By Walter Thirring

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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.