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

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

2011-05-27

arxiv.org/abs/1105.5618v2

Mathematics

Classical Analysis and ODEs

13 pages, 1 figure

Scientific paper

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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