Chaos, Complexity and Transport: Theory and Applications by Xavier Leoncini, Cristel Chandre, George Zaslavsky

By Xavier Leoncini, Cristel Chandre, George Zaslavsky

This ebook goals to supply the readers with a large landscape of other elements with regards to Chaos, Complexity and shipping. It includes a suite of contributions starting from utilized arithmetic to experiments, offered through the CCT'07 convention (Marseilles, June 4A-8, 2007). The booklet encompasses various conventional fields of physics and arithmetic whereas attempting to preserve a typical language one of the fields, and pursuits a nonspecialized viewers.

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J. Peebles, The Large Scale Structure ofthe Universe, (Princeton University Press, Princeton, NJ, 1980). 2. C. Benedetti, S. Rambaldi, G. H. Chavanis, Eur. Phys. J. B 52, 61 (2006). 3. J. B a d , T. Dauxois, G. de Ninno, D. Fanelli, S. Ruffo, Phys. Rev E 69, 045501(R) (2004). 4. T. , Dynamics and Thermodynamics of Systems with Long Range Interactions, Lect. Not. Phys. 602, Springer (2002). 5. A. Rapisarda, A. Pluchino, Europhysics News 36,202 (2005); F. Bouchet, T. Dauxois and S. Ruffo, Europhysics News 37,9 (2006).

Singularities (umbilics) also cause problems. l4 For our application-flow down a curved substrate-the Monge representation of a surface15 is the most convenient. The Monge representation is a glorified name for a parametrisation of the substrate by 44 in three-dimensional Cartesian space. Following standard notation,15 we define The unnormalised, nonorthogonal tangents el and el = a1X = (1 O P ) T , e2 e2 are = ~ Z X= (0 1 4 ) T , and their normalised cross product gives the normal to the substrate, 1 T ( - p -4 1) &3 = W , The corresponding covectors are 1 e1 = 3 ((1+ q 2 ) -P4 P ) T 7 1 T e2 = 3 (-P4 (1+ P 2 ) 4 ) and 63 is its own covector.

__ -333 -1000 Fig. 2 . 22. where (19) are phases in the rotational frame of reference. The invariant structure of the SW can be obtained from (18) if we present H , in the split form H, = V, + ABg(t) (20) where V, is time independent, and perform an averaging over time. ejl + U) + cos(u + b ~ejz)] . (21) j1jz=l where ejl,zare unit vectors ej1,2 = (cos(27rj1,2/q), sin(27rj1,2/q), j1,z = 1, * * . 14 (22) r is a vector in (u, u ) space and R is a vector in (Y,2)space. 30 Expression (21) defines two coupled degrees of freedom and their interaction could, in general, lead to chaotic trajectories.

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