Lyapunov Exponents from Random Fibonacci Sequences to the Lorenz Equations

Lyapunov exponents give a way to capture the central features of chaos and of stability in both deterministic and stochastic systems using just a few real numbers. However, exact analytic determination of Lyapunov exponents is rarely possible, and as we will show, even an accurate numerical computation is not a trivial task. One of the principal results of this thesis is about random Fibonacci sequences. Random Fibonacci sequences are defined by $t\sb1=t\sb2=1$ and $t\sb{n}={\pm}t\sb{n-1}\pm t\sb{n-2}$ for $n>2,$ where each $\pm$ sign is independent and either + or $-$ with probability 1/2. Using Stern-Brocot sequences, we prove that$${\root n\of{\vert t\sb{n}\vert}}\to1.13198824\...\ {\rm as}\ n\to\infty$$with probability 1. Other contributions of this thesis include formulas for condition numbers of random triangular matrices and an accurate computation of the Lyapunov exponents of the Lorenz equations.