The Evolution of a Self-Sustained Oscillation in a Nonlinear Continuous System

Abstract

We consider a gas-filled tube into which there is an input of energy due to a pressure sensitive heat source. The system is linearly unstable to perturbations about the initial equilibrium state. Within nonlinear theory a disturbance grows until a shock forms. The shock can then act as a dissipative mechanism so that ultimately a time periodic oscillation may result. The small amplitude disturbance in the pipe is represented as the superposition of two simple waves traveling in opposite directions, and without interaction. Thereby, the problem is reduced to solving a nonlinear difference equation subject to given initial conditions. Then not only is the final periodic state described but also its evolution from the prescribed initial perturbation. The concept of critical points of a nonlinear difference equation is introduced which allows the direct computation of the periodic state. The effects of dissipation and of a retarded heater response are also treated.