Innovated by the nonlinear modes concept in the
vibrational dynamics, the vertical periodic orbits around the
triangular libration points are revisited for the Circular Re-
stricted Three-body Problem. The ζ -component motion is
treated as the dominant motion and the ξ and η-component
motions are treated as the slave motions. The slave motions
are in nature related to the dominant motion through the
approximate nonlinear polynomial expansions with respect
to the ζ -position and ζ -velocity during the one of the peri-
odic orbital motions. By employing the relations among the
three directions, the three-dimensional system can be trans-
ferred into one-dimensional problem. Then the approximate
three-dimensional vertical periodic solution can be analyti-
cally obtained by solving the dominant motion only on ζ -
direction. To demonstrate the effectiveness of the proposed
method, an accuracy study was carried out to validate the
polynomial expansion (PE) method. As one of the appli-
cations, the invariant nonlinear relations in polynomial ex-
pansion form are used as constraints to obtain numerical
solutions by differential correction. The nonlinear relations
among the directions provide an alternative point of view to
explore the overall dynamics of periodic orbits around libra-
tion points with general rules.

EB00000003877K

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• Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions