Arno Berger

One-Dimensional Dynamical Systems and Benford's Law

One-dimensional projections of (at least) almost all orbits of many multi- dimensional dynamical systems are shown to follow Benfords law, i.e. their (base b) mantissa distribution is asymptotically logarithmic, typically for all bases b. As a generalization and uniflcation of known results it is proved that under a (generic) non-resonance condition on A 2C d£d , for every z 2C d real and imaginary part of each non-trivial component of (A n z)n2N0 and (e At z)t‚0 follow Benfords law. Also, Benford behavior is found to be ubiquitous for several classes of non-linear maps and difierential equations. In particular, emergence of the logarithmic mantissa distribution turns out to be generic for complex analytic maps T with T(0) = 0, jT0(0)j < 1. The results signiflcantly extend known facts obtained by other, e.g. number-theoretical methods, and also generalize recent flndings for one-dimensional systems.

A basic theory of Benford’s Law

Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benfords Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.

An Improved Maximum Allowable Transfer Interval for

An elementary self-contained proof is given for an improved bound on the maximum allowable transfer interval that guarantees Lp-stability in networked control systems with disturbances.

L^{p}

1991 Mathematics Subject Classification. Primary Primary 37B10, 37A35, 43A60; Secondary 37B20, 54H20.

-Stability of Networked Control Systems

Mantissa distributions generated by dynamical processes continue to attract much interest. In this article, it is demonstrated that one-dimensional projections of (at least) almost all orbits of many multi-dimensional nonautonomous dynamical systems exhibit a mantissa distribution that is a convex combination of a trivial point mass and Benfords Law, i.e. the mantissa distribution of the non-trivial part of the orbit is asymptotically logarithmic, typically for all bases. Both linear and power-like systems are considered, and Benford behaviour is found to be ubiquitous for either class. The results unify previously known facts and extend them to the nonautonomous setting, with many of the conclusions being best possible in general.

On almost automorphic dynamics in symbolic lattices

A generalized shadowing lemma is used to study the generation of Benford sequences under non-autonomous iteration of power-like maps Tj : x ↦ αjxβj (1 - fj(x)), with αj, βj > 0 and fj ∈ C1, fj(0) = 0, near the fixed point at x = 0. Under mild regularity conditions almost all orbits close to the fixed point asymptotically exhibit Benfords logarithmic mantissa distribution with respect to all bases, provided that the family (Tj) is contracting on average, i.e. . The technique presented here also applies if the maps are chosen at random, in which case the contraction condition reads 𝔼 log β > 0. These results complement, unify and widely extend previous work. Also, they supplement recent empirical observations in experiments with and simulations of deterministic as well as stochastic dynamical systems.

On the distribution of mantissae in nonautonomous difference equations

Numerical data generated by dynamical processes often obey Benfords law of logarithmic mantissa distributions. For non-autonomous difference equations this article presents necessary as well as sufficient conditions for to conform with Benfords law in its strongest form: The proportion of values in with base b mantissa less than t tends to as , for all integer bases . The assumptions on , viz. asymptotic convexity and eventual expansivity on average, are very mild and met, e.g. by practically all polynomial, rational and exponential maps and any combinations thereof. The results complement, extend and unify previous work.

BENFORD'S LAW IN POWER-LIKE DYNAMICAL SYSTEMS

A solution of a nonautonomous ordinary differential equation is finite-time hyperbolic, i.e. hyperbolic on a compact interval of time, if the linearisation along that solution exhibits a strong exponential dichotomy. In analogy to classical asymptotic facts, it is shown that finite-time hyperbolicity is robust, that is, it persists under small perturbations. Eigenvalues and -vectors may be misleading with regards to hyperbolicity. This is demonstrated by means of simple examples.

Some dynamical properties of Benford sequences

Abstract The model of an inextensible uniform string subject to constant gravitation is used to study the propagation of transversal waves in one-dimensional continua. Perturbation analysis of the equations of motion yields as a result the local representation of small waves in terms of a normalized Riemann function. By means of the latter, shape and speed of propagating waves may be discussed. A refined analysis confirms that on first order, small waves travel along characteristics of the unperturbed equilibrium configuration. An explicit power law for the waves’ amplitudes is given, and the findings are supported by the numerical results.

More on finite-time hyperbolicity

Benford’s Law (BL), a notorious gem of mathematics folklore, asserts that leading digits of numerical data are usually not equidistributed, as might be expected, but rather follow one particular logarithmic distribution. Since first recorded by Newcomb in 1881, this apparently counter-intuitive phenomenon has attracted much interest from scientists and mathematicians alike. This article presents a comprehensive overview of the theory of BL for autonomous linear difference equations. Necessary and sufficient conditions are given for solutions of such equations to conform to BL in its strongest form. The results extend and unify previous results in the literature. Their scope and limitations are illustrated by numerous instructive examples.