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

By Xavier Leoncini, Cristel Chandre, George Zaslavsky

This e-book goals to supply the readers with a large landscape of alternative features on the topic of Chaos, Complexity and shipping. It includes a set of contributions starting from utilized arithmetic to experiments, provided throughout the CCT'07 convention (Marseilles, June 4A-8, 2007). The booklet encompasses varied conventional fields of physics and arithmetic whereas attempting to hold a standard language one of the fields, and pursuits a nonspecialized viewers.

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2. Substrate Coordinates For most applications in the literature of thin films and shallow layers, orthonormal coordinates have been the coordinates of choice. This is because the main substrate shapes that have been treated are planes, cylinders, and spheres, where orthonormal coordinates are readily available. For a general substrate shape, orthonormal coordinates are difficult to construct and require numerical integration. Singularities (umbilics) also cause problems. l4 For our application-flow down a curved substrate-the Monge representation of a surface15 is the most convenient.

In both cases the web has co-dimension two in the phase space. Keywords: Chaos; Stochastic web; Quasicrystal symmetry, Coupled oscillators. 1. Introduction Weak perturbation of integrable system that leads to interaction of the system’s degrees of freedom can generate in the phase space a thin net of channels inside of which trajectories are chaotic. The net is called stochastic web (SW), and its emergence is an important physical phenomenon that imposes the transport properties of systems. Arnold web is a paradigm example of the SW, and the corresponding transport is known as Arnold Arnold web is a universal characteristic of nonlinear systems.

Small-parameter Expansion Now we assume that the fluid layer is shallow, so that y is proportional to E . After replacing y by E Y , Eq. (14b) becomes + 2P + ~ - ~ ( d ~+c p ) ~2 9 . ( X + ~ P (GOip O ( E ) ) a,cpapcp - We also expand cp in powers of cp(z 1 ~ 6 =32H. ) (17) E, ,32 2 Y) = V(0) + E P(1)+ 2 P(2) 3- . . 7 (18) The leading-order term in (17) occurs a t order E - ~ , and gives d , ~ p ( ~=) 0. Hence, we have cp(o) = @ ( x 1 , x 2 independent ) of y, The next nontrivial terms are a t order E O , GapaOi(a dp(a 2P + (d,(p(1))2+ P - 2 g .