Study of woodwind-like systems through nonlinear differential equations. Part I. Simple geometry

Abstract

Idealized woodwind models assume a well-localized nonlinearity coupled to a linear bore whose mathematical description is usually expressed through a convolution product. For the case of the simplest bore geometries, cylindrical and conical, the convolution can be transformed into a delayed differential equation. When including in it the nonlinearity, the usual concepts of nonlinear dynamical systems allow a better understanding of the system’s evolution. In this paper they are applied to the cylindrical and conical geometries and some characteristics of their behaviors are analyzed at the threshold of oscillation through an analytical study. It is shown that their dynamics cannot be reduced to a finite number of degrees of freedom (effective dynamics). Finally numerical simulations reveal peculiar characteristics of the direct and inverse bifurcations involved in such simple systems for quadratic and cubic nonlinearities respectively.