Computing and Visualizing Stable and Unstable Manifolds of Halo Orbits

In order to compute and visualize the stable and unstable manifolds of the halo orbits, I computed and stored an array of orbit points, and their corresponding stable and unstable eigendirections. I then took each point on the manifold and stepped off in the desired eigendirection a small bit. Integrating forwards off the periodic orbit in the unstable eigendirection produced an orbit lying on the approximation to the unstable manifold. Likewise trajectories were integrated backwards in the stable eigendirection to produce trajectories on the approximate stable manifold. A Mesh was generated representing the manifolds by connecting neighboring manifolds at set time intervals in the integration. The mesh was then piped into GeomView where the manifolds can be viewed and manipulated in real time. The following figures are screen snapshots of manifolds rendered by our software in GeomView.

A number of color schemes are available for the user to select and apply to generated manifolds. Included among them are

total velocity,

directional velocity,


time parameterization,

initial condition parameterization.

In addition to these coloring schemes are utilities for finding the six-dimensional coordinates on the manifold.