## Séminaire Alain Haraux

Jeudi 7 décembre 2017 (14h30, Salle G001)

Alain Haraux, Laboratoire Jacques-Louis Lions, CNRS et Université Pierre et Marie Curie.

Titre : Bornes ultimes de l'énergie des solutions de certaines équations d'évolution du second ordre avec amortissement non linéaire et terme source borné .

Résumé : Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which insure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddot{u}(t) + g(\dot{u}(t))+ Au(t)=h(t),\quad t\in\R^+$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $C(1+ ||h||^4)$ where $||h||$ stands for the $L^\infty$ norm of $h$ with values in $H$ and the actual growth of $g$ does not seem to play any role as long as we are in the non-resonant situation. If $g$ behaves like a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ independently of the power and this result is optimal. If $h$ is anti-periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.

## Séminaire Jonathan Touboul

Jeudi 23 novembre 2017 (15h00, Salle G001)

Jonathan Touboul, Collège de France.

Titre : Around the dynamics of nonlinear integrate-and-fire neurons.

Résumé : Neurons communicate information through the emission of stereotyped impulses, called spikes, which are emitted in response to increases of the electrical voltage of the cell. Integrate-and-fire models, a central class of models of neurons, decompose nerve cells activity into an integration phase and the spike emission with an instantaneous reset. They thus form a class of hybrid dynamical systems that combine a nonlinear differential equation and a discrete dynamical systems associated to spikes. I will present the analysis of the dynamics of these models of neurons. I will start by focusing on the geometry of the continuous (subthreshold) dynamics, a key to understand the way neurons respond to an input. I will next show that sequences of spikes can be described as iterates of a discrete map, called the firing map, which is a continuous unimodal map when the subthreshold dynamics has no fixed points, and which may be discontinuous with infinite left- and right-derivatives at the discontinuity points otherwise. I will show the relationship between fixed points, periodic and chaotic orbits of the firing map and regular spiking, bursting and chaotic spiking of the neuron model. Furthermore, I will exhibit the purely geometric hybrid mechanism supporting the emergence of MMOs in these systems (in the absence of explicit slow-fast structure) or a period-adding bifurcation structure and chaos. This talk relies on joint works with R. Brette, J. Rubin, J. Signerska and A. Vidal.

## Séminaire Juliette Bouhours

Jeudi 9 novembre 2017 (14h00, Salle G001)

Juliette Bouhours, Ecole Polytechnique.

Titre : Reaction-diffusion equations in ecology: extinction and spreading of a species under the joint influence of climate change and a weak Allee effect.

Résumé : In this presentation we will be interested in the effect of climate change on species ranges and its consequences on population persistence. To do so we will use reaction-diffusion equations to model the dynamics of a population density with respect to time and space. After explaining the ecological framework and introducing the history of reaction-diffusion equations in ecology, I will describe how from a classical homogeneous reaction-diffusion equation, we can include the effect of climate change and weak Allee effect in the model. Then the main question will be to understand whether the population vanishes or spreads depending on the values of some parameters of the problem and prove that the behaviour of the solution is different from what is known in the case of reaction-diffusion equations with no Allee effect.

## Séminaire Yao Li

Jeudi 22 juin 2017 (14h00, Salle G001)

Yao LI, University of Massachusetts Amherst.

Titre : Polynomial convergence rate to nonequilibrium steady-state.

Résumé : In this talk, I will present my recent result about the ergodic properties of nonequilibrium steady-state (NESS) for a stochastic energy exchange model. The energy exchange model is numerically reduced from a billiards-like deterministic particle system that models the microscopic heat conduction in a 1D chain. By using a technique called the induced chain method, I proved the existence, uniqueness, polynomial speed of convergence to the NESS, and polynomial speed of mixing for the stochastic energy exchange model. All of these are consistent with the numerical simulation results of the original deterministic billiards-like system.

## Séminaire Mahdi Achache

Jeudi 15 juin 2017 (14h00, Salle G001)

Mahdi ACHACHE, Institut Mathématique de Bordeaux.

Titre : Sur quelques problèmes de Cauchy non-autonomes.

Résumé : On considère le problème de la régularité maximale pour les problèmes de Cauchy non-autonomes: \dot{u}+B(t)A(t)u(t)+P(t)u(t)=f(t), où les opérateurs dépendant du temps A (t) sont associés à une Famille de formes sesquilinéaires et les perturbations multiplicatives gauche B(t) ainsi que la perturbation additive P(t) sont des familles des opérateurs bornés sur l'espace de Hilbert considéré. On prouve la Lp-régularité maximale et d'autres propriétés de régularité pour les solutions du problème précédent sous des hypothèses de régularité minimale sur les formes et les perturbations.