On the approximation of the canard explosion point in epsilon-free systems

TL;DR

This paper introduces an iterative algorithm to accurately determine the canard explosion point in general slow-fast systems without explicit small parameters, validated through applications to classic and biological models.

Contribution

The paper presents a novel iterative method for estimating the canard explosion point in systems lacking an explicit small parameter, expanding applicability beyond traditional singular perturbation frameworks.

Findings

Algorithm accurately estimates canard explosion points.

Good agreement with numerical simulations in tested models.

Applicable to systems without explicit small parameters.

Abstract

A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of non-linear 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 and a Templator model for a self-replicating system with no explicit small parameter and obtain very good agreement with results from numerical simulations.