Chaotic Dynamics of the Kepler Problem with Oscillating Singularity

Alessandro Margheri 1  and Pedro J. Torres 2
  • 1 Faculdade de Ciências da Universidade de Lisboa e CMAF-CIO, Campo Grande, Edifício C6, piso 2, P-1749-016 Lisboa, Portugal
  • 2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Alessandro Margheri and Pedro J. Torres


We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.

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Advanced Nonlinear Studies (ANS)  is aimed at publishing scholarly articles on nonlinear problems, particularly those involving Differential and Integral  Equations, Dynamical Systems, Calculus of Variations, and related areas. It will also publish novel and interesting applications of these areas to problems in biology,  engineering,  materials sciences,  physics and other  sciences.