Floquet bundles for tridiagonal competitive-cooperative systems with Applications

TL;DR

This paper develops Floquet bundle theory for time-dependent tridiagonal competitive-cooperative systems, enabling analysis of their long-term dynamics, especially on hyperbolic omega-limit sets, with applications to almost periodic and automorphic systems.

Contribution

It introduces canonical Floquet invariant bundles for general time-dependent systems, extending classical Floquet theory to non-periodic, recurrent structures.

Findings

Floquet bundles are exponentially separated in skew-product flows.

Reduction to standard Floquet space in time-periodic cases.

Application to dynamics on hyperbolic omega-limit sets.

Abstract

For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. Such Floquet bundles naturally reduce to the standard Floquet space when the system is assumed to be time-periodic. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.