Reduction and Reconstruction of the Oscillator in 1:1:2 Resonance plus an Axially Symmetric Polynomial Perturbation

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2489-2532, September 2024.
Abstract.We consider a family of perturbed Hamiltonian systems with Hamiltonian [math] in 1:1:2 resonance, where [math] is a polynomial which is axially symmetric with respect to the [math]-axis. Here, [math] is a homogeneous polynomial of degree [math], and we note that our analysis is carried out considering the polynomials [math] and [math]. We initially perform a Lie–Deprit normalization (truncation of the higher-order terms), and a singular reduction by the oscillator symmetry is done. Considering the averaging method for Hamiltonian systems, the existence and an approximation of two families of periodic solutions are proved together with their linear stability. A third family of periodic solutions is found by using the Lyapunov center theorem. In addition, the existence of KAM 3-tori is obtained by enclosing the stable periodic solutions. After that, since the Hamiltonian is axially symmetric, we carry out another reduction induced by this exact symmetry. Studying its Poisson vector field on the reduced space by the exact symmetry, we show the existence of two equilibrium points. We reconstruct these points as two families of periodic solutions of the complete Hamiltonian system together with their linear stability. Next, we make a second singular reduction using the axial symmetry. A geometrical study of the twice-reduced space is done to characterize the singularities. Precisely, we study the critical points (relative equilibria) on the twice-reduced space together with the stability, and parametric bifurcations are determined. The equilibria occurring in the twice-reduced space are reconstructed as 3-tori filled by quasi-periodic solutions of the full system. Our analysis permits us to determine the main representative parameters of the cubic ([math]) and quartic ([math]) terms to get our results. Important differences with the case of resonance 1:1:1 are detected.