Time-dependent invariant manifolds in the restricted four-body problem

The lagrangian coherent structure (LCS) is defined as ridges of finite-time Lyapunov exponent (FTLE) fields, and it is demonstrated that an understanding of time-dependent invariant manifold (TDIM) can be obtained by use of LCS. Taking Sun-Earth-Moon bicircular model (BCM) as an example and LCS as a tool, the property of the TDIM of restricted 4-body problem (R4BP) is demonstrated numerically that TDIM is invariant set of orbits and acts as separatrix. Dichotomy is then used to extract the LCS on the Poincare section, and the configuration of TDIM on specified section is illustrated by a series of LCS with regularly spaced energy. Finally, low energy transfer from the Earth to the Moon is constructed in BCM directly.