CSDC Seminar: Nonautonomous and slow-fast systems

Multiple time-scale and nonautonomous nonlinear dynamics share many physical features, e.g., a slowly drifting variable with constant speed directly leads to a nonautonomous system. Furthermore, a system with time-dependent forcing can be made formally autonomous by augmenting time as a phase space variable. If the forcing is very fast or very slow, a natural multiple time scale structure emerges. This leads to the natural approach that one might want to interpolate between these two system classes in the hope of relating results from their respective analysis.

In this talk, I follow this approach to investigate its advantages but also its substantial challenges. I focus on one of the simplest paradigmatic nonlinear dynamics models and consider a parametric linear interpolation between a FitzHugh-Nagumo model (including the classic Van der Pol system) and a skewed nonautonomous differential problem sharing the same fundamental structure. Moreover, I show how classic methods from geometric singular perturbation theory fail in the nonautonomous framework and new alternatives are necessary.

I.P. Longo, E. Queirolo, C. Kuehn: On the transition between autonomous and nonautonomous systems: the case of FitzHugh-Nagumo's model. To appear in Chaos (2024). arXiv: arXiv:2408.12256 [math.DS]

I.P. Longo, R. Obaya, A.M. Sanz: Tracking nonautonomous attractors in singularly perturbed systems of ODEs with dependence on the fast time. Journal of Differential  Equations 414 (2025), 609-644.