Traveling pulses with oscillatory tails, figure-eight stack of isolas, and dynamics in heterogeneous media

The interplay between 1D traveling pulses with oscillatory tails (TPO) and heterogeneities of bump type is studied for a generalized three-component FitzHugh-Nagumo equation. We first present that stationary pulses with oscillatory tails (SPO) forms a “snaky" structure in homogeneous space, then TPO branches take a form of "figure-eight-like stack of isolas" located close to the snaky structure of SPO. Here we adopt voltage-difference as a bifurcation parameter. A drift bifurcation from SPO to TPO can be found by introducing another parameter at which these two solution sheets merge. As for the heterogeneous problem, in contrast to monotone tail case, there appears a nonlocal interaction between the TPO and the heterogeneity that creates infinitely many saddle solutions. The response of TPO shows a variety of dynamics including pinning and depinning processes in addition to penetration and rebound. Stable/unstable manifolds of these saddles interact with TPO in a complex way, which causes a subtle dependence on the initial condition and a difficulty to predict the behavior after collision even in one-dimensional space. Nevertheless, for 1D case, a systematic global exploration of solution branches (HIOP) induced by heterogeneities, and the reduction method to finite-dimensional ODEs allow us to clarify such a subtle dependence of initial condition and detailed mechanism of the transitions from penetration to pinning and pinning to rebound from dynamical system view point. It turns out that the basin boundary between two different outputs against the heterogeneities forms an infinitely many successive reconnections of heteroclinic orbits among those saddles as the height of the bump is changed, which causes the subtle dependence of initial condition. This is a joint work with Takeshi Watanabe.