Different faces of chaos in FRW models with scalar fields -- geometrical point of view

TL;DR

This paper explores the complex dynamical behavior of FRW cosmologies with conformally coupled scalar fields using a geometrical approach based on geodesics of the Jacobi metric, revealing insights into chaos, periodic orbits, and recurrence properties.

Contribution

It introduces a geometrical framework analyzing the dynamics via geodesics of the Jacobi metric, including classification of periodic orbits and recurrence analysis in FRW models.

Findings

Singular set contains information about dynamical complexity.

The model exhibits recurrence and complex trajectories.

Existence of unstable periodic orbits (UPO) is demonstrated.

Abstract

FRW cosmologies with conformally coupled scalar fields are investigated in a geometrical way by the means of geodesics of the Jacobi metric. In this model of dynamics, trajectories in the configuration space are represented by geodesics. Because of the singular nature of the Jacobi metric on the boundary set $\partial\mathcal{D}$ of the domain of admissible motion, the geodesics change the cone sectors several times (or an infinite number of times) in the neighborhood of the singular set $\partial\mathcal{D}$. We show that this singular set contains interesting information about the dynamical complexity of the model. Firstly, this set can be used as a Poincar{\'e} surface for construction of Poincar{\'e} sections, and the trajectories then have the recurrence property. We also investigate the distribution of the intersection points. Secondly, the full classification of periodic orbits in the configuration space is performed and existence of UPO is demonstrated. Our general conclusion is that, although the presented model leads to several complications, like divergence of curvature invariants as a measure of sensitive dependence on initial conditions, some global results can be obtained and some additional physical insight is gained from using the conformal Jacobi metric. We also study the complex behavior of trajectories in terms of symbolic dynamics.