Period-Doubling Bifurcation
Author: Tianjiang Says
Website: https://cislunarspace.cn
Definition
Period-doubling bifurcation is a bifurcation phenomenon in nonlinear dynamical systems where the period of a periodic orbit suddenly doubles as a system parameter changes. In cislunar orbital mechanics, period-doubling bifurcation is an important characteristic in the evolution of DRO orbit families.
Dynamic Mechanism
In the DRO orbit family, as the orbital energy (Jacobi constant) changes:
- Stable region: Low-energy DRO orbits are typically stable, with monodromy matrix eigenvalues on the unit circle
- Bifurcation point: When the energy reaches a specific value, eigenvalues pass through , and the orbital period doubles
- Unstable region: Orbits after bifurcation become unstable, and period-doubled orbits appear
Period-doubling bifurcation and stability:
| Before Bifurcation | Bifurcation Point | After Bifurcation |
|---|---|---|
| Stable periodic orbit | Eigenvalues pass | Unstable periodic orbit |
| Period | Critical state | Period |
Manifestation in DRO Orbit Families
During the continuation of DRO orbit families, period-doubling bifurcation manifests as:
- Orbit configuration changes from simple symmetric to complex asymmetric
- Stability indices undergo sudden changes
- Orbit family branches appear
Related Concepts
References
- Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.