Existence and Stability of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem

Abstract

We extend our previous analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized fully symmetric equal mass four-body problem to the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for $m$ between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when $0.54\le m\le 1$, and are linearly unstable when $0<m\le 0.53$.

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References in corpus (3)

• Chaos in the one-dimensional gravitational three-body problem
• Linear Stability for Some Symmetric Periodic Simultaneous Binary Collision Orbits in the Four-Body Problem
• Periodic Solutions with Alternating Singularities in the Collinear Four-body Problem

Cited by in corpus (6)

• Linear Stability for Some Symmetric Periodic Simultaneous Binary Collision Orbits in the Four-Body Problem
• On the $C^{8/3}$-Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem
• Regularisation for Planar Vector Fields
• On the $C^{8/3}$-Regularisation of Simultaneous Binary Collisions in the Collinear 4-Body Problem
• Linear Stability Analysis of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem
• The Eight-Body Cubic Collision-Based Periodic Orbit