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Linear stability analysis of the figure-eight orbit in the three-body problem

Published online by Cambridge University Press:  01 December 2007

GARETH E. ROBERTS
GARETH E. ROBERTS*
Affiliation:
Department of Mathematics and Computer Science, 1 College Street, College of the Holy Cross, Worcester, MA 01610, USA (email: groberts@radius.holycross.edu)
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Abstract

We show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2×2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge–Kutta–Fehlberg algorithm. From this, we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1947 - 1963
Copyright
Copyright © Cambridge University Press 2007

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