Bifurcations of periodic solutions in two-parameter autonomous systems

Abstract

In this article, we study the problem of bifurcations in a two-parameter nonlinear autonomous system with a homogeneous nonlinearity. It is shown that various scenarios of bifurcations are possible in such a system. Namely, along withthe classical Andronov-Hopf bifurcation, there can also be scenarios for the emergence of periodic solutions branchingoff from some cycle of a linearized system, the occurrence of cycles of large amplitudes, etc. Applications to the analysisof the dynamics of complex nonlinear systems are considered. In the theory of nonlinear oscillations, one of the mostinteresting is the Andronov-Hopf bifurcation problem – the problem of the occurrence of nonstationary periodic oscillations of small amplitude from a complex focus of a dynamical system. Numerous studies have been devoted to the study of the Andronov-Hopf bifurcation. The classical formulation of the Andronov-Hopf bifurcation problem is one-parametric, and the bifurcation itself has a codimension equal to one. At the same time, many applications lead to multiparametric problems, while the bifurcation co-dimension may be more than two. The study of such bifurcations has become one of the popular directions in modern nonlinear dynamics. In this paper, we study two-parameter problems leading to codimension two bifurcation.