Monodromy Matrix

Author: Tianjiang Says

Website: https://cislunarspace.cn

Definition

The monodromy matrix is the state transition matrix evaluated over one complete orbital period of a periodic orbit:

M=Φ(t0+T,t0)M = \Phi(t_0 + T, t_0)

where TT is the orbital period. It describes how small perturbations evolve after one full revolution and is the fundamental tool for analyzing periodic orbit stability.

Eigenvalue Structure

For a Hamiltonian system like the CR3BP, the monodromy matrix has special eigenvalue structure:

  • Eigenvalues come in reciprocal pairs: (λ,1/λ)(\lambda, 1/\lambda)
  • For a 6-dimensional system, there are 6 eigenvalues
  • Two eigenvalues are always +1 (corresponding to the direction along the orbit and the energy integral)
  • The remaining four eigenvalues determine orbital stability
Eigenvalue ConfigurationStability
All on unit circle ($\lambda
Real pair off unit circleUnstable (hyperbolic)
Complex pair off unit circleUnstable (complex hyperbolic)

Applications

The monodromy matrix is used for:

  • Stability classification: Determining whether a periodic orbit is stable, unstable, or conditionally stable
  • Floquet analysis: Decomposing perturbations into stable and unstable modal components
  • Orbit continuation: Identifying bifurcation points where orbit families change character
  • Station-keeping design: Floquet mode-based control strategies target unstable components

References

  • Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.