SLOW MANIFOLDS OF SOME CHAOTIC SYSTEMS WITH APPLICATIONS TO LASER SYSTEMS

Abstract

In this work we deal with slow–fast autonomous dynamical systems. We initially define them as being modeled by systems of differential equations having a small parameter multiplying one of their velocity components. In order to analyze their solutions, some being chaotic, we have proposed a mathematical analytic method based on an iterative approach [Rossetto et al., 1998]. Under some conditions, this method allows us to give an analytic equation of the slow manifold. This equation is obtained by considering that the slow manifold is locally defined by a plane orthogonal to the tangent system's left fast eigenvector. In this paper, we give another method to compute the slow manifold equation by using the tangent system's slow eigenvectors. This method allows us to give a geometrical characterization of the attractor and a global qualitative description of its dynamics. The method used to compute the equation of the slow manifold has been extended to systems having a real and negative eigenvalue in a large domain of the phase space, as it is the case with the Lorenz system. Indeed, we give the Lorenz slow manifold equation and this allows us to make a qualitative study comparing this model and Chua's model. Finally, we apply our results to derive the slow manifold equations of a nonlinear optical slow–fast system, namely, the optical parametric oscillator model.