Existence of generalized homoclinic orbits for one parameter families of flows

Abstract:

We consider a one parameter family of flows generated by $\dot x = f\left ( {x,\lambda } \right )$, where $\lambda \in \left [ {0,1} \right ]$. We, also, assume that there exists an isolated invariant set, $S$, which continues across the interval $\left [ {0,1} \right ]$, and that we know the connection matrices for $\dot x = f\left ( {x,1} \right )$ and $\dot x = f\left ( {x,0} \right )$. We then give conditions under which a Morse decomposition of $S$ cannot continue across $\left [ {0,1} \right ]$. Furthermore, using the language of chain recurrence we define a generalized homoclinic orbit and give conditions under which such objects exist.