Heterogeneity is one of the most important and ubiquitous types of external perturbations. We study a spontaneous pulse-generating mechanism in an excitable medium with jump-type heterogeneity. Such a pulse generator (PG) has attracted considerable interest due to the computational potential of pulse waves in physiological signal processing. We first investigate the conditions for the onset of robust-type PGs, and then we show the global bifurcation structure of heterogeneity-induced patterns, including the complex ordered sequence of pulse-generating manners. We devise numerical frameworks to trace the long-term behavior of PGs as periodic solutions, and we detect the associated terminal homoclinic orbits that are homoclinic to a special type of heterogeneity-induced ordered pattern with a hyperbolic saddle. These numerical approaches assist us in identifying a candidate for the organizing center, and producing a variety of PGs as a codimension-two gluing bifurcation, in which two homoclinic trajectories associated with pulse emission and breathing motions form a butterfly configuration.
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