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

• Main Authors: Constantin Buse, Dorel Barbu, Afshan Tabassum
• Format: Article
• Language: English
• Published: University of Szeged 2015-09-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: differential equations; dichotomy; hyers-ulam stability
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=3918

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.