Adiabatic elimination and Ginzburg-Landau form description for steps on creep curve

Abstract

We consider a model proposed earlier by us for explaining the phenomenon of jumps on creep curve. It consists of three types of dislocations and some transformations between them, leading to a coupled set of nonlinear differential equations for the densities of the dislocations. The mathematical mechanism has been shown to be Hopf bifurcation with respect to drive parameters. Here, we present two approaches of obtaining a Ginsburg-Landau form representation of the order parameter beyond the Hopf bifurcation up to quintic terms. The present analysis shows that over a certain range of one of the parameters we find a 'first order' (subcritical bifurcation) transition consistent with our earlier approximate solutions. In addition, we found another range of the parameter over which the transition is 'second order' (supercritical bifurcation). This method also allows us to map the theoretical parameters to the experimentally observed macroscopic quantities.