padberg-CDS-Talk-2003

padberg-CDS-Talk-2003 - Lyapunov exponents, invariant...

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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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padberg-CDS-Talk-2003 - Lyapunov exponents, invariant...

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