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

Abstract

A family of perturbed Hamiltonians Hε = 1 2 (x 2 + X 2) − 1 2 (y 2 + Y 2) + 1 2 (z 2 + Z 2) + ε 2 [α(x 4 + y 4 + z 4) + β(x 2 y 2 + x 2 z 2 + y 2 z 2)] in 1:−1:1 resonance depending on two real parameters is considered. We show the existence and stability of periodic solutions using reduction and averaging. In fact, there are at most thirteen families for every energy level h < 0 and at most twenty six families for every h > 0. The different types of periodic solutions for every nonzero energy level, as well as their bifurcations, are characterised in terms of the parameters. The linear stability of each family of periodic solutions, together with the determination of KAM 3-tori encasing some of the linearly stable periodic solutions is proved. Critical Hamiltonian bifurcations on the reduced space are characterised. We find important differences with respect to the dynamics of the 1:1:1 resonance with the same perturbation as the one given here. We end up with an intuitive interpretation of the results from a cosmological viewpoint.

Horseshoes and invariant tori in cosmological models with a coupled field and non-zero curvature

“…In [50], Palacián, Vidal, Vidarte & Yanguas study a 3 degree-of-freedom version of the hamiltonian (2) for k = 1 distinct from the one in the previous paragraph. Similarly, however, the paper focuses on the critical point at the origin and uses multi-scale KAM theory to prove the existence of invariant 3 tori near that critical point.…”

Horseshoes and invariant tori in cosmological models with a coupled field and non-zero curvature *

“…In [51], Palacián, Vidal, Vidarte and Yanguas study a 3 degree-of-freedom version of the Hamiltonian (2) for k = 1 distinct from the one in the previous paragraph. Similarly, however, the paper focuses on the critical point at the origin and uses multi-scale KAM theory to prove the existence of invariant 3 tori near that critical point.…”

“…The results here are intended to be illustrative rather than exhaustive. The paper by Palacián, et al [51] contains more expansive results in a more specialized context.…”