Computable examples of the maximal Lyapunov exponent

Abstract

Some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.

Persistence of structured populations in random environments

“…Two notable exceptions are Roerdink's formula (1987) for γ for a two age class model which is discussed in Section 5.3 and random matrices that share a reproductive value or share a stable structure (Tuljapurkar 1986(Tuljapurkar ,1990. Other computable examples for random products of 2× 2 matrices can be found in (Key 1987, Mannion 1993). Tuljapurkar states that the random matrices A(e, 0) share a stable structure if there is a positive vector v and Borel function λ :…”

An upper bound for the largest Lyapunov exponent of a Markovian product of nonnegative matrices

“…(1) is well known [14,24,28,30]; further, it is easy to see that the limit does not depend on the choice of matrix norm. The exact determination of is well known to be a difficult problem [1,2,[6][7][8]10,13,21,25,27,29]. The purpose of this paper is to derive an upper bound for .…”

A Review of Parrondo's Paradox

“…[12,[16][17][18][19][20]. However, Parrondo's games have gained particular attention because: (i) they are the first game-theoretic realization of such processes, (ii) in their original form, they can be directly mapped onto the workings of a flashing Brownian ratchet and (iii) the effect is strikingly counterintuitive and relatively simple to analyze.…”