Saddle Point

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A fixed point on the Poincaré map with mixed stability: one eigenvalue of its monodromy matrix has modulus greater than one and the other less than one, corresponding to unstable and stable directions respectively. Saddle points are key hubs for transport in phase space: their unstable manifolds guide orbits away and stable manifolds guide orbits toward them, with intersections of the two manifold families forming lobe transport structures. The paper identifies five sets of saddle points (orders 4, 4, 5, 7, 8) in the Sun-Jupiter PCR3BP, where the order-5 unstable manifold intersects Mars's orbit and the order-4 stable manifold intersects Earth's orbit, with manifold switching enabling the long-time Mars-to-Earth transport.

Application Value

In low-thrust trajectory optimization, minimum-time trajectories serve as a prerequisite, first determining the minimum flight time, then using it as the terminal time constraint for the fuel-optimal problem. This hierarchical optimization strategy significantly reduces the computational complexity of multi-objective trajectory design.

Related Concepts

• Jacobi Integral
• Coriolis Theorem
• Velocity Function

References

• Ren 等 - 2011 - On the mechanisms of natural transport in the solar system