Horseshoes for the nearly symmetric heavy top |
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Authors: | G H M van der Heijden Kazuyuki Yagasaki |
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Institution: | 1. Centre for Nonlinear Dynamics and its Applications, Department of Civil, Environmental and Geomatic Engineering, University College London, Gower Street, London, WC1E 6BT, UK 2. Mathematics Division, Department of Information Engineering, Niigata Univeristy, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan 3. Geometric and Algebraic Analysis Group, Department of Mathematics, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, 739-8521, Japan
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Abstract: | 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. |
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