Slow-Fast Autonomous Dynamical Systems

Abstract

In this paper, we consider a class of slow-fast autonomous dynamical systems, i.e. systems having a small parameter ∊ multiplying a component of velocity. At first, the singular perturbation method (∊ = 0+) is recalled. Then we consider the case ∊ ≠ 0. Starting from a working hypothesis and particularly in the case of a singular approximation, our purpose is to show that there exists slow manifolds which can be defined as the slow manifolds of a so-called tangent linear system. The method allowed us to plot the slow manifold and to go further into the qualitative study and the geometric characterization of attractors. As an example, we give the explicit slow manifold equation of the van der Pol limit cycle. The value of the parameter corresponding to bifurcations is computed. Other third order systems are also treated. The method is extended to dynamical systems with no small parameter, and, therefore, which have no singular approximations, but have at least one real and negative eigenvalue in a large domain. It is numerically shown from the Lorenz model and from a laser model that there exists slow manifolds which can be defined as the slow manifods of a so-called tangent linear system, as in the previous cases. The implicit equation of these slow manifolds has been calculated too.