Collection Colloque Scientifique International Poincaré 100
Organisateur(s)
Date(s)
29/04/2024
Limit cycles in the strongly nonlinear limit
De Yves Pomeau
The discovery of the idea of limit cycle by Poincare was based mostly on geometrical considerations. The application to ordinary differential equations can be done analytically in the opposite limits of weak and of strong nonlinearity. In the latter, one finds what are called relaxation oscillations, namely periodic oscillations with two widely separated time scales: a drift on a slow manifold is interrupted by fast jumps out of this manifold followed later by a return to it, etc. The classical example of such relaxation oscillations is provided by the van der Pol model in the strongly nonlinear limit. , and can be generalized to a whole class of systems where the transition slow to fast is by a dynamical saddle-node bifurcation. By looking at an explicit model of Earthquake dynamics, we found another class of model of relaxation oscillations where the slow to fast transition is by a finite time singularity of the slow dynamics. This type of transition occurs in flow in three dimension at least. In concrete physical systems with relaxation dynamics there is a way of knowing the type of transition by analysing the response to external noise, something possible in many noisy physical systems.
