Supercritical bifurcation theory and stability of point attractors

Abstract

A novel approach is employed for studying a paramagnetic/ferromagnetic phase transition. Here, the Fokker–Planck transport equation is used to describe the time dependence of the spin distribution function (order parameter) for the XY model in mean-field theory. The evolution of the phase-space trajectories from initial nonequilibrium states is obtained. This is then applied in explaining the otherwise well-known behavior of the XY model using a fully dynamical-systems approach. The attractors of this infinite-dimensional dynamical system are determined and their stability for any value of a system parameter δ, that plays the role of the absolute temperature, is obtained. A supercritical bifurcation occurs for δ=1/2, and this bifurcation corresponds to the paramagnetic/ferromagnetic phase transition. For an arbitrary initial spin density, a unique equilibrium magnetization is shown as being due to a continuum of fixed points existing in the temperature range 0<δ<1/2. These fixed points attract all the phase-space trajectories, except those lying on the stable manifold of a trivial fixed point. The trivial fixed point at the origin is stable if δ≳1/2, otherwise it is a saddle point. These two types of fixed points determine the limit behavior of the dynamical system and therefore also the equilibrium state of the XY model in the approximation used here.