Hill's Equation with Random Forcing Parameters: Determination of Growth Rates through Random Matrices

TL;DR

This paper analyzes the growth rates of solutions to the random Hill's equation by deriving expressions for the growth of 2x2 matrices with randomly varying parameters, covering various stability regimes.

Contribution

It introduces new formulas for growth rates of solutions to the random Hill's equation considering fluctuating parameters and different stability scenarios.

Findings

Derived explicit expressions for growth rates in various regimes.

Analyzed cases with highly unstable, near-stability, and stable without fluctuations.

Provided insights into the stability behavior under random forcing parameters.

Abstract

This paper derives expressions for the growth rates for the random 2 x 2 matrices that result from solutions to the random Hill's equation. The parameters that appear in Hill's equation include the forcing strength and oscillation frequency. The development of the solutions to this periodic differential equation can be described by a discrete map, where the matrix elements are given by the principal solutions for each cycle. Variations in the forcing strength and oscillation frequency lead to matrix elements that vary from cycle to cycle. This paper presents an analysis of the growth rates including cases where all of the cycles are highly unstable, where some cycles are near the stability border, and where the map would be stable in the absence of fluctuations. For all of these regimes, we provide expressions for the growth rates of the matrices that describe the solutions.