Random Hill’s Equations, Random Walks, and Products of Random Matrices

Abstract

Hill’s equations arise in a wide variety of physical problems, and are specified by a natural frequency, a periodic forcing function, and a forcing strength parameter. This classic problem can be generalized by allowing the forcing strength q _k , the frequency λ _k , and the period (Δ τ) _k of the forcing function to vary from cycle to cycle. The growth rates for the solutions are then given by the growth rates of a matrix transformation, under matrix multiplication, where the elements vary from cycle to cycle. Simplified models of such problems are given by products of 2 ×2 random matrices drawn from a given class.

This paper analyzes two simple classes of models of 2 ×2 random matrices where the growth rates (Lyapunov exponents) can be computed in an explicit form. Both models are special cases of random products involving random similarity transformations. The first of these corresponds to the random Hill’s equation in a regime where the solutions are highly unstable. This model is a product of random similarity transformations of a fixed singular matrix. The second class of models is a two parameter class that studies products of 2 ×2 random symmetric matrices of a special form which are conjugated by random orthogonal similarities. These matrices are nonsingular in general, but in a special case they give rank one matrices, which may be compared with the first model. In the latter case the two models have different growth rate behavior, which arises from the different nature of the allowed similarity transformations in the models.