Canard cycles in global dynamics

TL;DR

This paper explores canard cycles in fast-slow dynamical systems, especially near transcritical bifurcations, revealing new effects like amplification and rapid recovery with potential ecological implications.

Contribution

It introduces novel phenomena related to canard cycles near transcritical points and discusses their relevance to ecological resilience in complex systems.

Findings

Discovered amplification of canards near transcritical points

Identified exceptionally fast recovery phenomena

Linked canard dynamics to ecological resilience

Abstract

Fast-slow systems are studied usually by "geometrical dissection". The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation and bursting oscillations. More recently, in relation both with theory of dynamical systems and with applications to physiology, a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered when the transition points are folds, in this article, we focus on the case where some transition points are instead transcritical. We present several new surprising effects like the "amplification of canards" or the "exceptionally fast recovery" on both (1+1)-systems and (2+1)-systems associated with tritrophic food chain dynamics. Finally, we also mention their possible relevance to the notion of resilience which has been coined out in ecology.