Best Ulam constants for two-dimensional non-autonomous linear differential systems

TL;DR

This paper derives the best Ulam stability constants for two-dimensional non-autonomous linear differential systems, including those without exponential dichotomy, with broad applicability to systems with finite-time blow-up and non-periodic cases.

Contribution

It introduces the first derivation of optimal Ulam constants for non-autonomous systems beyond periodic cases, including systems with finite-time blow-up and generalized Jordan forms.

Findings

Derived explicit formulas for best Ulam constants.

Applicable to systems with solutions on infinite or finite intervals.

Includes examples and approximations for different system types.

Abstract

This study deals with the Ulam stability of non-autonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional non-autonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on $(-\infty,\infty)$, but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for non-autonomous systems other than periodic systems.