Horseshoes for the nearly symmetric heavy top



We prove the existence of horseshoes in the nearly symmetric heavy top. This problem was previously addressed but treated inappropriately due to a singularity of the equations of motion. We introduce an (artificial) inclined plane to remove this singularity and use a Melnikov-type approach to show that there exist transverse homoclinic orbits to periodic orbits on four-dimensional level sets. The price we pay for removing the singularity is that the Hamiltonian system becomes a three-degree-of-freedom system with an additional first integral, unlike the two-degree-of-freedom formulation in the classical treatment. We therefore have to analyze three-dimensional stable and unstable manifolds of periodic orbits in a six-dimensional phase space. A new Melnikov-type technique is developed for this situation. Numerical evidence for the existence of transverse homoclinic orbits on a four-dimensional level set is also given.

Mathematics Subject Classification (2010)

Primary 37J45 70H08 70K44 Secondary 34C37 37C29 70H09 


Horseshoe Heavy top Chaos Nonintegrability Melnikov method 


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© Springer Basel 2013

Authors and Affiliations

  • G. H. M. van der Heijden
    • 1
  • Kazuyuki Yagasaki
    • 2
    • 3
  1. 1.Centre for Nonlinear Dynamics and its Applications, Department of Civil, Environmental and Geomatic EngineeringUniversity College LondonLondonUK
  2. 2.Mathematics Division, Department of Information EngineeringNiigata UniveristyNiigataJapan
  3. 3.Geometric and Algebraic Analysis Group, Department of MathematicsHiroshima UniversityHigashi-HiroshimaJapan

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