From first lyapunov coefficients to maximal canards

Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with a surprisingly rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space. A first-order asymptotic expansion of this location was found by Krupa and Szmolyan [2001a, 2001b, 2001c] in the framework of a "canard point"-normal-form for systems with one fast and one slow variable. We show how to compute the coefficients in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding the use of this normal form. Our results connect the theory of canard explosions with existing numerical software, enabling easier calculations of where canard explosions occur.