Hill's Equation with Small Fluctuations: Cycle to Cycle Variations and Stochastic Processes

TL;DR

This paper extends Hill's equations to include small random and stochastic variations, analyzing their stability and growth rates, and finds that such variations generally induce instability even in classically stable regimes.

Contribution

It introduces a generalized framework for Hill's equations with small random and stochastic parameter variations, deriving analytic growth rate approximations and stability conditions.

Findings

Random Hill's equations are typically unstable.

Growth rates are proportional to the square of small parameter variations.

Analytic formulas for instability growth rates are provided.

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 q_k, the oscillation frequency \lambda_k, and the period \tau_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 \xi. This paper considers both random and stochastic Hill's equations with small parameter variations, so that p_k=q_k-<q_k>, \ell_k=\lambda_k-<\lambda_k>, and \xi are all O(\epsilon), where \epsilon<<1. We show that random Hill's equations and stochastic Hill's equations have the same growth rates when the parameter variations p_k and \ell_k obey certain constraints given in terms of the moments of \xi. 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 \lambda-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 \gamma = O(\epsilon^2). Using the relationship between the (\ell_k,p_k) and the \xi, this result for \gamma can be used to find growth rates for stochastic Hill's equations.