Stability of Broucke's isosceles orbit

Skyler Simmons

doi:10.3934/dcds.2021015

Article Contents

2021, Volume 41, Issue 8: 3759-3779. Doi: 10.3934/dcds.2021015

This issue Previous Article Martin boundary of brownian motion on Gromov hyperbolic metric graphs Next Article Chaotic Delone sets

Stability of Broucke's isosceles orbit

Skyler Simmons

Mathematics Department, Utah Valley University, 800 W University Parkway, Orem, UT 84058, USA

Received: June 30, 2020

Revised: September 30, 2020

Early access: January 2021

Published: August 2021

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Abstract

We extend the result of Yan to Broucke's isosceles orbit with masses $ m_1 $, $ m_1 $, and $ m_2 $ with $ 2m_1 + m_2 = 3 $. Under suitable changes of variables, isolated binary collisions between the two mass $ m_1 $ particles are regularizable. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is 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 the interval $ 0.595 < m_1 < 0.715 $, whereas the two-degrees-of-freedom orbit is linearly stable for a much wider interval.

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

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Figure 1. The coordinates for Broucke's Orbit

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Figure 2. Broucke's orbit as it evolves in time. One-quarter period is shown. The remainder of the orbit is obtained by a symmetric extension

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Figure 3. A plot of the value of $ Q_4(0) $. The value of $ m_1 $ is plotted on the horizontal axis

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Figure 4. A plot of the numerically-computed upper-left entry $ k_{11} $ from the matrix $ K $ in Equation 11 (vertical) against $ m_1 $ (horizontal)

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Figure 5. A vertically rescaled plot of the graph from Figure 4

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Figure 6. A plot of the value of $ e $ from Equation (11) corresponding to stability in the 2DF setting

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Figure 7. A vertically rescaled plot of the value of $ e $. The curve appears to be asymptotic to the line $ y = 1 $ as $ m_1 \to 0^- $

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Figure 8. A plot of the second non-trivial eigenvalue of $ K $

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Figure 9. A vertically rescaled plot of the value of the second non-trivial eigenvalue of $ K $. For the values of $ m_1 \in [0.595, 0.715] $, this second eigenvalue lies within the interval $ [-1, 1] $

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Figure 10. Both eigenvalue plots superimposed. The two graphs cross at roughly $ m_1 = 0.71 $, where the difference between the two is numerically $ 0.000429 $

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