By J. R. Dorfman

This publication is an advent to the functions in nonequilibrium statistical mechanics of chaotic dynamics, and likewise to using thoughts in statistical mechanics very important for an knowing of the chaotic behaviour of fluid structures. the basic ideas of dynamical platforms idea are reviewed and straightforward examples are given. complex subject matters together with SRB and Gibbs measures, risky periodic orbit expansions, and purposes to billiard-ball structures, are then defined. The textual content emphasises the connections among shipping coefficients, had to describe macroscopic homes of fluid flows, and amounts, similar to Lyapunov exponents and Kolmogorov-Sinai entropies, which describe the microscopic, chaotic behaviour of the fluid. Later chapters give some thought to the jobs of the increasing and contracting manifolds of hyperbolic dynamical structures and the big variety of debris in macroscopic platforms. routines, specified references and proposals for additional interpreting are incorporated.

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Indeed, by again viewing ε as a dummy dynamic variable, and using the fast time s = t/ε, the slow–fast system can be rewritten as x = f (x, y) , y = εg(x, y) , ε =0. 23) around such a point has the structure ⎞ ⎛ 0 A (y) ∂y f (x (y), y) ⎝ 0 0 g(x (y), y)⎠ . 24) 0 0 0 4 There might also exist equilibrium points (x (y), y, ε) with ε > 0, namely if g(x (y), y) = 0. At these points, slow and adiabatic manifold co¨ıncide for all ε. 1 Slow Manifolds 25 Mε M x y2 y1 Fig. 3. An adiabatic manifold Mε associated with a uniformly asymptotically stable slow manifold.

23) around such a point has the structure ⎞ ⎛ 0 A (y) ∂y f (x (y), y) ⎝ 0 0 g(x (y), y)⎠ . 24) 0 0 0 4 There might also exist equilibrium points (x (y), y, ε) with ε > 0, namely if g(x (y), y) = 0. At these points, slow and adiabatic manifold co¨ıncide for all ε. 1 Slow Manifolds 25 Mε M x y2 y1 Fig. 3. An adiabatic manifold Mε associated with a uniformly asymptotically stable slow manifold. Orbits starting in its vicinity converge exponentially fast to an orbit on the adiabatic manifold. Hence it admits m + 1 vanishing eigenvalues, while the eigenvalues of A (y) are bounded away from the imaginary axis by assumption.

6) 2 Let us assume that the friction √ coeﬃcient γ is large, and set ε = 1/γ . With respect to the slow time t = εs, the dynamics is governed by the slow–fast system εx˙ = y − x , y˙ = −∇U (x) . 7) The slow manifold is given by x (y) = y. 8) or, equivalently, x˙ = −∇U (x). This relation is sometimes called Aristotle’s law , since it reﬂects the fact that at large friction, velocity is proportional to force, as if inertia were absent. 4. 1) that the right-hand side does not explicitly depend on ε.