Numerical continuation of families of homoclinic connections of periodic orbits in the RTBP

The goal of this paper is the numerical computation and continuation of homoclinic connections of the Lyapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L1, L2 and L3, of the planar circular restricted three-body problem (RTBP). We describe the method used that allows us to follow individual families of homoclinic connections by numerical continuation of a system of (nonlinear) equations that has as unknowns the initial condition of the p.o., the linear approximation of its stable and unstable manifolds and a point in a given Poincare section in which the unstable and stable manifolds match. For the L3 case, some comments are made on the geometry of the manifold tubes and the possibility of obtaining trajectories with prescribed itineraries.

[1]  Daniel Wilczak,et al.  Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three Body Problem. Part II , 2004 .

[2]  Alan R. Champneys,et al.  Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem , 1997 .

[3]  G. Gómez,et al.  The invariant manifold structure of the spatial Hill's problem , 2005 .

[4]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[5]  Shane D. Ross,et al.  Connecting orbits and invariant manifolds in the spatial restricted three-body problem , 2004 .

[6]  Kenneth R. Meyer,et al.  Introduction to Hamiltonian Dynamical Systems and the N-Body Problem , 1991 .

[7]  J. Llibre,et al.  The motion of Saturn coorbital satellites in the restricted three-body problem , 2001 .

[8]  G. Gómez,et al.  The dynamics around the collinear equilibrium points of the RTBP , 2001 .

[9]  C. C. Conley,et al.  Low Energy Transit Orbits in the Restricted Three-Body Problems , 1968 .

[10]  J. Masdemont,et al.  The scattering map in the planar restricted three body problem , 2006 .

[11]  Bernd Krauskopf,et al.  A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits , 2008 .

[12]  Shane D. Ross,et al.  Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. , 2000, Chaos.

[13]  P. Chodas,et al.  Discovery of an asteroid and quasi‐satellite in an Earth‐like horseshoe orbit , 2002 .

[14]  Josep J. Masdemont,et al.  Geometry of homoclinic Connections in a Planar Circular Restricted Three-Body Problem , 2007, Int. J. Bifurc. Chaos.

[15]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[16]  J. Mondelo,et al.  Families of Cycler Trajectories in the Earth-Moon System , 2008 .

[17]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Homoclinic Orbits for Maps , 1997 .

[18]  Josep González,et al.  Contribution to the study of Fourier methods for quasi-periodic functions and the vicinity of the collinear libration points , 2001 .

[19]  Jaume Llibre,et al.  Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem , 1985 .

[20]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[21]  M. Torner,et al.  Invariant manifolds of L_3 and horseshoe motion in the restricted three-body problem , 2005 .

[22]  Victor Szebehely,et al.  Theory of Orbits. , 1967 .

[23]  Yuri A. Kuznetsov,et al.  Continuation of Connecting orbits in 3D-ODES (I): Point-to-Cycle Connections , 2007, Int. J. Bifurc. Chaos.

[24]  S. Mikkola,et al.  Families of periodic horseshoe orbits in the restricted three-body problem , 2005 .

[25]  Robert W. Farquhar,et al.  Libration Point Missions, 1978 - 2002 , 2003 .

[26]  Bernd Krauskopf,et al.  Computing Invariant Manifolds via the Continuation of Orbit Segments , 2007 .

[27]  V. Szebehely.  Theory of Orbits: The Restricted Problem of Three Bodies , 1968 .

[28]  Josep J. Masdemont,et al.  Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems , 2005 .

[29]  Josep J. Masdemont,et al.  Dynamics in the center manifold of the collinear points of the restricted three body problem , 1999 .

[30]  M. Torner,et al.  The motion of Saturn coorbital satellites in the restricted three-body problem , 2001 .

[31]  Bernd Krauskopf,et al.  Numerical Continuation Methods for Dynamical Systems , 2007 .

[32]  Some Zero Cost Transfers between Libration Point Orbits , 2000 .

[33]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[34]  Kazuyuki Yagasaki.  NUMERICAL DETECTION AND CONTINUATION OF HOMOCLINIC POINTS AND THEIR BIFURCATIONS FOR MAPS AND PERIODICALLY FORCED SYSTEMS , 1998 .

[35]  D. Wilczak,et al.  Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three-Body Problem – A Computer Assisted Proof , 2002, math/0201278.

[36]  M. Ollé,et al.  Invariant manifolds of L3 and horseshoe motion in the restricted three-body problem , 2006 .

[37]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[38]  Randy C. Paffenroth,et al.  Elemental Periodic orbits Associated with the libration Points in the Circular Restricted 3-Body Problem , 2007, Int. J. Bifurc. Chaos.

[39]  Yuri A. Kuznetsov,et al.  Continuation of Connecting orbits in 3D-ODES (II) : Cycle-to-Cycle Connections , 2008, Int. J. Bifurc. Chaos.

[40]  K. Howell,et al.  Trajectory Design Using a Dynamical Systems Approach With Application to Genesis , 1997 .

[41]  ’ K.C.IIowcII,et al.  TRAJECTORY DESIGN USING A DYNAMICAL SYSTEMS APPROACH WITH APPLICATION TO GENESIS , 1999 .

[42]  L. Dieci,et al.  Point-to-Periodic and Periodic-to-Periodic Connections , 2004 .

[43]  Gerard Gómez,et al.  Dynamics and Mission Design Near Libration Points: Volume IV: Advanced Methods for Triangular Points , 2001 .