  We present a systematic approach to deriving normal forms and related
amplitude 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 amplitude
equation and the associated transformation from amplitudes to physical space.
At any order, the relation provides explicit expressions for all the
nonvanishing coefficients of the amplitude equation together with
straightforward linear equations for the coefficients of the transformation.
The recurrence relation therefore provides all the machinery needed to solve a
given 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 approaches in terms of
multiple time scales. We illustrate the use of the method by deriving amplitude
equations and associated transformations for the most common simple
bifurcations in flows and iterated maps. The results are expressed in tables in
a form that can be immediately applied to specific problems.
