Hill's Equation with Random Forcing Terms

TL;DR

This paper analyzes solutions to Hill's equation with randomly varying forcing terms and period changes, deriving growth rates, stability conditions, and bounds for the solutions in various regimes, including the delta function limit.

Contribution

It provides exact solutions, bounds, and estimates for Hill's equation with random and cycle-varying parameters, extending previous deterministic analyses.

Findings

Derived growth rates and stability conditions for random Hill's equation.

Established bounds and estimates for products of random 2x2 matrices.

Provided analytic solutions in the delta function limit for cycle-to-cycle matching.

Abstract

Motivated by a class of orbit problems in astrophysics, this paper considers solutions to Hill's equation with forcing strength parameters that vary from cycle to cycle. The results are generalized to include period variations from cycle to cycle. The development of the solutions to the differential equation is governed by a discrete map. For the general case of Hill's equation in the unstable limit, we consider separately the case of purely positive matrix elements and those with mixed signs; we then find exact expressions, bounds, and working estimates for the growth rates. We also find exact expressions, estimates, and bounds for the infinite products of several $2 \times 2$ matrices with random variables in the matrix elements. In the limit of sharply spiked forcing terms (the delta function limit), we find analytic solutions for each cycle and for the discrete map that matches solutions from cycle to cycle; for this case we find the growth rates and the condition for instability in the limit of large forcing strength, as well as the widths of the stable/unstable zones.