Hill's equation with small fluctuations: Cycle to cycle variations and stochastic processes

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 is generalized here in two ways: (a) to Random Hill's equations which allow the forcing strength qk, the oscillation frequency λk, and the period (Δτ)k of the forcing function to vary from cycle to cycle, and (b) to Stochastic Hill's equations which contain (at least) one additional term that is a stochastic process ξ. This paper considers both random and stochastic Hill's equations with small parameter variations, so that pk = qk−⟨qk⟩, ℓk = λk−⟨λk⟩, and ξ are all \documentclass[12pt]{minimal}\begin{document}${\cal O}(\epsilon )$\end{document}O(ε), where ε ≪ 1. We show that random Hill's equations and stochastic Hill's equations have the same growth rates when the parameter variations pk and ℓk obey certain constraints given in terms of the moments of ξ. For random Hill's equations, the growth rates for the solutions are given by the growth rates of a matrix transformation, under matrix multiplication, where the matrix elements vary from cycle to cycle. Unlike classic Hill's equations where the parameter space (the λ-q plane) displays bands of stable solutions interlaced with bands of unstable solutions, random Hill's equations are generically unstable. We find analytic approximations for the growth rates of the instability; for the regime where Hill's equation is classically stable, and the parameter variations are small, the growth rate γ = \documentclass[12pt]{minimal}\begin{document}${\cal O}(p_k^2)$\end{document}O(pk2) = \documentclass[12pt]{minimal}\begin{document}${\cal O}(\epsilon ^2)$\end{document}O(ε2). Using the relationship between the (ℓk, pk) and the ξ, this result for γ can be used to find growth rates for stochastic Hill's equations.