Geometric singular perturbation theory

Advanced models of fuel droplet heating and evaporation

Mostafa ADIMY Directeur de Recherches à l’INRIA Dir. de thèse Ionel S. CIUPERCA Mâıtre de Conférence à l’Université Lyon 1 Examinateur Michael C. MACKEY Directeur de Recherche à l’Université Mc GIll Dir. de thèse Sylvie MÉLÉARD Professeur à l’Ecole Polytechnique Examinatrice Sophie MERCIER Professeur à l’Université de Pau et des Pays de l’Adour Rapportrice Laurent PUJO-MENJOUET Mâıtre de Conférence à l’Université Lyon 1 Dir. de thèse Marta TYRAN-KAMIŃSKA Professeur à l’University of Silesia Examinatrice Bernard YCART Professeur à l’Université de Grenoble Rapporteur

Sous la direction de

We investigate the interaction of an excitable system with a slow oscillation. Under robust and general assumptions compatible with the more stringent assumptions usually made about excitable systems, we show that such a coupled system can display bursting, i.e. a stable solution in which some variable undergoes rapid oscillations followed by a period of quiescence, with both oscillation and quiescence continually repeated. Under a further weak condition, the bursting is “parabolic”, i.e. the local frequency of the fast oscillation increases and then decreases within a burst. The technique in this paper involves nonlinear changes of coordinates which transform the equations into ones which are closely related to Hill’s equation.

Parabolic bursting in an excitable system coupled with a slow oscillation

Systems linking people and nature, known as social-ecological systems, are increasingly understood as complex adaptive systems. Essential features of these complex adaptive systems – such as nonlinear feedbacks, strategic interactions, individual and spatial heterogeneity, and varying time scales – pose substantial challenges for modeling. However, ignoring these characteristics can distort our picture of how these systems work, causing policies to be less effective or even counterproductive. In this paper we present recent developments in modeling social-ecological systems, illustrate some of these challenges with examples related to coral reefs and grasslands, and identify the implications for economic and policy analysis.

Neil Fenichel

Papers

Geometric singular perturbation theory for ordinary differential equations

Hadamard's generalization of hyperbolicity, with applications to the hopf bifurcation problem