Numerical Computation of Canards

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example. Mathematics Department, Cornell University, Ithaca, NY 14853 Mathematics Department, University of Maryland, Baltimore County, Baltimore, MD 21250 Mathematics Department, University of Michigan, Ann Arbor MI 48109 1