Testing the accuracy of the overlap criterion

TL;DR

This paper evaluates the accuracy of the overlap criterion in predicting chaos in near-integrable models, confirming that considering resonances up to second order in perturbation theory yields reliable results in 2D and 3D systems.

Contribution

It demonstrates that the overlap criterion's effectiveness is validated when resonances are computed up to second order, through both theoretical estimates and numerical simulations in 2D and 3D models.

Findings

Good agreement between theoretical and numerical critical values.

Resonances up to second order are sufficient for the overlap criterion.

Validation in both 2D and 3D systems.

Abstract

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}