Project R-12627

Slow-fast systems with non-invertible first return maps (Research)

Slow-fast systems have two separate time scales: a fast dynamics together with an underlying slow drift. The slow drift can cause changes in stability of the fast system at so-called contact points (tipping points in applications), possibly resulting in periodic behaviour. Geometric singular perturbation theory studies such system from its limit, where the ratio of time scales tends to 0. This may cause singular behaviour and cause for example that the return maps to be non-invertible. These first return maps determine how a system's state changes after one iteration along its periodic behaviour, and in continuous deterministic models, such return maps are always invertible, so that at least theoretically the change in system state can be reversed, say by following orbits in negative time. It is no longer the case in the limit of a slow-fast system, and this opens the door for embedding well-known non-invertible maps in a flow of a slow-fast system. Chaotic behaviour such as the one found in the logistic map is typically caused by non-invertibility of the map, so this leads the way to obtaining chaotic slow-fast systems. We aim to prove that we can embed any non-invertible map in such a flow, using tangencies between the two limiting flows in the slow-fast system. We continue by giving a so-called exchange lemma that is able to cope with such tangencies.

01 January 2022 - 31 December 2024