Bifurcation and Center Conditions for Several Kind of Differertial Autonomous Systems
|School||Zhejiang Normal University|
|Keywords||Singular point value Bifurcation of Limit Cycles Traveling wave solutions Solitary wave Periodic wave Kink wave|
This paper studies the determination of the types of planar polynomial differential systems center focus and bifurcation of limit cycles, and a class of nonlinear equations of traveling wave solutions, the text of four chapters: The first chapter is the central focus of the planar polynomial differential systems reviewed the historical background and current status of problem with bifurcation of limit cycles, and the work done in this article a brief introduction of the second chapter of a class of five system singular point and center conditions, to study the origin singular point value, with embryo transform infinity into the origin (elementary singular point), with computer algebra system Mathenatics in the polynomial system origin 6 singularity and infinity 12 singular point value , and thus has been the center of the conditions of the origin and the point at infinity. third chapter of a class of quasi cubic systems Singular Point, the central focus of determination and limit cycles, first through the appropriate transformation of the system of origin (infinite far point) into the origin, and through the transformation to be complex system, the origin of the system before 21 singularity with Mathenatics software on the computer, further export centered at the origin conditions and order fine focus ( fine singularity) conditions, and were given the origin and INFINITY spending four limit cycle instance. Chapter 4, with the deepening of social progress and scientific research, in engineering and natural sciences subdiscipline even the social sciences have emerged a large number of non-linear mathematical model, wait for the scientists of various disciplines to study linear problems, nonlinear problems in general difficult to get the exact solution. solitary wave, including various types of Finite Travelling Waves is a very important research topic, there are a large number of research results and solving methods and techniques, including reflection method, Darboux transformation method, Hirota bilinear method, tanh method The basic idea of ??these methods that equations into the form of relatively easy-to-solve through a variety of transformation techniques, in particular in certain circumstances special exact solution of the equation of soliton is obtained so far at home and abroad mathematical physics researchers used these methods has received a large number of research results However, these methods in addition to identify exact solution in the particular case, the equation parameters changes the existence of soliton solutions and special limited solution could not give a more complete answer at home and abroad in recent years, the latest research shows that qualitative theory of differential equations and dynamical systems bifurcation theory can make up for the lack of the exact solution of the demand in this regard, some of the known exact solution can even use the point of view of the power system to provide a deeper understanding of this chapter of a class of non- linear simultaneous Schrodinger equations, the use of dynamical systems, bifurcation theory, to prove that the system of equations exist Peakons the kink wave solutions, infinite smooth periodic wave solutions and under different conditions, given Peakons kink wave solutions, periodic wave solutions of infinite smoothness various sufficient conditions, and gives order to all of the above display exact traveling wave solutions.