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Statistics of finite-time Lyapunov exponents in the Ulam map

Celia Anteneodo
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Abstract

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length $n$. The variance anomalously decays as $1/n^2$. The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting $2^{n-1}$ spikes that narrow and accumulate close to the mean value with increasing $n$. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small $n$, where the finiteness of bin size trimmes the sharp peaks.

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Celia Anteneodo. 2003-10-31. Statistics of finite-time Lyapunov exponents in the Ulam map. https://doi.org/10.1103/physreve.69.016207

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