Bifurcations of Liouville tori of a two fixed center problem

  • F. M. El-Sabaa
  • M. Hosny
  • S. K. Zakria
Original Article


A complete description of the real phase topology of a two fixed center problem is introduced and studied. Moreover, all generic bifurcations of Liouville tori are determined theoretically and the periodic solution is presented. At the end of this paper, the phase portrait is studied.


Hamilton–Jacobi’s equations Bifurcations of Liouville tori Topology of the level sets Momentum maps Periodic solution Elliptic functions Phase portrait 



The authors would like to express their sincere thanks to the editor and to anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper.


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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationAin Shams UniversityCairoEgypt

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