Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

Qinglong Zhou; Yongchao Zhang

doi:10.3934/dcds.2017074

Article Contents

2017, Volume 37, Issue 3: 1763-1787. Doi: 10.3934/dcds.2017074

This issue Previous Article Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations Next Article

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

Qinglong Zhou^1, , and
Yongchao Zhang^2,

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China
2.
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, Hebei, China

* Corresponding author
* Corresponding author

Received: April 2016

Revised: October 2016

Published: May 2017

The first author is supported by NSFC (No. 11501330, No. 11425105) and CPSF (Grant No. 2015M582071). The second author is supported by the Fundamental Research Funds for the Central Universities (Grant No. N142303010).

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Abstract

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [13], which is used to model small oscillations of the earth's inner core taking into account the moon's attraction. Earlier results for the linear stability of the elliptic equilibrium point in Robe's restricted problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the range of the mass parameter. We first establish some relations of the linear stability problem to the properties of some symplectic paths and some corresponding linear operators. Then using the Maslov-type $ω$ -index theory of symplectic paths and the theory of linear operators, we compute $ω$ -indices and obtain certain properties of the linear stability of the elliptic equilibrium point of Robe's restricted three-body problem.

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Mathematics Subject Classification: Primary:70F07, 70H14;Secondary:34C25.

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Figure 1. The Robe's restricted three-body problem considered: $m_1$ is a spherical shell filled with a fluid of density $\rho_1$; $m_2$ a mass point outside the shell and $m_3$ a small solid sphere of density $\rho_3$ inside the shell.

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Figure 2. Stability bifurcation diagram of elliptic equilibrium point of the Robe's restricted three-body problem in the $(\mu,e)$ rectangle $[0,1]\times [0,1)$.

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