American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - S, 2008, 1 (4) : 557-587. doi: 10.3934/dcdss.2008.1.557
[1]

Davide L. Ferrario, Alessandro Portaluri. Dynamics of the the dihedral four-body problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 925-974. doi: 10.3934/dcdss.2013.6.925
[2]

Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541
[3]

Hichem Chtioui, Hichem Hajaiej, Marwa Soula. The scalar curvature problem on four-dimensional manifolds. Communications on Pure & Applied Analysis, 2020, 19 (2) : 723-746. doi: 10.3934/cpaa.2020034
[4]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009
[5]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
[6]

Snir Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. Journal of Modern Dynamics, 2018, 13: 43-113. doi: 10.3934/jmd.2018013
[7]

Frederic Gabern, Àngel Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 143-182. doi: 10.3934/dcdsb.2001.1.143
[8]

Nicola Soave, Susanna Terracini. Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3825-3829. doi: 10.3934/dcds.2013.33.3825
[9]

Hsin-Yuan Huang. Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1763-1774. doi: 10.3934/dcds.2012.32.1763
[10]

Duokui Yan, Tiancheng Ouyang, Zhifu Xie. Classification of periodic orbits in the planar equal-mass four-body problem. Conference Publications, 2015, 2015 (special) : 1115-1124. doi: 10.3934/proc.2015.1115
[11]

Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909
[12]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[13]

Giancarlo Benettin, Massimiliano Guzzo, Anatoly Neishtadt. A new problem of adiabatic invariance related to the rigid body dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 959-975. doi: 10.3934/dcds.2008.21.959
[14]

Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621
[15]

George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43
[16]

Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301
[17]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[18]

Antonio Giorgilli, Stefano Marmi. Convergence radius in the Poincaré-Siegel problem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 601-621. doi: 10.3934/dcdss.2010.3.601
[19]

Jose S. Cánovas, Tönu Puu, Manuel Ruiz Marín. Detecting chaos in a duopoly model via symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 269-278. doi: 10.3934/dcdsb.2010.13.269
[20]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences