Analysis of critical and post-critical behaviour of non-linear dynamical systems by the normal form method, Part I: Normalization formulae

Abstract

A method, based on normal form theory, is presented to study the dynamical behaviour of a system in the neighbourhood of a nearly critical equilibrium state associated with a bifurcation condition. Explicit formulae for the normalization procedure are derived. These formulae can be numerically programmed, avoiding usual complicated algebraic calculations and making the method effectively applicable for n-dimensional systems. Rather general bifurcations can be included: e.g., non-linear flutter (Hopf bifurcation), divergence and internal resonance.