Stability Index

Author: Tianjiang Says

Website: https://cislunarspace.cn

Definition

The stability index is a quantitative measure of the stability of a periodic orbit, derived from the eigenvalues of the monodromy matrix. Two common stability indices are used:

ν\nu Stability Index

For eigenvalues on the unit circle (λ=eiθ\lambda = e^{i\theta}):

ν=∣cos⁡θ∣=12∣λ+1/λ∣\nu = |\cos \theta| = \frac{1}{2}|\lambda + 1/\lambda|

  • ν<1\nu < 1: orbit is linearly stable
  • ν=1\nu = 1: marginal stability (bifurcation point)
  • ν>1\nu > 1: orbit is unstable

LL Stability Index

L=12(∣λmax∣+1∣λmax∣)L = \frac{1}{2}\left(|\lambda_{max}| + \frac{1}{|\lambda_{max}|}\right)

where λmax\lambda_{max} is the eigenvalue with the largest magnitude.

  • L=1L = 1: orbit is stable
  • L>1L > 1: orbit is unstable; larger LL means more unstable

Applications

Stability indices are used throughout periodic orbit analysis:

ApplicationHow Stability Index is Used
Orbit family characterizationPlotting ν\nu or LL vs. orbit parameter reveals stable/unstable regions
Bifurcation detectionν=1\nu = 1 or L=1L = 1 indicates bifurcation points
Station-keeping planningHigher instability requires more frequent or larger corrections
Orbit selectionPrefer orbits with lower stability indices for long-duration missions

References

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