Narrow Results

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

The dynamical behavior of a simple pendulum hanging from a rotating arm has been investigated. The system is invariant under rotations around the axis and can be formulated as a two-degrees of freedom integrable Hamiltonian system in the absence of external forcing. The bifurcation diagram is organized around the relative equilibria (solutions that are invariant under the symmetry) and bridges connecting different bifurcation points. Special attention has been given to those solutions that could shed some light into the stabilization of the upside down solution and the control problem.

We present an overview of detailed computational results for families of periodic orbits that emanate from the five libration points in the Circular Restricted 3-Body Problem, as well as for various secondary bifurcating families. Our extensive overview covers all values of the mass-ratio parameter, and includes many known families that have been studied in the past. The numerical continuation and bifurcation algorithms employed in our study are based on boundary value techniques, as implemented in the numerical continuation and bifurcation software AUTO.

We illustrate how numerical boundary value techniques can be used to obtain a rather complete classification of certain types of periodic solutions of the Circular Restricted 3-Body Problem (CR3BP), for all values of the mass-ratio parameter.