Resonance in excitable systems under step-function forcing II. Subharmonic solutions and persistence

In an earlier paper (Othmer and Watanabe, 1994) we studied the existence and stability of harmonic solutions in a Fitzhugh-Nagumo type model under step-function forcing. In this paper we study subharmonic solutions, and show that a number of phenomena arise which do not occur in invertible circle maps. In particular, we show that the rotation number may be non-monotonic or discontinuous in one-parameter families, and that stable periodic solutions with different rotation numbers coexist on open sets in parameter space. The latter result shows how an experimental observation first due to Mines (1913) can be understood in the context of a flow. We also show that the 'stable' results for the singular system persist in the non-singular system, in a sense made precise later.