Control of Chaos in Mechanical System With Time Delay

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This paper studies1. bifurcation set of a nonlinear system with time delay.2. equivalent nonlinearization of a nonlinear system with time delay. We construct the equivalent system.3. optimal feedback gains in the time-delayed feedback system.It is known that it is difficult to investigate these systems by analytical methods, because these systems are described by a difference-differential equation which is usually difficult to solve. So, it is difficult to determine an appropriate feedback gain theoretically. 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, and construction the equivalent system. And, a coefficient of this system is determined that behavior of the averaged system of the equivalent system coincide with that of the original system. We compare the equivalent system with the original system by the phase portrait and resonance curve. Where, we regard the system whose motion equation is described the difference-differential equation as an original system, and an assumed system without time delay as an equivalent system.The result shows1. This system has 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 has a Bogdanov-Takens bifurcation point from which the Hopf bifurcation curve, a saddle-node bifurcation curve and homoclinic bifurcation curve are derived.2. Behavior of the equivalent system is in good agreement with behavior of the original system.3. A relation between the appropriate feedback gain and the revolution around an unstable periodic orbit.