Symbolic-numerical methods for the computation of normal forms of PDEs

The center manifold and the normal forms are effective tools for the study of local bifurcations occurring in evolution equations. The computation of the center manifold and the normal form amounts, after more or less complex algebraic transformations, to solve in a recursive way a hierarchy of linear equations. We present a method and computer programs for the computation of normal forms of some nonlinear parabolic PDEs. These computations are performed using the symbolic algebra system Maple, Matlab and exploiting the compatibility of these two systems. Here the linear equations to be solved are infinite dimensional and we use the finite element method for this purpose. The use of the finite element method allows to consider problems with complex shape domains. In our programs, Maple takes care of the algebraic manipulations delivering the set of linear equations to be solved and writes some parts of the Matlab code for their resolution. We give three applications: a pitchfork bifurcation in a semilinear parabolic equation, a Hopf bifurcation and a bifurcation to rotating and standing waves in a reaction–diffusion system.