On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

ABSTRACT

A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.

Keywords:

• Singular perturbation theory
• canards
• iterative methods

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. We could in principle take any satisfying .

2. In fact, any for C sufficiently large would do.