Static and dynamic bifurcations associated with a double-zero eigenvalue

Abstract

This paper is concerned with the static and dynamic bifurcation phenomena exhibited by a nonlinear autonomous system in the vicinity of a double-zero eigenvalue. It is demonstrated analytically that such a coincident critical point is often located on the intersection of the divergence and flutter boundaries of the system. The analysis is performed via a new ‘unification technique’ which is based on an intrinsic perturbation procedure. The new approach is capable of yielding information about stability of solutions as well as incipient and secondary bifurcations. Indeed, the stability properties of the system and secondary Hopf bifurcations are discussed conveniently, and explicit results concerning post-critical solutions are presented. An illustrative example is analysed.