On the detuned 2:4 resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.Comment: 31 pages, 7 figures: On line first on Journal of Nonlinear Science (2020

Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach

We perform a bifurcation analysis of normal–internal resonances in parametrised families of quasi–periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the ‘backbone’ system; forced, the system is a skew–product flow with a quasi–periodic driving with basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The averaged system turns out to have the same structure as in the well–known case of periodic forcing ; for a real analytic system, the non–integrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasi–periodic –dimensional tori in the averaged system, filling normal–internal resonance ‘gaps’ that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of ‘gaps within gaps’ makes the quasi–periodic case more complicated than the periodic case

Quasi-periodic Motions of a Rigid Body. A case study on perturbations of superintegrable systems

Iedereen heeft ooit wel eens met een bromtol gespeeld en daarmee een typisch voorbeeld van een star lichaam leren kennen. De bromtol is niet alleen 'star' (bij zijn beweging verandert de afstand van elk tweetal van zijn punten niet), maar ook rotatie-symmetrisch (op de voor het brommen nodige gaatjes na). De symmetrie- of figuur-as eindigt in de punt van de tol die op de grond geplaatst wordt nadat we de tol in snelle rotatie hebben gebracht. ... Zie: Samenvatting.

A Galilean dance 1:2:4 resonant periodic motions and their librations of Jupiter and his Galilean moons

The four Galilean moons of Jupiter were discovered by Galileo in the early 17th century, and their motion was first seen as a miniature solar system. Around 1800 Laplace discovered that the Galilean motion is subjected to an orbital 1:2:4-resonance of the inner three moons Io, Europa and Ganymedes. In the early 20th century De Sitter gave a mathematical explanation for this in a Newtonian framework. In fact, he found a family of stable periodic solutions by using the seminal work of Poincaré, which at the time was quite new. In this paper we review and summarize recent results of Broer, Hanßmann and Zhao on the motion of the entire Galilean system, so including the fourth moon Callisto. To this purpose we use a version of parametrised Kolmogorov-Arnol'd-Moser theory where a family of multi-periodic isotropic invariant three-dimensional tori is found that combines the periodic motions of De Sitter and Callisto. The 3-tori are normally elliptic and excite a family of invariant Lagrangean 8-tori that project down to librational motions. Both the 3- and the 8-tori occur for an almost full Hausdorff measure set in the product of corresponding dimension in phase space and a parameter space, where the external parameters are given by the masses of the moons