Numerical results on some generalized random Fibonacci sequences

Random Fibonacci sequences are stochastic versions of the classical Fibonacci sequence fn+1=fn+fn−1fn+1=fn+fn−1 for n>0n>0, and f0=f1=1f0=f1=1, obtained by randomizing one or both signs on the right side of the defining equation and/or adding a “growth parameter.” These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent. Following the techniques presented by Embree and Trefethen in their numerical paper Embree and Trefethen (1999) [2], we study the behavior of the Lyapunov exponents as a function of the probability pp of choosing ++ in the sign randomization.