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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 μalmost all initial points • in practice: finite time Lyapunov exponents  depend on initial condition and the integration time Russian stamp (1957) on the occasion of Lyapunov’s 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 ε in the initial condition x . With y = x + ε we obtain y 1 = x 1 + ε 1 = g ( y ) = g ( x ) + Dg ( x ) · ε + h.o.t. , where Dg ( x ) is the total derivative of g at x . ⇒ ε 1 = Dg ( x ) · ε . 6 • For k ∈ N it follows inductively ε k +1 = Dg ( x k ) · ε k = k Y i =0 Dg ( x i ) · ε = Dg k ( x ) · ε . • So the evolution of a small displacement ε 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 λ 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 μ almost all x , where μ is an ergodic measure....
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This note was uploaded on 01/04/2012 for the course CDS 280 taught by Professor Marsden,j during the Fall '08 term at Caltech.
 Fall '08
 Marsden,J

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