State Space Decomposition for Nonautonomous Dynamical Systems

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.