Variation of Lyapunov exponents on a strange attractor

SummaryWe introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL → ∞ and argue from our numerical work on several chaotic systems that this approach is asL−v. In our examplesv ≈ 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.

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