The mechanism of the hard appearance of a two-frequency oscillation mode in the case of Andronov-Hopf reverse bifurcation
This paper deals with the sudden birth of a two-frequency periodic solution in an m-dimensional autonomous ordinary differential equation, \(m>2\), when an equation parameter crosses through a critical value. This parameter value corresponds to a bifurcation leading to the disappearing of a saddle limit cycle, when it merges into a node equilibrium point (Andronov-Hopf reverse bifurcation). After the bifurcation, locally it remains only the equilibrium point which has turned into a saddle. When \(m=3\), the unstable invariant manifold \(W^ u\) of this saddle point is a two-dimensional one with unstable focus type trajectories. The stable invariant manifold \(W^ S\) has the dimension one. In this situation a phase trajectory coming from a neighborhood of \(W^ u\) can be injected in a neighborhood of \(W^ S\), this giving rise to a stable closed trajectory in the three-dimensional phase space. The projection of this trajectory on a plane is a curve with a certain number of loops related to the rank of a higher harmonic resonance. When the phase trajectory is injected in \(W^ S\), and so returns to the equilibrium point, a limit case, corresponding to infinitely many loops, is obtained, and one has a homoclinic situation. Sufficient conditions of existence of such periodic solutions are given for \(m=3\), and \(m=4\).
scientific article; zbMATH DE number 4209158