Abstract
Stability analysis of periodic orbits rely 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 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, coefficients of which can be compared to standard bifurcation equations to gain insight into higher order stability. The motivating example are 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 easily creating parametrised Poincare maps which can be used to get a bifurcation normal form via Lyapunov-Schmidt Method.