Approximation of probability density functions propagated through the perturbed two-body problem

This paper presents the details associated with the development of approximate probability density function (PDF) expressions for its evolution along a batch of trajectories. The exact value of the PDF is determined using the transformation of variables formula, in conjunction with the Abel's identity for the evolution of the determinant of the state transition matrix along the trajectory of the dynamical system of interest. Global Local Orthogonal MAPping (GLO-MAP) process is employed to construct the approximate PDFs from the exact values that are scattered in the phase fluid. The methodology is applied to representative dynamical systems, including the perturbed two body problem to demonstrate the effectiveness of the tools developed in this paper.