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