Long-Time Behavior of Some ODEs with Partial Damping

Abstract

This thesis examines some partially damped ODEs with a conservative bilinear term, a damping matrix term with a nontrivial kernel, and a deterministic forcing term. We prove that, when forcing is absent, the condition that the bilinear term has no invariant sets in the kernel of the damping term is sufficient to show convergence of all solutions to the origin. We then consider the case that invariant sets exist in the kernel of the damping term and include forcing to escape the invariant sets. We show that solutions diverge under certain symmetries and give a partial proof of boundedness with hyperbolic equilibria in the kernel of the damping term.