From First Lyapunov Coefficients to Maximal Canards

TL;DR

This paper links the calculation of maximal canards in fast-slow systems to the first Lyapunov coefficient at Hopf bifurcations, simplifying the process and integrating it with existing numerical tools.

Contribution

It introduces a method to compute the canard explosion location directly from the Lyapunov coefficient, bypassing the need for normal form transformations.

Findings

Derived a formula connecting Lyapunov coefficient to canard location

Enabled easier numerical computation of canard explosions

Bridged theoretical analysis with practical software tools

Abstract

Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with surprising 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 in the framework of a "canard point"-normal-form for systems with one fast and one slow variable. We show how to compute the coefficient in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding 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.