The distribution of Lyapunov exponents: Exact results for random matrices

Charles M. Newman

doi:10.1007/BF01464284

The distribution of Lyapunov exponents: Exact results for random matrices

Charles M. Newman

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N^-1/2dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.

Original language	English (US)
Pages (from-to)	121-126
Number of pages	6
Journal	Communications In Mathematical Physics
Volume	103
Issue number	1
DOIs	https://doi.org/10.1007/BF01464284
State	Published - Mar 1986

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/BF01464284

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abstract = "Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N-1/2dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.",

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AB - Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N-1/2dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.

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The distribution of Lyapunov exponents: Exact results for random matrices

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