Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 63, pp. 1-42.

Title: Periodic orbits of the spatial anisotropic Kepler problem with anisotropic perturbations
       
Authors: Mengyuan Li (Guilin Univ. of Electronic Tech., Guilin, China)
         Qihuai Liu  (Guilin Univ. of Electronic Tech., Guilin, China)

Abstract:
 In this article, we study the periodic orbits of the spatial anisotropic Kepler problem
 with anisotropic perturbations on each negative energy surface, where the perturbations
 are homogeneous functions of arbitrary integer degree p.
 By choosing the different ranges of a parameter β, we show that there exist
 at least 6 periodic solutions for p>1, while there exist at least 2 periodic
 solutions for p≤1 on each negative energy surface.
 The proofs of main results are based on symplectic Delaunay coordinates, residue theorem,
 and averaging theory. 

Submitted April 12, 2020. Published July 08, 2021.
Math Subject Classifications: 37N05, 37G15, 70F05.
Key Words: Periodic orbit; averaging theory; residue theorem; spatial anisotropic Kepler problem.