Chaotic dynamics of a bouncing ball |
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| Affiliation: | 1. Department of Physics, Florida International University, Miami, FL 33199, USA;2. Department of Natural Sciences, Miami Dade College, 627 SW 27th Ave., Miami, FL 33135, USA;3. Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, Apartado Postal 47586, Caracas 1041-A, Venezuela;4. Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Chile;5. Departamento de Física, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080-A, Venezuela;6. Departamento de Matemática Aplicada a las TT.II., E.T.S.I. Telecomunicación, Universidad Politécnica de Madrid, 28040-Madrid, Spain;7. Departamento de Matemática Aplicada, Facultad de Informática, Universidad Complutense de Madrid, 28040-Madrid, Spain;1. Department of Civil and Environmental Engineering, Rice University, Houston, TX 77005, United States;2. Department of Mechanical Engineering, Rice University, Houston, TX 77005, United States;1. Bashkir State University, 32, Validy Str., Ufa, 450076, Russia;2. Institute of Molecule and Crystal Physics Ufa Research Centre of Russian Academy of Sciences, Prospekt Oktyabrya 151, Ufa, 450075, Russia |
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| Abstract: | A detailed study of a mapping on a two-dimensional manifold is made. The mapping describes a system subject to periodic forcing, in particular an imperfectly elastic ball bouncing on a vibrating platform. Quasiperiodic motion on a one-dimensional manifold is proven, and observed numerically, at low forcing, while at higher forcing Smale horseshoes are present. We examine the evolution of the attracting set with changing parameter. Spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises. A new type of chaos, in which a trajectory alternates between two distinct chaotic regions, is described and explained in terms of manifold collisions. Throughout we are concerned to examine the behaviour of Lyapunov exponents. Typical behaviour of Lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed. At higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given. |
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