High-order amplitude equation for steps on the creep curve

We consider a model proposed by one of the authors for a type of plastic instability found in creep experiments which reproduces a number of experimentally observed features. The model consists of three coupled nonlinear differential equations describing the evolution of three types of dislocations. The transition to the instability has been shown to be via Hopf bifurcation, leading to limit cycle solutions with respect to physically relevant drive parameters. Here we use a reductive perturbative method to extract an amplitude equation of up to seventh order to obtain an approximate analytic expression for the order parameter. The analysis also enables us to obtain the bifurcation (phase) diagram of the instability. We find that while supercritical bifurcation dominates the major part of the instability region, subcritical bifurcation gradually takes over at one end of the region. These results are compared with the known experimental results. Approximate analytic expressions for the limit cycles for different types of bifurcations are shown to agree with their corresponding numerical solutions of the equations describing the model. The analysis also shows that high order nonlinearities are important in the problem. This approach further allows us to map the theoretical parameters to the experimentally observed macroscopic quantities.