RESONANT PERTURBATIONS OF HIGHER-ORDER HARMONIC OSCILLATION

Abstract

Abstract A higher-order extension of the perturbed Simple Harmonic Oscillator is discussed. The differential equation is on [a, ∞), where D = d/dx, the constants aj are real and distinct, and r(x) is a real-valued perturbation of the form r(x) = ξ(x)p(λx). Here ξ(x) is o(1) as x → ∞, p(s) has period 2π in s and λ is a real parameter. A general condition is obtained under which a solution y has a large amplitude factor when λ takes certain specified values. This situation is known as resonance. The nature of this condition is analysed in some detail in the case of fourth-order equations and when p(s) is either sins or a periodic step-function. It is found that there are some new features and new problems which are not present in the second-order case.