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arXiv · 1105.5618

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

Lennard F. Bakker·Scott C. Mancuso·Skyler C. Simmons
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Abstract

We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four-body problem with a mass $m\in(0,1]$ as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a $3\times 3$ matrix for each $m\in(0,1]$ obtained from numerical integration of the linearized regularized equations along only the first one-eighth of each regularized periodic orbit. The results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as $m$ varies over $(0,1]$, with linear instability for $m$ close or equal to 0.01, and linear stability for $m$ close or equal to 1.

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Lennard F. Bakker, Scott C. Mancuso, Skyler C. Simmons. 2011-11-11. Linear Stability Analysis of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem. https://doi.org/10.1007/s10569-012-9402-6

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