We present a generalized methodology for constructing local orbital elements (LOEs) in the neigh-

borhood of periodic orbits in the circular restricted 3-body problem (CR3BP). Earlier work developed

LOEs around the Earth-Moon Lagrange points by exploiting Hamiltonian normal forms of equilibirum

points of flows. Although this flow-based normal form framework can, in principle, be extended to peri-

odic orbits by constructing a Hamiltonian description in a neighborhood of the periodic trajectory, this

process requires generating an orbit-dependent Hamiltonian and performing the resulting normalizing

transformations–an approach that is computationally intensive and technically delicate. In contrast, the

discrete-time approach developed here uses normal forms of fixed points of Poincar´e maps, providing a

flexible and efficient alternative that eliminates the need for constructing new Hamiltonians while retain-

ing the fidelity of the local dynamics. The remaining LOE corresponding to motion along the periodic

orbit is recovered through numerical Taylor integration, enabling a complete local parametrization.


Our construction begins by selecting a Poincar´e section transverse to the periodic orbit and computing

the associated return map as a high-order polynomial. This is accomplished using jet transport and

Taylor integration techniques, which propagate high-order derivatives of the dynamical flow and thereby

yield a validated Taylor expansion of the return map about its fixed point. A systematic sequence of

near-identity coordinate transformations then brings this polynomial map into a nonlinear normal form,

clarifying the geometry of the transverse dynamics. The resulting simplified representation provides

a natural set of LOEs encoding phase, amplitude, and hyperbolic invariant manifolds of not only the

underlying periodic orbit, but also a collection of quasi-periodic orbits in its vicinity.


We present initial results for representative periodic orbits in the Earth-Moon CR3BP and discuss

how these LOEs enable the joint parametrization of families of nearby quasi-periodic orbits together

with their associated stable and unstable manifolds. More broadly, we outline potential applications to

the characterization of local invariant structures for space situational awareness, for example, as well as

applications to cislunar mission design and control for satellite constellations.