Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

Abstract

We study the dynamics of a family of perturbed three-degreesof-freedom (3-DOF) Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction of the two symmetries leads to a one-degrees-offreedom system that is completely analysed in the twice reduced space. All the relative equilibria, together with the stability and parametric bifurcations are determined. Moreover the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduce space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.

Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

“…To improve the procedure, we use a type of symplectic variables (L, Q, l, P) as in Palacián et al, 18 where the authors construct these variables such that the unperturbed Hamiltonian in (2) in the new coordinates depends only on L. They make a particularization for the 1 : 1 : 1 resonance of the construction of local symplectic maps for resonant Hamiltonian systems with n degrees of freedom (see Carrasco et al 19 and Meyer et al 20 ). In our work, we use these symplectic variables in two degrees of freedom given by…”

Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential

“…. More details on how to build these maps appear in [4]. The coordinate is an angle whereas L corresponds to its conjugate action.…”

“…In order to apply the theory of [18] we introduce an S 1 -symmetry to the non-linear normal form. This step was not needed in [19,4] as the systems tackled in those papers already enjoyed that symmetry. In our problem, if we try to build a normal form through a change of coordinates introduced by a generating function defined as a polynomial of degree four we notice that it is impossible.…”

“…Multiplying the coefficients of x 2 1 +x 2 2 and ε 1/2 (y 2 1 +y 2 2 ) 2 in H jet 4 we get 4 3 h 2 (14δ 3 +94δ 2 +191δ +123) which becomes 16h 2 when δ is replaced by −1. As h < 0 we conclude that the bifurcation is supercritical.…”

“…More precisely, we make use of Han, Li and Yi's Theorem A. 4 (see details in [17]) that applies in the case of Hamiltonian systems with high-order proper degeneracy.…”