From First Lyapunov Coefficients to Maximal Canards

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.

preprint version - for final version see journal reference

References in corpus (3)

• Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System
• Computing Slow Manifolds of Saddle Type
• Homoclinic Orbits of the FitzHugh-Nagumo Equation: The Singular-Limit

Cited by in corpus (4)

• Efficient Gluing of Numerical Continuation and a Multiple Solution Method for Elliptic PDEs
• From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation
• Global bifurcation map of the homogeneus states in the Gray-Scott model
• Discretized Fast-Slow Systems with Canards in Two Dimensions