If a planar non-circular periodic orbit is non-degenerate, then it persists under small 3D electromagnetic perturbations, and for fixed period at least five distinct spatial periodic solutions bifurcate.
Abbondandolo, Morse theory for Hamiltonian systems, vol
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Bifurcation from periodic solutions of central force problems in the three-dimensional space
If a planar non-circular periodic orbit is non-degenerate, then it persists under small 3D electromagnetic perturbations, and for fixed period at least five distinct spatial periodic solutions bifurcate.