Attractor properties of physical branches of the solution to the renormalization group equation

We investigate the global phase-portrait structure of a local version of th e exact renormalization group (RG) equation for a fluctuating scalar field of the order parameter. All the physical branches of the RG equation soluti on for the fixed points belong to the attractor subspace to which the local density of the Ginzburg-Landau-Wilson functional is attracted for largely arbitrary initial configurations. The solution of the RG equation correspon ding to the nontrivial fixed point determining the critical behavior under the second-order phase transition is a fixed saddle point of this attractor subspace separating the attraction domains of two stable solutions corresp onding to the high- and low-temperature thermodynamic regimes.