ALMOST-PERIODIC BIFURCATIONS FOR 2-DIMENSIONAL DEGENERATE HAMILTONIAN VECTOR FIELDS

In this paper, we develop almost-periodic tori bifurcation theory for 2-dimensional degenerate Hamiltonian vector fields. With KAM theory and singularity theory, we show that the universal unfolding of completely degenerate Hamiltonian N(x, y) = x2y+yl and partially degenerate Hamiltonian M(x, y) = x2 + yl, respectively, can persist under any small almost-periodic time-dependent perturbation and some appropriate non-resonant conditions on almost-periodic frequency omega = (center dot center dot center dot, omega i, center dot center dot center dot )i is an element of Z is an element of RZ. We extend the analysis about almost-periodic bifurcations of one-dimensional degenerate vector fields considered in [21] to 2-dimensional degenerate vector fields. Our main results (Theorem 2.1 and Theorem 2.2) imply infinite-dimensional degenerate umbilical tori or normally parabolic tori bifurcate according to a generalised umbilical catastrophe or generalised cuspoid catastrophe under any small almost-periodic perturbation. For the proof in this paper we use the overall strategy of [21], which however has to be substantially developed to deal with the equations considered here.

Automated summary

In this paper, an almost-periodic tori bifurcation theory for 2-dimensional degenerate Hamiltonian vector fields is developed. The universal unfolding of completely degenerate Hamiltonian N(x, y) = x2y+yl and partially degenerate Hamiltonian M(x, y) = x2 + yl can persist under small almost-periodic time-dependent perturbation and certain non-resonant conditions on almost-periodic frequency omega. The main results of this study show that infinite-dimensional degenerate umbilical tori or normally parabolic tori can bifurcate according to a generalized umbilical catastrophe or generalized cuspoid catastrophe under small almost-periodic perturbation.