Numerical approach for bifurcation and orbital stability analysis of periodic motions of a 2-DOF autonomous Hamiltonian system

Abstract

In Spaceflight Dynamics it is often necessary to obtain periodic motions of conservative mechanical systems and analyze their stability and bifurcation. These conservative systems can be described using Hamiltonian equations. We consider bifurcation and orbital stability problem for periodic motions of a 2-DOF autonomous Hamiltonian system. Since it is not possible to obtain analytical solutions to the aforementioned problem for all admissible values of its parameters a two-step numerical approach is proposed. On the first step the so-called base solutions are obtained analytically for particular values of problem’s parameters. The base solutions are then continued to the borders of their existence domains using a numerical algorithm. In course of computation bifurcation points are identified and orbital stability is studied. On the second step new base solutions are identified in the neighborhood of bifurcation points and the continuation process is repeated. Finally, orbital stability and bifurcation diagrams of the resulting families of periodic motions are constructed. Poincare sections are also computed in the neighborhoods of bifurcation points to verify the results. To illustrate this approach, we computed the bifurcation and orbital stability diagrams for families of short-periodic motions originating from Regular precessions of a dynamically-symmetric satellite.