  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. First, we present that stationary pulses with
oscillatory tails (SPO) form a ``snaky'' structure in homogeneous spaces, after
which TPO branches form a ``figure-eight-like stack of isolas'' located
adjacent to the snaky structure of SPO. Herein, we adopted input resources as a
bifurcation parameter. A drift bifurcation from SPO to TPO was observed by
introducing another parameter at which these two solution sheets merged. In
contrast to the monotone tail case, a nonlocal interaction appeared in the
heterogeneous problem between the TPO and heterogeneity, which created
infinitely many saddle solutions and finitely many stable stationary solutions
distributed across the entire line. TPO displayed a variety of dynamics
including pinning and depinning processes along with penetration (PEN) and
rebound (REB). The stable/unstable manifolds of these saddles interacted with
the TPO in a complex manner, which created a subtle dependence on the initial
condition, and a difficulty to predict the behavior after collision. A
systematic global exploration of solution branches induced by heterogeneities
enabled us to find all the asymptotic states after collision to predict the
solution results without solving the PDEs. The reduction method of
finite-dimensional ODE system allowed us to clarify the detailed mechanism of
the transitions from PEN to pinning and pinning to REB based on a dynamical
system perspective. Consequently, the basin boundary between two distinct
outputs against the heterogeneities yielded an infinitely many successive
reconnection of heteroclinic orbits among those saddles, because the strength
of heterogeneity increased and caused the aforementioned subtle dependence of
initial condition.
