Principal Asymptotics in the Problem on the Andronov–Hopf Bifurcation and Their Applications

Abstract

New formulas are obtained for the principal asymptotics of bifurcation solutions in the problem on the Andronov–Hopf bifurcation, leading to new algorithms for studying bifurcations in the general setting. The approach proposed in the paper allows one to consider not only the classical problems about bifurcations of codimension one but also some problems concerning bifurcations of codimension two. A new approach to the analysis of bifurcations of cycles in systems with homogeneous nonlinearities is proposed. As an application, we consider the problem on the bifurcation of periodic solutions of the van der Pol equation.