Stability Switching Phenomenon And Bifurcation Analysis Of Positively Damped And Delayed Position Feedback Oscillators

This paper takes into consideration a damped harmonic oscillator model with delayed feedback.After transforming the model into a system of first-order delayed differential equations with a single discrete delay,the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation.Particularly,in view of the normal form method and the center manifold reduction for retarded functional differential equations,the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given.In order to check the rationality of our theoretical results,numerical simulations for some specific examples are also carried out by means of the MATLAB software package.The first chapter summarizes the research background,significance and current situation of modified oscillators with positive damping and delayed position feedback,and the main contents of this paper are pointed out.Chapter 2 mainly studies the local asymptotic stability of the equilibrium point and the Hopf bifurcation problem.In the third chapter,we discusses the direction of Hopf bifurcation and the stability of bifurcation periodic solution of the system studied in this paper.In chapter 4,the theoretical results of absolute stability,single stability switching and multiple stability switching are verified by numerical simulation.