Qualitative properties of systems of 2 complex homogeneous ODE's: a connection to polygonal billiards

TL;DR

This paper establishes a broad correspondence between certain complex polynomial ODE systems and polygonal billiards, enabling transfer of known billiard properties to these differential systems, revealing their intricate dynamics.

Contribution

It introduces a general link between 2 complex homogeneous polynomial ODEs and polygonal billiards, allowing the application of billiard theory results to ODE systems.

Findings

Transfer of ergodicity and periodic orbit results

Identification of conservation laws and discontinuities

Revelation of complex dynamics in simple ODE systems

Abstract

A correspondence between the orbits of a system of 2 complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general, in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind. This allows to transfer results well-known from the theory of polygonal billiards, such as ergodicity, the existence of periodic orbits, the absence of exponential divergence, the existence of additional conservation laws, and the presence of discontinuities in the dynamics, to the corresponding systems of ordinary differential equations. It also shows that the considerable intricacy known to exist for polygonal billiards, also attends these apparently simpler systems of ordinary differential equations.