Weak Disorder in Fibonacci Sequences

TL;DR

This paper investigates how small random perturbations in Fibonacci sequences influence their growth, deriving the Lyapunov exponent and distribution of ratios, with various recursion modifications analyzed.

Contribution

It introduces a stochastic model of Fibonacci sequences with weak disorder and derives analytical expressions for growth rates and distributions.

Findings

Lyapunov exponent calculated for weak disorder limit

Distribution of ratios of consecutive elements derived

Various recursion modifications analyzed

Abstract

We study how weak disorder affects the growth of the Fibonacci series. We introduce a family of stochastic sequences that grow by the normal Fibonacci recursion with probability 1-epsilon, but follow a different recursion rule with a small probability epsilon. We focus on the weak disorder limit and obtain the Lyapunov exponent, that characterizes the typical growth of the sequence elements, using perturbation theory. The limiting distribution for the ratio of consecutive sequence elements is obtained as well. A number of variations to the basic Fibonacci recursion including shift, doubling, and copying are considered.