Qualitative Analysis Of The Nos(?)-hoover Oscillator And Its Generalization

Since the three-dimensional Nos(?)-Hoover oscillator was proposed in 1986,it has been widely concerned due to its physical background and rich dynamic phenomena.This paper mainly considers the Nos(?)-Hoover oscillator and its generalization.In view of the computer simulation results of the solutions of these two equations by other scholars,the solutions of the Nos(?)-Hoover oscillator and its generalization are qualita-tively analyzed by using mathematical analysis techniques and dynamical system theo-ry,and the important information about the behavior of the orbits of the corresponding systems is obtained.This Ph.D.thesis consists of four chapters.In Chapter 1,the background and the development status of the Nos(?)-Hoover os-cillator and its generalization are introduced,then the summary of the research content of this paper is given.In Chapter 2,the Nos(?)-Hoover oscillator is considered.Using dynamical system theory,the global dynamic behavior of the solution of the Nos(?)-Hoover oscillator is analyzed,and it is proved that every invariant set of this oscillator is neither in the region{(x,y,z)?R~3|z>0}nor in the region{(x,y,z)?R~3|z<0},that is,it must intersect the xOy plane.In Chapter 3,a generalized Nos(?)-Hoover oscillator is considered.In a specific range of parameters of the generalized equation,the global dynamic behavior of the solution of this equation is studied qualitatively,it is proved that if a solution of this generalized equation will not tend to the one-dimensional invariant manifold{(x,y,z)?R~3|x=0,y=0},it must intersect the xOy plane infinite times.Especially,every invariant set of this generalized equation must have nonempty intersection with the xOy plane.In addition,it is also proved that if a solution of the generalized Nos(?)-Hoover oscillator is quasi-periodic,it must pass through at least five quadrants of the phase space R~3provided it has no intersection with the coordinate axis,otherwise,it must pass through at least four quadrants of R~3.In Chapter 4,the conclusion of this paper and prospect of the research is given.