Stability of the rhomboidal symmetric-mass orbit

Lennard Bakker; Skyler Simmons

doi:10.3934/dcds.2015.35.1

Article Contents

2015, Volume 35, Issue 1: 1-23. Doi: 10.3934/dcds.2015.35.1

This issue Previous Article Next Article Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics

Stability of the rhomboidal symmetric-mass orbit

Lennard Bakker^1, and
Skyler Simmons^1,

1.
275 TMCB, Brigham Young University, Provo, UT 84602

Received: July 31, 2013

Revised: May 31, 2014

Published: December 2014

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Abstract

We study the rhomboidal symmetric-mass $1$, $m$, $1$, $m$ four-body problem in the four-degrees-of-freedom setting, where $0 < m \leq 1$. Under suitable changes of variables, isolated binary collisions at the origin are regularizable. Analytic existence of the orbit in the four-degrees-of-freedom setting is established. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is analytically reduced to computing three entries of a $4 \times 4$ matrix related to the monodromy matrix. Additionally, it is shown that the four-degrees-of-freedom setting has a two-degrees-of-freedom invariant set, and linear stability results in the subset comes ``for free'' from the calculation in the full space. The final numerical analysis shows that the four-degrees-of-freedom orbit is linearly unstable except for a very small interval about $m = 0.4$, whereas the two-degrees-of-freedom orbit is linearly stable for all but very small values of $m$.

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Mathematics Subject Classification: Primary: 70F16; Secondary: 37N05, 37J25.

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