Chaos in the Hill system

TL;DR

This paper investigates the dynamical behavior of the Hill system, especially its potential for chaos, through analytical and numerical methods, including explicit computation of the Poincaré-Melnikov function and numerical evidence.

Contribution

It introduces a specific Hill system modeling gravitational interactions and provides explicit calculations and numerical evidence for chaos in this system.

Findings

Explicit Poincaré-Melnikov function computed

Zeros of the Melnikov function identified

Numerical evidence supports chaotic behavior

Abstract

We define the general Hill system and briefly analyze its dynamical behavior. A particular Hill system representing the interaction of a Keplerian binary system with a normally incident circularly polarized gravitational wave is discussed in detail. In this case, we compute the Poincar\'e-Melnikov function explicitly and determine its zeros. Moreover, we provide numerical evidence in favor of chaos in this system. The partially averaged equations for the Hill system are used to predict the regular behavior of the Keplerian orbit at resonance with the external radiation.