[Paper Review] Chaos in the Hill system

Carmen Chicone, Bahram Mashhoon+1 more|ArXiv.org|Jul 1, 1999

Quantum chaos and dynamical systems11 references3 citations

TL;DR

This paper investigates chaos in the Hill system under the influence of a normally incident circularly polarized gravitational wave, using the Poincaré-Melnikov function to analytically identify chaotic behavior and providing numerical evidence for chaos. The study establishes that resonance between the Keplerian orbit and the external wave leads to chaotic dynamics, with explicit computation of the Melnikov function zeros confirming the onset of chaos.

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 Poincaré-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.

Motivation & Objective

• To analyze the dynamical behavior of the general Hill system under periodic external perturbations.
• To investigate the emergence of chaos in a Keplerian binary system interacting with a circularly polarized gravitational wave.
• To compute the Poincaré-Melnikov function explicitly for the specific Hill system model.
• To determine the zeros of the Melnikov function as a criterion for chaotic behavior.
• To provide numerical evidence supporting the presence of chaos in the system.

Proposed method

• The general Hill system is defined as a perturbed Hamiltonian system with a time-periodic external forcing.
• The specific case involves a Keplerian binary system perturbed by a normally incident, circularly polarized gravitational wave.
• The Poincaré-Melnikov function is derived analytically to assess the splitting of separatrices in the phase space.
• Zeros of the Melnikov function are computed to identify conditions under which chaotic dynamics arise.
• Partially averaged equations are used to predict regular behavior at resonance with the external wave.
• Numerical simulations are performed to support the analytical findings and confirm chaotic behavior.

Experimental results

Research questions

• RQ1 Under what conditions does the Hill system exhibit chaotic dynamics when perturbed by a circularly polarized gravitational wave?
• RQ2 How does the Poincaré-Melnikov function predict the onset of chaos in this system?
• RQ3 What is the role of resonance between the Keplerian orbital frequency and the gravitational wave frequency in inducing chaos?
• RQ4 How do the partially averaged equations describe the system's behavior at resonance?
• RQ5 What numerical evidence supports the analytical prediction of chaotic motion?

Key findings

• The Poincaré-Melnikov function is computed explicitly for the Hill system under a circularly polarized gravitational wave.
• Zeros of the Melnikov function are found, indicating the presence of transverse homoclinic orbits and thus chaotic dynamics.
• Numerical simulations provide strong evidence for chaotic behavior in the system, particularly near resonance.
• The partially averaged equations predict regular, non-chaotic motion when the Keplerian orbit is in resonance with the external wave.
• The analytical and numerical results together confirm that the system exhibits chaotic behavior under specific conditions of frequency and amplitude.
• The study establishes a clear link between the Melnikov function's zeros and the onset of chaos in this class of dynamical systems.