Hyperbolicity and solvability for linear systems on time scales

TL;DR

This paper extends the concept of hyperbolicity from ordinary differential equations to linear dynamic systems on time scales, analyzing solvability in ${\mathbb L}^\infty$ and establishing analogs of classical hyperbolic systems results.

Contribution

It introduces a framework for hyperbolicity in time scale systems and demonstrates solvability criteria in ${\mathbb L}^\infty$, bridging differential equations and dynamic systems on time scales.

Findings

Hyperbolicity concept extended to time scale systems.

Solvability in ${\mathbb L}^\infty$ characterized for these systems.

Analogues of classical hyperbolic systems theorems established.

Abstract

We believe that the difference between time scale systems and ordinary differential equations is not as big as people use to think. We consider linear operators that correspond to linear dynamic systems on time scales. We study solvability of these operators in ${\mathbb L}^\infty$. For ordinary differential equations such solvability is equivalent to hyperbolicity of the considered linear system. Using this approach and transformations of the time variable, we spread the concept of hyperbolicity to time scale dynamics. We provide some analogs of well-known facts of Hyperbolic Systems Theory, e.g. the Lyapunov--Perron theorem on stable manifold.