Floquet theory based on new periodicity concept for hybrid systems involving $q$-difference equations

TL;DR

This paper introduces a unified Floquet theory for hybrid systems with a new shift-based periodicity concept, enabling stability analysis on non-additive domains including $q$-difference systems.

Contribution

It develops a Floquet theory based on shift periodicity, extending analysis to hybrid and non-additive domains, and applies it to $q$-difference systems and stability analysis.

Findings

Constructed a Floquet decomposition theorem for hybrid systems.

Established a spectral mapping theorem for Floquet multipliers and exponents.

Demonstrated the theory's application to stability analysis on shift-periodic time scales.

Abstract

Using the new periodicity concept based on shifts, we construct a unified Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous, discrete or hybrid structure. New periodicity concept based on shifts enables the construction of Floquet theory on hybrid domains that are not necessarily additive periodic. This makes periodicity and stability analysis of hybrid periodic systems possible on non-additive domains. In particular, this new approach can be useful to know more about Floquet theory for linear $q$-difference systems defined on $\overline{q^{\mathbb{Z}}}:=\{q^{n}% :n\in\mathbb{Z}\} \cup \{0\}$ where $q>1$. By constructing the solution of matrix exponential equation we establish a canonical Floquet decomposition theorem. Determining the relation between Floquet multipliers and Floquet exponents, we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts. Finally, we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts.