Nonlinear Dynamics

, Volume 93, Issue 2, pp 909–932 | Cite as

High-order solutions of motion near triangular libration points for arbitrary value of \({\varvec{\mu }}\)

  • Yuying Liang
  • Ming Xu
  • Shijie Xu
Original Paper


In this paper, a new methodology is proposed to derive the high-order approximations of motions near them in three cases, i.e., the mass ratio is greater than, smaller than and equal to Gascheau and Routh critical value. A preliminary analysis on the phase space structure of triangular libration points in the first case is accomplished from the point of view of the dynamical system theory, demonstrating that they have two-dimensional stable/unstable manifolds and zero-dimensional center manifolds, referred to as \(2+2+0\) type. Further investigations show the topological type of triangular libration points evolves from center–center type to \(2+2+0\) type as mass ratio increases above Gascheau and Routh critical value. The high-order approximations of motion near triangular libration points are constructed using a modified method of variation of parameters. The methodology developed in this paper can deal with triangular libration points of all three topological types, which remains unsolved by the traditional Lindstedt–Poincaré method. The simulation results demonstrate that the expansions up to 11th order are a good replacement of numerical computation with a tolerate error. The validity regions of initial position deviation are presented, and a modification of the algorithm is performed to guarantee the continuity near Gascheau and Routh critical value. Furthermore, the high-order solutions of two-dimensional stable/unstable manifolds are employed to search for homoclinic connections of triangular libration points of 2577 Litva.


Triangular libration points Invariant manifolds Phase space structure Homoclinic/heteroclinic connections Circular restricted three-body problem 



Yuying Liang acknowledges the finical supports from Academic Excellence Foundation of BUAA for PhD Students and Shanghai Space Science and Technology Innovation Foundation (SAST2017-033). Ming Xu acknowledges the supports from the National Natural Science Foundation of China (11772024 and 11432001). Shijie Xu acknowledges the supports from Aerospace Science Foundation by China Aerospace Science and Industry Corporation (170134) and the Fundamental Research Funds for the Central Universities (YWF-16-BJ-Y-10).


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of AstronauticsBeihang UniversityBeijingChina

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