Classifying Flow-kick Equilibria: Reactivity and Transient Behavior in the Variational Equation

In light of concerns about climate change, there is interest in how sustainable management can maintain the resilience of ecosystems. We use flow-kick dynamical systems to model ecosystems subject to a constant kick occurring every τ time units. We classify the stability of flow-kick equilibria to determine which management strategies result in desirable long-term characteristics. To classify the stability of a flow-kick equilibrium, we classify the linearization of the time-τ map given by the time-τ map of the variational equation about the equilibrium trajectory. Since the variational equation is a non-autonomous linear differential equation, we conjecture that the asymptotic stability classification of each instantaneous local linearization along the equilibrium trajectory indicates the stability of the variational time-τ map. In Chapter 3, we prove this conjecture holds when all of the asymptotic and transient behavior of the instantaneous local linearizations is the same. To explore whether the conjecture holds in general, we ask: To what degree can transient behavior differ from asymptotic behavior? Under what conditions can this transient behavior accumulate asymptotically? In Chapter 4, we develop the radial and tangential velocity framework to characterize transient behavior in autonomous linear systems. In Chapter 5, we use this framework to construct an example of a non-autonomous linear system whose time-τ map has asymptotic behavior that differs from the asymptotic behavior of each instantaneous linear system that composes it. Future work seeks to determine whether this constructed example can arise as a variational equation, and thus provide a counterexample for our conjecture.