Asymmetric Mathieu equations

Abstract

An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible). However, solutions are scaleable: if x(t) is a solution, then so is αx(t) for any real α>0. We numerically study the stability regions in the parameter plane of this system for a fixed degree of asymmetry in the elastic restraints. A Lyapunov-like exponent is defined and numerically evaluated to find these regions of stable and unstable behaviour. These numerics indicate that there are infinitely many possibilities of instabilities in this system that are missing in the usual or symmetric Mathieu equation. We find numerically that there are periodic solutions at the boundaries of stable regions in the parameter plane, analogous to the symmetric Mathieu equation. We compute and plot several of these solution branches, which provide a relatively simpler means of computing the stability transition curves of this system. We prove theoretically that such periodic solutions must exist on all stability boundaries. Our theoretical results apply to the asymmetric Hill's equation, of which the pendulum system is a special case. We demonstrate this with numerical studies of a more general asymmetric Mathieu equation.