Horseshoes for the nearly symmetric heavy top
- First Online:
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
Unable to display preview. Download preview PDF.
- 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