Stability of Elliptic Periodic Points with an Application to Lagrangian Equilibrium Solutions

Yongxia Hua, Zhihong Xia*

*Corresponding author for this work

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We study the stability of elliptic rest points and periodic points of Hamiltonian systems of two degrees of freedom. We try to understand to what extent the linear stability would imply nonlinear stability. In case of one and one half degrees of freedom, linear stability, most of the times, do imply nonlinear stability provided that certain conditions on the eigenvalues are satisfied. This is a result of Herman. However, this is no longer the case for two degrees of freedom. We will analyse this case and it turns out that we can still say a lot about the nonlinear stability even in this case. As an example, we consider the Lagrange solutions (L4 and L5) of circular restricted three body problem. For certain mass ratios of the primaries, the Lagrangian solution is elliptic and the high order term in the normal form is non-degenerate, and therefore the Lagrangian point is stable from the standard KAM theory. However, there is one particular value, famously named μ0, where the fourth order term in the normal form happens to be degenerate. We will apply our result to this case.

Original languageEnglish (US)
Pages (from-to)243-253
Number of pages11
JournalQualitative Theory of Dynamical Systems
Volume12
Issue number1
DOIs
StatePublished - Apr 15 2013

Fingerprint

Equilibrium Solution
Nonlinear Stability
Periodic Points
Degree of freedom
Linear Stability
Normal Form
KAM Theory
Restricted Three-body Problem
Imply
Term
Lagrange
Hamiltonian Systems
Fourth Order
Higher Order
Eigenvalue
Hamiltonians

Keywords

  • KAM theory
  • Restricted Three Body Problem
  • Stability

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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Stability of Elliptic Periodic Points with an Application to Lagrangian Equilibrium Solutions. / Hua, Yongxia; Xia, Zhihong.

In: Qualitative Theory of Dynamical Systems, Vol. 12, No. 1, 15.04.2013, p. 243-253.

Research output: Contribution to journalArticle

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AB - We study the stability of elliptic rest points and periodic points of Hamiltonian systems of two degrees of freedom. We try to understand to what extent the linear stability would imply nonlinear stability. In case of one and one half degrees of freedom, linear stability, most of the times, do imply nonlinear stability provided that certain conditions on the eigenvalues are satisfied. This is a result of Herman. However, this is no longer the case for two degrees of freedom. We will analyse this case and it turns out that we can still say a lot about the nonlinear stability even in this case. As an example, we consider the Lagrange solutions (L4 and L5) of circular restricted three body problem. For certain mass ratios of the primaries, the Lagrangian solution is elliptic and the high order term in the normal form is non-degenerate, and therefore the Lagrangian point is stable from the standard KAM theory. However, there is one particular value, famously named μ0, where the fourth order term in the normal form happens to be degenerate. We will apply our result to this case.

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