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The interplay between 1D traveling pulses with oscillatory tails (TPO) andheterogeneities of bump type is studied for a generalized three-componentFitzHugh-Nagumo equation. First, we present that stationary pulses withoscillatory tails (SPO) form a ``snaky'' structure in homogeneous spaces, afterwhich TPO branches form a ``figure-eight-like stack of isolas'' locatedadjacent to the snaky structure of SPO. Herein, we adopted input resources as abifurcation parameter. A drift bifurcation from SPO to TPO was observed byintroducing another parameter at which these two solution sheets merged. Incontrast to the monotone tail case, a nonlocal interaction appeared in theheterogeneous problem between the TPO and heterogeneity, which createdinfinitely many saddle solutions and finitely many stable stationary solutionsdistributed across the entire line. TPO displayed a variety of dynamicsincluding pinning and depinning processes along with penetration (PEN) andrebound (REB). The stable/unstable manifolds of these saddles interacted withthe TPO in a complex manner, which created a subtle dependence on the initialcondition, and a difficulty to predict the behavior after collision. Asystematic global exploration of solution branches induced by heterogeneitiesenabled us to find all the asymptotic states after collision to predict thesolution results without solving the PDEs. The reduction method offinite-dimensional ODE system allowed us to clarify the detailed mechanism ofthe transitions from PEN to pinning and pinning to REB based on a dynamicalsystem perspective. Consequently, the basin boundary between two distinctoutputs against the heterogeneities yielded an infinitely many successivereconnection of heteroclinic orbits among those saddles, because the strengthof heterogeneity increased and caused the aforementioned subtle dependence ofinitial condition.

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