Towards a Theory of Chaos Explained as Travel on Riemann Surfaces

TL;DR

This paper explains the transition from regular to chaotic motions in a simple model using travel on Riemann surfaces, providing a new perspective on deterministic chaos through complex analysis.

Contribution

It introduces a simple, solvable model that illustrates chaotic behavior and connects it to travel on Riemann surfaces, advancing understanding of chaos in dynamical systems.

Findings

Explicit solutions for isochronous cases with known periods

Chaotic motions confined but aperiodic in certain parameter regimes

Quantitative link between complex-time solutions and Riemann surface travel

Abstract

This paper presents a more complete version than hitherto published of our explanation of a transition from regular to irregular motions and more generally of the nature of a certain kind of deterministic chaos. To this end we introduced a simple model analogous to a three-body problem in the plane, whose general solution is obtained via quadratures all performed in terms of elementary functions. For some values of the coupling constants the system is isochronous and explicit formulas for the period of the solutions can be given. For other values, the motions are confined but feature aperiodic (in some sense chaotic) motions. This rich phenomenology can be understood in remarkable, quantitative detail in terms of travel on a certain (circular) path on the Riemann surfaces defined by the solutions of a related model considered as functions of a complex time. This model is meant to provide a paradigmatic first step towards a somewhat novel understanding of a certain kind of chaotic phenomena.