Systematic derivation of amplitude equations and normal forms for dynamical systems

We present a systematic approach to deriving normal forms and related ampli tude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general, explicit recurrence relation that completely determines the ampli tude equation and the associated transformation from amplitudes to physical space. At any order, the relation provides explicit expressions for all th e nonvanishing coefficients of the amplitude equation together with straigh tforward linear equations for the coefficients of the transformation. The r ecurrence relation therefore provides all the machinery needed to solve a g iven physical problem in physical terms through an amplitude equation. The new result applies to any local bifurcation of a flow or map for which all the critical eigenvalues are semisimple (i.e., have Riesz index unity). The method is an efficient and rigorous alternative to more intuitive approach es in terms of multiple time scales. We illustrate the use of the method by deriving amplitude equations and associated transformations for the most c ommon simple bifurcations in flows and iterated maps. The results are expre ssed in tables in a form that can be immediately applied to specific proble ms. (C) 1998 American Institute of Physics. [S1054-1500(98)00404-2].