Canard cycles in Global Dynamics

Fast-slow systems are studied usually by “geometrical dissection” [4]. 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 oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [11] and with applications to physiology [10, 26], 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 one or several transition points or “jumps” 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 [19, 22, 23]. Introduction Systems are often complex because their evolution involves different time scales. Purpose of this article is to present several phenomena which can be observed numerically and analyzed mathematically via bifurcation theory. A first approximation for the time evolution of fast-slow dynamics is often seen as follows. A generic orbit quickly reaches the vicinity of an attractive invariant set of the fast dynamics. It evolves then slowly close to this attractive part until, under the influence of the slow dynamics, this attractive part bifurcates into a repulsive one. Then, the generic orbit quickly reaches the vicinity of another attractive invariant set until it also loses its stability. This approach is a quite meaningful approximation because it explains many phenomena like hysteresis cycles, relaxation oscillations, bursting oscillations [13, 27, 28] and 1991 Mathematics Subject Classification. Primary 34C29, 34C25,58F22.