mathbb{C}^m \end{equation} is Hyers-Ulam stable if and only if the monodromy matrix associated to the family $\{A(t)\}_{t\ge 0}$ posses a discrete dichotomy, i.e. its spectrum does not intersect the unit circle.

Keywords

• differential equations
• dichotomy
• hyers-ulam stability