Minimum Transition Values and the Dynamics of Subcritical Bifurcation

Perturbation and asymptotic methods are presented for analyzing a class of subcritical bifurcation problems whose solutions possess minimum transition values. These minimum transition values are determined. In addition, the dynamics of the transitions from the basic state to the larger amplitude bifurcation states are obtained. The effects of imperfections on the response of the systems are also investigated. The method is presented for two model problems. However, it is valid for a wide class of problems in elastic and hydrodynamic stability, in reaction-diffusion systems and in other applications. In the first problem we obtain subcritical steady bifurcation states for a one-dimensional nonlinear diffusion problem. In the second problem we consider the subcritical Hopf bifurcation of periodic solutions for a higher order van der Pol–Duffing oscillator.