Time-regularized bifurcation framework for constructing tulip orbits in the CRTBP

Tulip orbits, a recently identified family of complex periodic solutions in the Earth-Moon circular restricted three-body problem (CRTBP), exhibit near-neutral stability, diverse geometrical structures, and multi-directional coverage of the secondary body, offering promising prospects for lunar observation and space domain awareness. Although prior studies suggest that these orbits can emerge through period-multiplying bifurcations from the L2 near-rectilinear halo orbit (NRHO) subset, the singular dynamics associated with extremely low perilune distances hinder conventional bifurcation analysis to a high order. This study introduces a time-regularized bifurcation framework to systematically construct tulip orbits, particularly those bifurcated from extremely low-perilune NRHOs, and evaluate their potential in deep-space missions. A key innovation is the integration of a time regularization scheme based on the Sundman transformation into the bifurcation analysis. Specifically, the equations of motion for the CRTBP under the Sundman transformation, along with their first-order variational forms, are derived to facilitate accurate computation of state transition and monodromy matrices. This enables the numerical continuation of the L2 NRHO subset to extremely low perilune distances, followed by the identification of a series of high-order period-multiplying bifurcations. Utilizing an improved orbit family switching algorithm, robust construction and continuation of tulip orbits with up to twenty petals are achieved. Furthermore, this framework is applied to various representative three-body systems, exhibiting consistent dynamical connections between tulip orbits and NRHO bifurcations. Finally, preliminary mission-level assessments under a high-fidelity ephemeris model suggest that, compared with the widely used 9:2 NRHO and 2:1 distant retrograde orbit, selected tulip orbits can achieve complete lunar communication coverage with shorter communication gaps, while requiring comparable station-keeping costs.