Bifurcation map of periodic and quasi-periodic orbits around the asteroid 433 Eros using the harmonic balance method

Abstract

This study applies the harmonic balance method (HBM) to compute and analyze periodic and quasi-periodic orbits around the asteroid 433 Eros. Traditional time-integration techniques are often computationally intensive and prone to numerical drift, particularly in the presence of instabilities or during the continuation of complex orbit families. By working directly in the frequency domain, HBM avoids these issues and offers a global, efficient framework for orbit computation. The use of the polyhedron gravity model ensures accurate modeling of Eros’s irregular gravitational field, while Hill’s method provides insight into the stability of periodic solutions. The study presents a dense bifurcation map highlighting over 100 bifurcations, including period-doubling and Neimark–Sacker types. A detailed spectral analysis of resonant and bifurcated orbits is conducted via normalized Fourier coefficients, offering new perspectives on orbital transitions and instabilities. The multi-harmonic extension (MHBM) is also employed to compute quasi-periodic orbits, with validation against time-domain solutions. The results demonstrate that HBM is a powerful alternative to traditional techniques, enabling accurate, efficient, and insightful analysis of complex orbital dynamics in asteroid environments.