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)=x^2y+y^l $ and partially degenerate Hamiltonian $ M(x, y)=x^2+y^l, $ respectively, can persist under any small almost-periodic time-dependent perturbation and some appropriate non-resonant conditions on almost-periodic frequency $ \omega=(\cdots, \omega_i, \cdots)_{i\in \mathbb{Z}}\in \mathbb{R}^\mathbb{Z}. $ 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.