An extensive search for stable periodic orbits of the equal-mass zero angular momentum three-body problem

TL;DR

This work analyzes a special 2D initial-condition domain of the equal-mass zero-angular-momentum planar three-body problem to map stability regions and enumerate stable, collisionless periodic orbits. Leveraging a high-precision Newton solver together with a high-order Taylor Series Method and a half-period symmetry approach, the authors locate 971 verified linearly stable orbits with $T^*<800$, distributed across four distinct stability regions and associated with region-specific syzygy patterns. They compute monodromy matrices with extreme precision, extract the stability frequencies $(\omega_1,\omega_2)$, and verify non-resonance to argue that these orbits are candidates for KAM-stability, while noting that a full twist condition check remains for future work. The authors provide extensive high-precision data for initial conditions, periods, and stability properties, and discuss implications for the existence of rich, measurably stable regions generated by many stable periodic orbits, with potential extensions to varying mass ratios and higher-dimensional domains.

Abstract

A special 2D initial conditions' domain of the equal-mass zero angular momentum planar three-body problem, which has been formerly studied, is analyzed to deepen the knowledge of the stability regions in it. The decay times in the domain are carefully computed. Four stability regions are established. 971 verified initial conditions for linearly stable periodic collisionless orbits are found. Many of these identified initial conditions are new ones. The periodic orbits of each stability region are characterized by a certain pattern in their syzygy sequences. Additional computations show that the orbits found should be considered as candidates for KAM-stable orbits.

Figures (6)

• Figure 1: Initial conditions
• Figure 2: The 2D initial conditions' domain bounded by $v_x=0$ and $v_y=0$ axis and $E=0$ curve.
• Figure 3: A colour map of the decay times.
• Figure 4: Distribution of the stable i.c.s. The stability regions ($t^*_{decay}>10000$) are in yellow.
• Figure 5: Bumblebee Suvakov:2013 -- left and Moth I Suvakov:2013 -- right