From First Lyapunov Coefficients to Maximal Canards
Abstract
Hopf bifurcations in fastslow 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 firstorder asymptotic expansion of this location was found by Krupa and Szmolyan in the framework of a "canard point"normalform 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.
 Publication:

International Journal of Bifurcation and Chaos
 Pub Date:
 2010
 DOI:
 10.1142/S0218127410026617
 arXiv:
 arXiv:1201.6595
 Bibcode:
 2010IJBC...20.1467K
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Numerical Analysis;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 preprint version  for final version see journal reference