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December  2008, 1(4): 557-587. doi: 10.3934/dcdss.2008.1.557

Oscillatory motions in the rectangular four body problem

Ernesto A. Lacomba 1, and  Mario Medina 2,

1. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana–Iztapalapa, Av. San Rafael Atlixco 186, C.P. 09340 México, D.F., Mexico
2. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, México DF, CP 09340, Mexico

Received  March 2008 Revised  August 2008 Published  September 2008

In this paper we describe a symbolic dynamics for the rectangular four body problem by applying blow ups at total collisions and at infinity, studying the homoclinic or heteroclinic orbits obtained as intersection of corresponding two dimensional invariant submanifolds in a 3 dimensional energy level plus a convenient Poincaré map. With this tool we show the existence of a very rich dynamics and obtain the Main Theorem of this article. It gives the transition matrix for the symbolic dynamics of the images of conveniently chosen rectangles in the Poincaré section of the flow.
Keywords: symbolic dynamics, Poincaré section., Four body problem, invariant manifolds.
Mathematics Subject Classification: Primary: 70F07; Secondary: 70F1.
Citation: Ernesto A. Lacomba, Mario Medina. Oscillatory motions in the rectangular four body problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 557-587. doi: 10.3934/dcdss.2008.1.557
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