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Hill's equations arise in a wide variety of physical problems, and arespecified by a natural frequency, a periodic forcing function, and a forcingstrength parameter. This classic problem is generalized here in two ways: [A]to Random Hill's equations which allow the forcing strength q_k, theoscillation frequency \lambda_k, and the period \tau_k of the forcing functionto vary from cycle to cycle, and [B] to Stochastic Hill's equations whichcontain (at least) one additional term that is a stochastic process \xi. Thispaper considers both random and stochastic Hill's equations with smallparameter 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'sequations and stochastic Hill's equations have the same growth rates when theparameter variations p_k and \ell_k obey certain constraints given in terms ofthe moments of \xi. For random Hill's equations, the growth rates for thesolutions are given by the growth rates of a matrix transformation, undermatrix 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 analyticapproximations for the growth rates of the instability; for the regime whereHill'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 growthrates for stochastic Hill's equations.

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