Zeitschrift für angewandte Mathematik und Physik

, Volume 65, Issue 2, pp 221–240

Horseshoes for the nearly symmetric heavy top


  • G. H. M. van der Heijden
    • Centre for Nonlinear Dynamics and its Applications, Department of Civil, Environmental and Geomatic EngineeringUniversity College London
    • Mathematics Division, Department of Information EngineeringNiigata Univeristy
    • Geometric and Algebraic Analysis Group, Department of MathematicsHiroshima University

DOI: 10.1007/s00033-013-0319-z

Cite this article as:
van der Heijden, G.H.M. & Yagasaki, K. Z. Angew. Math. Phys. (2014) 65: 221. doi:10.1007/s00033-013-0319-z


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