High Order Continuation With Automatic Bifurcation Identification in the Circular Restricted Three Body Problem

Stability analysis of periodic orbits relies on linear analysis tools such as Floquet Multipliers or the Broucke stability diagram, typically aided by Monte Carlo simulations. However, these techniques fail to potentially capture the nonlinear behaviour of the system near bifurcations of linearly stable orbits. In this paper, a novel approach is used to expand upon continuation methods to automatically gain insight into the nonlinear stability of periodic orbits and identify bifurcation via Differential Algebra and Lyapunov–Schmidt Reduction. Differential algebra enables the representation of functions as Taylor polynomial series with mathematical operations defined on them, enabling a locally dense semi-analytical solution. The Lyapunov–Schmidt method can give a bifurcation equation, the coefficients of which can be compared to the standard bifurcation equation to gain insight into higher-order stability. The motivating example is the planar Quasi Satellite Orbits of the Mars Moon eXploer mission, which exhibited linear stability but were nonlinearly unstable near the 1:3 subharmonic bifurcation. Results show that the Differential Algebra is capable of creating parameterised Poincaré maps which can be used to get a bifurcation normal form via the Lyapunov-Schmidt Method.