Parameter functionalization and its application to the problem of local bifurcations in dynamic systems

The method of parameter functionalization suggested by M. A. Krasnosel'skii for solving problems with parameters and continuums of fixed points is considered. A general scheme of constructing the functionals in the bifurcation problem of small solutions to operator equations is suggested. As an application, we consider problems of local bifurcations in dynamic systems that are topical for the control theory: bifurcations of double equilibrium and forced oscillations and bifurcations of cycles of discrete systems. New sufficient criteria of bifurcations are indicated, an iteration procedure for constructing the solutions and their asymptotic representations are elaborated, and new stability conditions are stated.