Structure and Convergence of Poincaré-like Normal Forms. Let Us Consider a System of Nonlinear Ordinary Differential Equations

The general term of the Poincaré normalizing series is explicitly constructed for non-resonant systems of ODE's in a large class of equations. In the resonant case, a non-local transformation is found, which exactly linearizes the ODE's and whose series expansion always converges in a finite domain. Examples are treated. where the (purely nonlinear) functions f i (x) are analytic in the real variables x 1 ...x N. We suppose, for the simplicity of the purpose, that the linear part has been diagonalised. The study of the qualitative behavior of the system starts with a linear stability analysis of the fixed points [1]. This determines the values of the control parameters for which the system bifurcates (i.e. for which some stable fixed point becomes unstable, or vice-versa.). A further analysis gives the nonlinear behavior of the solutions in the neighborhood of the fixed points. Among the methods to compute this last step, the most often used is the normal form