  Numerical continuation techniques are powerful tools that have been
extensively used to identify particular solutions of nonlinear dynamical
systems and enable trajectory design in chaotic astrodynamics problems such as
the Circular Restricted Three-Body Problem. However, the applicability of
equilibrium points and periodic orbits may be questionable in real-world
applications where the uncertainties of the initial conditions of the
spacecraft and dynamical parameters of the problem (e.g., mass ratio parameter)
are taken into consideration. Due to uncertain parameters and initial
conditions, the spacecraft might not follow the reference periodic orbit owing
to growing uncertainties that cause the satellite to deviate from its nominal
path. Hence, it is crucial to keep track of the probability of finding the
spacecraft in a given region. Building on previous work, we extend numerical
continuation to moments of the distribution (i.e., stochastic continuation) by
directly continuing moments of the probability density function of the
spacecraft state. Only assuming normality of the initial conditions, and
leveraging moment-generating functions, Isserlis' theorem, and the algebra of
truncated polynomials, we propagate the distribution of the spacecraft state at
consecutive surface of section crossings while retaining a symbolic map of the
final moments of the distribution that depend on the initial mean and
covariance matrix only. The goal of the work is to offer a differential
algebra-based general framework to continue 3D periodic orbits in the presence
of uncertain dynamical systems. The proposed approach is compared against
traditional Monte Carlo simulations to validate the uncertainty propagation
approach and demonstrate the advantages of the proposed in terms of uncertainty
propagation computational burden and access to higher-dimensional problems.
