It is a known fact that irradiation with high-energy particles can significantly change the physical properties of solids. The reason for this is the radiation-induced evolution of the microstructure of the material which includes significantly different-scale, simultaneously occurring, and interconnected processes that form nonlinear feedbacks. As a result, the irradiated material, as an open nonequilibrium system, is sensitive to irradiation conditions and can evolve in qualitatively different ways due to their slight disturbance. The report discusses a theoretical approach dating back to Poincare, which allows us to identify and describe qualitatively different paths of evolution and the conditions under which they are realized. As examples, the evolution of a void lattice, self-oscillations of the material temperature under irradiation, and self-propagating thermal annealing of defects are considered.