Analytical Approximation And Computation For Periodic Solutions Of A Damped Oscillator With Delayed Feedback

Damped harmonic oscillator is a classical mathematical model which has been widely usedin mechanical engineering, aerospace, earthquake science, etc. Moreover, the delayed feedbackcontrol has become one of the main methods used in vibration control. In this thesis, we concen-trate on analysis and computation of a damped harmonic oscillator with delayed feedback. Weuse Liapunov-Schmidt reduction method to approximate periodic solutions of damped harmonicoscillator with delayed feedback analytically and use collocation method to solve the model nu-merically. Furthermore, we show the validity of our methods by means of comparing the twokinds of approximate results.Firstly, we investigate the Hopf bifurcation of a damped oscillator with delayed feedback. Wediscuss the physical parameter conditions when Hopf bifurcation occurs.Secondly, near the Hopf bifurcation point, we obtain the approximate analytic periodic so-lutions which bifurcated from the trivial solution of the damped oscillator by Liapunov-Schmidtreduction method.Finally, we compare the approximate analytic periodic solution with the numerical results,which are computed by the collocation method based on piecewise Hermite polynomials. Weuse low order approximate analytic solution as the initial value of Newton iterative method. Thenumerical results are in line with the high order approximate analytic solution. Moreover, thealgorithm is easy to use.