Bifurcation of Nonlinear System with Time Delay. Analysis of Fundamental Harmonic Response by Averaging Method.

Abstract

This paper studies bifurcation set of a nonlinear system with time delay. It is known that it is difficult to investigate this system by analytical methods, because this system is described by a difference-differential equation which is usually difficult to solve. To analyze this kind of a system, we have already introduced an averaging method for the functional differential equation. The previous work showed that this averaging method is effective for nonlinear system with time delay. Applying this averaging method, this paper studies bifurcation set of this kind of a system with a fundamental harmonic response. The result shows that this system have a simple harmonic motion, when a delay is small and an angular frequency of external force is large. Furthermore, it is shown that this system have a derivative point from which the Hopf bifurcation curve, a saddle-node bifurcation curve and homoclinic bifurcation curve are derived.