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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References
Arnold V.I.: Mathematical Methods of Classical Mechanics, 2nd ed. Springer, Berlin (1989)
Audin M.: Spinning Tops: A Course on Integrable Systems. Cambridge University Press, Cambridge (1996)
Champneys A.R., Lord G.J.: Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem. Phys. D 102, 232–269 (1997)
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/)
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)
Fenichel N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)
Goldstein H.: Classical Mechanics, 2nd ed. Addison-Wesley, Reading (1980)
Guckenheimer J., Holmes P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983)
Haller G.: Chaos Near Resonance. Springer, Berlin (1999)
Holmes P.J., Marsden J.E.: Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom. Commun. Math. Phys. 82, 523–544 (1982)
Holmes P.J., Marsden J.E.: Melnikov’s method and Arnol’d diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys. 23, 669–675 (1982)
Holmes P.J., Marsden J.E.: Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J. 32, 273–309 (1983)
Homburg A.J., Knobloch J.: Multiple homoclinic orbits in conservative and reversible systems. Trans. Am. Math. Soc. 358, 1715–1740 (2006)
Landau L.D., Lifshitz E.M.: Mechanics, Course of Theoretical Physics vol. 1. Pergamon Press, NY (1960)
Lin X.-B.: Using Melnikov’s method to solve Šilnikov’s problems. Proc. R. Soc. Edinb. Sect. A 116, 295–325 (1990)
Maciejewski A.J., Przybylska M.: Differential Galois approach to the non-integrability of the heavy top problem. Ann. Fac. Sci. Toulouse Math. 14, 123–160 (2005)
Meiss, J.D.: Differential Dynamical Systems. SIAM, Philadelphia (2007)
Melnikov V.K.: On the stability of a center for time-periodic perturbations. Trans. Moscow Math. Soc. 12, 1–57 (1963)
Mielke A., Holmes P.: Spatially complex equilibria of buckled rods. Arch. Ration. Mech. Anal. 101, 319–348 (1988)
Moser J.: Stable and Random Motions in Dynamical Systems. Ann. Math. Stud. No. 77. Princeton University Press, Princeton (1973)
Sakajo T., Yagasaki K.: Chaotic motion of the N-vortex problem on a sphere: I. Saddle-centers in two-degree-of-freedom Hamiltonians. J. Nonlinear Sci. 18, 485–525 (2008)
Smale S.: Differentiable dynamical systems, Bull. Am. Math. Soc. 73, 747–817 (1967)
Vanderbauwhede A., Fiedler B.: Homoclinic period blow-up in reversible and conservative systems. Z. Angew. Math. Phys. 43, 292–318 (1992)
van der Heijden G.H.M., Yagasaki K.: Nonintegrability of an extensible conducting rod in a uniform magnetic field. J. Phys. A 44, 495101 (2011)
Whittaker E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. Cambridge University Press, Cambridge (1937)
Wiggins S.: Global Bifurcations and Chaos: Analytical Methods. Springer, Berlin (1988)
Wiggins S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer, Berlin (1994)
Yagasaki K.: Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers. Arch. Rational Mech. Anal. 154, 275–296 (2000)
Yagasaki K.: Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom Hamiltonian systems with saddle-centres. Nonlinearity 18, 1331–1350 (2005)
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)
Ziglin S.L.: Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics, I. Funct. Anal. Appl. 16, 181–189 (1983)
Ziglin S.L.: Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics, II. Funct. Anal. Appl. 17, 6–17 (1983)
Ziglin S.L.: The absence of an additional real-analytic first integral in some problems of dynamics. Funct. Anal. Appl. 31, 3–9 (1997)
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KY was partially supported by the Japan Society for the Promotion of Sciences through Grant-in-Aid for Scientific Research (C) Nos. 21540124 and 22540180.
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van der Heijden, G.H.M., Yagasaki, K. Horseshoes for the nearly symmetric heavy top. Z. Angew. Math. Phys. 65, 221–240 (2014). https://doi.org/10.1007/s00033-013-0319-z
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DOI: https://doi.org/10.1007/s00033-013-0319-z