Chaotic phenomena for generalised N-centre problems

TL;DR

This paper extends the classical N-centre problem to Riemannian surfaces, demonstrating the existence of chaotic dynamics and symbolic encoding of collision-free periodic solutions at high energies.

Contribution

It introduces a new framework for analyzing generalized N-centre problems on surfaces, establishing symbolic dynamics and chaos without collision regularisation.

Findings

Existence of infinitely many collision-free periodic solutions

Symbolic dynamics encoding solutions as bi-infinite sequences

Chaotic invariant sets when the Jacobi-Maupertuis metric curvature is negative

Abstract

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.