2015/10/31 by Marta Ceccaroni, Alessandra Celletti, Giuseppe Pucacco · 25 citations
Engineering · Mathematics · Physics and Astronomy · #Astro and Planetary Science #Bifurcation #Center manifold #Center of mass (relativistic) #Classical mechanics #Displacement (psychology) #Halo #Hopf bifurcation #Mathematical analysis #Mathematics #Perturbation (astronomy) #Physics #Quantum mechanics #Resonance (particle physics) #Saddle-node bifurcation #Series (stratigraphy) #Spacecraft Dynamics and Control #Stellar, planetary, and galactic studies #math-ph #math.MP #nlin.CD
paper · pdf · doi:10.1016/j.physd.2015.12.004
published in Physica D Nonlinear Phenomena 317, 28-42 (Elsevier BV) · 35 pages, 3 figures, updated version accepted for publication on Physica D
arxiv created 2015/12/10 · openalex publication_date 2015/12/23 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We perform an analytical study of the bifurcation of the halo orbits around the collinear points L1, L2, L3 for the circular, spatial, restricted three--body problem. Following a standard procedure, we reduce to the center manifold constructing a normal form adapted to the synchronous resonance. Introducing a detuning, which measures the displacement from the resonance and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio. In most cases, the analytical results thus obtained are in very good agreement with the numerical expectations, providing the bifurcation threshold with good accuracy. Care must be taken when dealing with L3 for small values of the mass-ratio between the primaries; in that case, the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold.