State space decomposition for nonautonomous dynamical systems

TL;DR

This paper establishes a Conley type decomposition theorem for nonautonomous dynamical systems, allowing the separation of the state space into chain recurrent and gradient-like parts, applicable to both finite and infinite-dimensional systems.

Contribution

It extends the Conley decomposition to nonautonomous systems on non-compact, separable spaces, including PDEs, with practical examples like Lorenz and Navier-Stokes systems.

Findings

Decomposition into chain recurrent and gradient-like parts for nonautonomous systems

Applicable to both ODEs and PDEs on non-compact spaces

Demonstrated with examples such as Lorenz and Navier-Stokes systems

Abstract

Decomposition of state spaces into dynamically different components is helpful for the understanding of dynamical behaviors of complex systems. A Conley type decomposition theorem is proved for nonautonomous dynamical systems defined on a non-compact but separable state space. Namely, the state space can be decomposed into a chain recurrent part and a gradient-like part. This result applies to both nonautonomous ordinary differential equations on Euclidean space (which is only locally compact), and nonautonomous partial differential equations on infinite dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier-Stokes system, under time-dependent forcing.