Keywords: center manifold and normal forms,bifurcation point, dynamical system.
Jointly with my PhD advisor, Professor Jean-Jacques Gervais of Laval University Quebec (Canada), we have derived an algorithm to calculate center manifold and normal forms for a dynamical system. Normal forms are, in general, not uniquely defined and many techniques are proposed to calculate the center manifold and normal forms. The techniques used for our work have the advantage of being algorithmizable and lead to a unique normal form. This normal form is characterized by a symmetry propriety that is obtainable a priori, since it requires only the knowledge of the linearized system at the non-hyperbolic equilibrium point. A symbolic code (in Maple) has been developed to execute the algebraic manipulations needed to obtain this hierarchy of equations. The program is user friendly and can be downloaded by ftp. This project was sponsored by "Le Programme Canadien des Bourses de la Francophonie". We present in the figures below same applications to study the dynamics of parabolic partial differential equations close to a bifurcation point. Here, in addition of the computation of the normal form, we are dealing with the problem of solving infinite-dimensional linear equations. We use an analytical-numerical hybrid analysis technique. Using our code, we find all the hierarchy of linear systems allowing to find the coefficients of the normal form and the center manifold. These infinite-dimensional linear systems are discretized with the finite element method and then solved numerically. The knowledge of the coefficients of the normal form and the center manifold allows approximating the solution of the parabolic PDE near the equilibrium point.
Numerical Solution of a reaction diffusion system on Center Manifold obtained from normal form(Just click for animation)