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

Abstract

The interplay between 1D traveling pulses with oscillatory tails (TPOs) and heterogeneities of the bump type is studied to formulate a generalized three-component FitzHugh–Nagumo system of equations. We present that stationary pulses with oscillatory tails (SPOs) form a “snaky” structure in homogeneous spaces and then TPO branches form a “figure-eight-like stack of isolas” located adjacent to the snaky structure of the SPO. Here, we adopt input resources such as the voltage-difference as a bifurcation parameter. A drift bifurcation from the SPO to TPO is observed by introducing another parameter at which these two solution sheets merge. In contrast to the monotone tail case, in the heterogeneous problem, a nonlocal interaction appears between the TPO and the heterogeneity, creating infinitely many saddle solutions and finitely many stable stationary solutions distributed across the line. The response of the TPO shows various dynamics including pinning and depinning processes along with penetration (PEN) and rebound (REB). The stable/unstable manifolds of these saddles interact with the TPO in a complex manner, creating a subtle dependence on the initial condition, and a difficulty to predict the behavior after collision even in 1D space. For the 1D case, a systematic global exploration of solution branches induced by heterogeneities (heterogeneity-induced-ordered patterns; HIOPs) reveals that HIOPs contain all the asymptotic states after collision to facilitate the prediction of the solution results without solving the PDEs. The reduction method of finite-dimensional system enables the clarification of 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 yields infinitely many successive reconnections of heteroclinic orbits among those saddles when the strength of heterogeneity increases. This causes the aforementioned subtle dependence of initial condition.