Analytical and numerical bifurcation analysis of dislocation pattern formation of the Walgraef-Aifantis model

We analyze the pattern formation due to dislocations under cyclic loading resulting from the Walgraef-Aifantis model. The model consists of a set of partial differential equations of the reaction-diffusion type in the one dimensional finite space with two different diffusion-like coefficients, for the mobile (free to move when the applied resolved shear stress in the slip plane exceeds a certain threshold) and for the immobile (of slow movement or trapped) dislocations. We derive analytically the Turing spatial and Andronov-Hopf temporal instabilities emanating from the homogeneous solutions and construct the complete bifurcation diagram of the far-from-equilibrium spatio-temporal patterns, with respect to the applied stress and the size of the domain. Finally, we analyze the symmetric properties of all branches of both steady and oscillating far-from-equilibrium regimes.