Testing the accuracy of the overlap criterion

Here we investigate the accuracy of the overlap criterion when applied to a simple near-integrable model in both its 2D and 3D version. To this end, we consider respectively, two and three quartic oscillators as the unperturbed system, and couple the degrees of freedom by a cubic, non-integrable perturbation. For both systems we compute the unperturbed resonances up to order O(\epsilon^2), and model each resonance by means of the pendulum approximation in order to estimate the theoretical critical value of the perturbation parameter for a global transition to chaos. We perform several surface of sections for the bidimensional case to derive an empirical value to be compared to our theoretical estimation, being both in good agreement. Also for the 3D case a numerical estimate is attained that we observe matches the critical value resulting from theoretical means. This confirms once again that reckoning resonances up to O(\epsilon^2) suffices in order the overlap criterion to work out. Keywords: {Chaos -- Resonances -- Theoretical and Numerical Methods}Comment: 16 page

On the relevance of chaos for halo stars in the solar neighbourhood II

In a previous paper based on dark matter only simulations we show that, in the approximation of an analytic and static potential describing the strongly triaxial and cuspy shape of Milky Way-sized haloes, diffusion due to chaotic mixing in the neighbourhood of the Sun does not efficiently erase phase space signatures of past accretion events. In this second paper we further explore the effect of chaotic mixing using multicomponent Galactic potential models and solar neighbourhood-like volumes extracted from fully cosmological hydrodynamic simulations, thus naturally accounting for the gravitational potential associated with baryonic components, such as the bulge and disc. Despite the strong change in the global Galactic potentials with respect to those obtained in dark matter only simulations, our results confirm that a large fraction of halo particles evolving on chaotic orbits exhibit their chaotic behaviour after periods of time significantly larger than a Hubble time. In addition, significant diffusion in phase space is not observed on those particles that do exhibit chaotic behaviour within a Hubble time

Chirikov Diffusion in the Asteroidal Three-Body Resonance (5,-2,-2)

The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulations developed by Chirikov is applied to the Nesvorn\'{y}-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances (Jupiter-Saturn-asteroid system). In particular, we investigate the diffusion \emph{along} and \emph{across} the separatrices of the (5,-2,-2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type sympletic map and numerical integrations for times up to $10^{8}$ years.Comment: 27 pages, 6 figure

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

n this article we investigate numerically the spectrum of some representative examples of discrete one-dimensional Schrödinger operators with quasi-periodic potential in terms of a perturbative constant b and the spectral parameter a. Our examples include the well-known Almost Mathieu model, other trigonometric potentials with a single quasi-periodic frequency and generalisations with two and three frequencies. We computed numerically the rotation number and the Lyapunov exponent to detect open and collapsed gaps, resonance tongues and the measure of the spectrum. We found that the case with one frequency was significantly different from the case of several frequencies because the latter has all gaps collapsed for a sufficiently large value of the perturbative constant and thus the spectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the cases with one frequency considered, gaps are always dense in the spectrum, although some gaps may collapse either for a single value of the perturbative constant or for a range of values. In all cases we found that there is a curve in the (a, b)-plane which separates the regions where the Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve, which is b = 2 in the Almost Mathieu case, the measure of the spectrum is zero.Peer ReviewedPostprint (published version