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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Figure 5. A vertically rescaled plot of the graph from Figure 4
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The coordinates for Broucke's Orbit
Broucke's orbit as it evolves in time. One-quarter period is shown. The remainder of the orbit is obtained by a symmetric extension
A plot of the value of
A plot of the numerically-computed upper-left entry
A vertically rescaled plot of the graph from Figure 4
A plot of the value of
A vertically rescaled plot of the value of
A plot of the second non-trivial eigenvalue of
A vertically rescaled plot of the value of the second non-trivial eigenvalue of
Both eigenvalue plots superimposed. The two graphs cross at roughly