Bifurcations associated with a double zero of index two and a pair of purely imaginary eigenvalues

Abstract

The bifurcation and instability behaviour of a non-linear autonomous system in the vicinity of a compound critical point, involving a pair of purely imaginary eigenvalues and a two-fold zero eigenvalue of index two, are investigated. The analysis is carried out via a systematic perturbation method which embraces a unification technique, leading to a simplified set of differential equations for the analysis of local behaviour. Incipient and secondary bifurcations are explored. The static and dynamic instability boundaries are established. A non-linear control system drawn from electrical network theory is analysed in detail.