Statistics of finite-time Lyapunov exponents in the Ulam map

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.

arXiv comment6 pages, 4 figures, to appear in Phys. Rev. E

References in corpus (4)

• Measuring the Lyapunov exponent using quantum mechanics
• Characteristic distributions of finite-time Lyapunov exponents
• Two-dimensional turbulence of dilute polymer solutions
• Analytical results for coupled map lattices with long-range interactions

Cited by in corpus (7)

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• Remarkable similarities in distributions of dynamical observables in chaotic systems
• Finite-time Lyaponov analysis of a trained reservoir computer