An Implicit Function Method for Computing the Stability Boundaries of Hill's Equation

TL;DR

This paper introduces a novel, more accurate numerical method for computing the stability boundaries of Hill's equation, applicable to both damped and undamped systems, improving upon traditional matrix trace criteria.

Contribution

The paper presents an implicit function method that enhances accuracy and efficiency in determining stability regions of Hill's equation, including damped cases, extending existing techniques.

Findings

The new method improves computational efficiency over traditional approaches.

It provides a more accurate determination of stability boundaries.

The approach generalizes to damped Hill's equations, broadening its applicability.

Abstract

Hill's equation is a common model of a time-periodic system that can undergo parametric resonance for certain choices of system parameters. For most kinds of parametric forcing, stable regions in its two-dimensional parameter space need to be identified numerically, typically by applying a matrix trace criterion. By integrating ODEs derived from the stability criterion, we present an alternative, more accurate and computationally efficient numerical method for determining the stability boundaries of Hill's equation in parameter space. This method works similarly to determine stability boundaries for the closely related problem of vibrational stabilization of the linearized Katpiza pendulum. Additionally, we derive a stability criterion for the damped Hill's equation in terms of a matrix trace criterion on an equivalent undamped system. In doing so we generalize the method of this paper to compute stability boundaries for parametric resonance in the presence of damping.