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Unformatted text preview: Lecture 43: Chaos III (14 Dec 09) A. Extend: mechanics models 1. Refs: Goldstein Chapter 11, P. Cvitanovic, Universality in Chaos (reprint collection, 1984), FW Supplement. 2. Much of the language of modern turbulence theory arose from a few idealized models that have varying basis in physical theory but all share the feature that complex (rich) behaviors result from simple rules. 3. Integrable model: well-behaved constants of the motion additional to the total energy. Examples with phase space that has integrable solutions in some regions and non-integrable in others. 4. FPU (Fermi-Pasta-Ulam) system of coupled linear oscillators with a little cubic or quartic anharmonicity. 5. H enon-Heiles Hamiltonian (4 DOFs) based on a gravitational problem [G Sec. 11.6] H = 1 2 ( p 2 1 + q 2 1 + p 2 2 + q 2 2 ) + q 2 1 q 2- 1 3 q 3 2 with stability of trajectories degrading as the energy E increases. Poincar e surface of section (G Sec. 11.5). The phase space has 4 dimensions ( q 1 p 1 q 2 p 2 ), with 3D energy surface (4 - 1 constraint). Project the tra- jectory on the ( q 2 ,p 2 ) plane by marking point with q 1 = 0; p 1 0. (a) For low enough energy, E = 1 / 12, every trajectory followed in the computation led to smooth curves in the Poincar e surface of section ( q 2 ,p 2 ). Trajectories that are initially close propagate with separation that increases linearly with time....
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- Fall '09