Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci

Abstract

<p style='text-indent:20px;'>A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a <i>canard explosion</i>. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.</p>