The primary goal of this project is to implement some algorithms for space mission design based on chaotic dynamics in the three-body problem in Mathematica 10.4 and create an easy to use interface for demonstration purposes. This includes the so called tube dynamics using invariant manifolds from [1] and the method of weak stability boundary from [2].

CR3BP stands for the circular restricted three-body problem which is the mathematical model used.

• Algorithms from [1, Chapter 4] to find trajectories with prescribed itineraries in the restricted 3-body problem are implemented in the library CR3BP.wl. The only missing part is the algorithm to compute Lyapunov orbits of energy E+dE from the Lyapunov orbit of energy E. The following is the reason why it should be implemented in a future release:

  • The shooting algorithm with initial values coming from the linearized problem converges only for energies E slightly above the energy E(L) of the Lagrange point L. In order to get Lyapunov orbits at both L_1 and L_2 simultaneously for given E, we have to choose E>E(L_2)>E(L_1) slightly above E(L_2), so that both necks are open; however, depending on E(L_2)-E(L_1), it might happen that E is not slightly above E(L_1) anymore. Therefore, we have to find the Lyapunov orbit for an energy E(L_2)>E'>E(L_1) by shooting and then extend it to E = E'+dE by the algorithm in question.

• See Documents/CR3BP.pdf for a talk based [1] for explanation of the mathematics.

• The library CR3BP.wl is used by the Mathematica 10.4 notebook explorer.nb, which is a functioning prototype of the promised user interface. One can look for Lyapunov orbits, draw stable and unstable manifolds and search for their intersections at various Poincare sections. The explorer keeps a list of computed data whose visualization can be turned on and off. Zooming is possible. The following are some nice examples:

  • For \mu="Sun-Jupiter", the Lagrange point L_2 and dE=0.0015, one can find 1:1 homoclinic transverse interesection of stable and unstable manifolds at U_4 in the exterior realm.

• KNOWN ERRORS

  • The definitions of U_2 and U_3 are messed up.
  • When no Poincare section is found for given conditions, the program gives an error (it cannot handle empty lists).

• TODO

  • Write manual to explorer.nb
  • Implement the algorithm for Lyapunov orbits with higher energy.
  • Find some more examples, e.g. parameters E and \mu for and a complete homoclinic-heteroclinic chain!
  • Long term - Implement the methods of weak stability boundary from [2] and compare to [1]

• There are the following mathematical and algorithmic problems suitable for further optimization:

  • Computing invariant manifolds and Poincare sections to get a heteroclinic-homoclinic chain: integration in families until stop-conditions are satisfied, interpolating a set of points in plane with a closed smooth curve, computing the intersection of the interiors of two simple smooth curves in plane.
  • Computing Poincare map and representing symbolic dynamics: Having the homoclinic-heteroclinic chain, i.e., a collection of 4 planes in the phase space with a pair of intersecting simple closed curves on each of them, propagate families of initial conditions and compute intersections with the interiors of the curves. A suitable data structure has to be developed to represent this so called Poincare map and quickly answer questions about trajectories with prescribed itineraries.

[1] W.S. Koon, M.W. Lo, J.E. Marsden, S.D. Ross: Dynamical Systems, the Three-Body Problem and Space Mission Design (link http://www.cds.caltech.edu/~marsden/volume/missiondesign/KoLoMaRo_DMissionBk.pdf)

[2] E. Belbruno, Capture Dynamics and Chaotic Motions in Celestial Mechanics