Bifurcations of periodic oscillations in dynamical system with homogeneous nonlinearities

The paper is devoted to the study of cycle bifurcations and bifurcations at infinity for dynamical systems with a small parameter, the nonlinearities of which contain homogeneous polynomials of even or odd degree, and the unperturbed equation has a continuum of periodic solutions. We propose new necessary and sufficient conditions for these bifurcations, obtain the formulas for the approximate construction of bifurcation solutions, and analyze their stability. We show that cycle bifurcations are typical only for systems with homogeneities of odd degree, while the bifurcations at infinity are typical only for systems with homogeneities of even degree. We demonstrate the relationship between these bifurcations and the classical Andronov - Hopf bifurcation.