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Unformatted text preview: Lyapunov exponents, invariant manifolds and transport Kathrin Padberg University of Paderborn October 22, 2003 1 Motivation dominant Lyapunov exponent is a measure of the chaoticity of an at tractor global, asymptotic quantity  independent of the initial condition for almost all initial points in practice: finite time Lyapunov exponents  depend on initial condition and the integration time Russian stamp (1957) on the occasion of Lyapunovs 100th anniversary 2 Example: Ikeda attractor distribution of finite time Lyapunov exponents (N=10 iterates) some areas on the chaotic attractor are more predictable than others predictability analysis of this kind goes back to Lorenz (1965) 3 More examples distribution of finite time Lyapunov exponents (integration time T=5) for the Duffing system and a double well potential Duffing Double well potential detection of hyperbolic structures and their stable manifolds (e.g. Haller, 2001) 4 In this talk ... Basic ideas about Lyapunov exponents, local Lyapunov exponents and finite time Lyapunov exponents and their characteristics Why do finite time Lyapunov exponents pick up those particular struc tures? Many examples on how we can make use of this concept in a set oriented analysis of dynamical systems: detection extraction Expansion and graphs Future work and open questions 5 Evolution of a small perturbation Let g be a diffeomorphism on a compact manifold M of dimension l : x k +1 = g ( x k ) , x k M Consider a small (infinitesimal) perturbation in the initial condition x . With y = x + we obtain y 1 = x 1 + 1 = g ( y ) = g ( x ) + Dg ( x ) + h.o.t. , where Dg ( x ) is the total derivative of g at x . 1 = Dg ( x ) . 6 For k N it follows inductively k +1 = Dg ( x k ) k = k Y i =0 Dg ( x i ) = Dg k ( x ) . So the evolution of a small displacement is governed by the linearized dynamical system. 7 Lyapunov exponents Lyapunov exponents (LE) measure the convergence or divergence of infinitesimal perturbations in the initial condition i = lim k 1 k log  Dg k ( x ) v i  , v i T x M Their existence is guaranteed by the Multiplicative Ergodic Theorem . On an attractor LEs are independent of the initial condition x for  almost all x , where is an ergodic measure....
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 Fall '08
 Marsden,J

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