Horseshoes for the nearly symmetric heavy top
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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
KeywordsHorseshoe Heavy top Chaos Nonintegrability Melnikov method
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- 4.Doedel, E., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), Concordia University, 1997 (an upgraded version is available at http://cmvl.cs.concordia.ca/auto/)
- 5.Dovbysh, S.A.: Splitting of separatrices of unstable uniform rotations and nonintegrability of a perturbed Lagrange problem, Vestnik Moskov. Univ. Ser. I Mat. Mekh., no. 3, 70–77 (in Russian) (1990)Google Scholar
- 14.Landau L.D., Lifshitz E.M.: Mechanics, Course of Theoretical Physics vol. 1. Pergamon Press, NY (1960)Google Scholar
- 17.Meiss, J.D.: Differential Dynamical Systems. SIAM, Philadelphia (2007)Google Scholar
- 18.Melnikov V.K.: On the stability of a center for time-periodic perturbations. Trans. Moscow Math. Soc. 12, 1–57 (1963)Google Scholar
- 20.Moser J.: Stable and Random Motions in Dynamical Systems. Ann. Math. Stud. No. 77. Princeton University Press, Princeton (1973)Google Scholar
- 25.Whittaker E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. Cambridge University Press, Cambridge (1937)Google Scholar
- 30.Ziglin S.L.: Splitting of separatrices, branching of solutions and nonexistence of an integral in the dynamics of a solid body. Trans. Moscow Math. Soc. 41, 283–298 (1982)Google Scholar