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

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 ordinary differential equations 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.