The Stratifications Of Energy-Casimir Mapping For Three-dimensional Integrable Systems

In 2009,Razvan M.Tudoran et al.first proposed a method to describe the dynamic behavior of three-dimensional integrable systems by using the image set layering of Energy-Casimir mapping.firstly,the dynamic behavior of Rikitake system is studied by the method of dynamical system,and then the Energy-Casimir mapping is constructed from the Hamilton function and Casimir function of the system,and the image set layering of Energy-Casimir mapping of Rikitake system is obtained.Then the corresponding relationship between the stratification of Energy-Casimir image set and the dynamic behavior of the system is given in detail.In other words,it is a clear and intuitive description of the dynamical behavior of the system in the language of planar geometric images.Since 2009,there is a wealth of research work that uses the Energy-Casimir mapping method to characterize the dynamics of the system.The basic idea of the Energy-Casimir mapping method is to start from the system,analyze the structure of the system solution using the dynamic method,then draw the corresponding Energy-Casimir mapping stratifications of the system,and finally point out the correspondence between the fibers of the Energy-Casimir image set layering and the system solution.The main work of this paper is to start from the opposite direction,to obtain a Hamilton-Poisson representation of the system for an integrable threedimensional system,then to construct the Energy-Casimir mapping,to draw the hierarchy of the system,and to determine the boundaries of the EnergyCasimir mapping hierarchy and the correspondence between the main layer and the dynamic behavior of the system.In this paper,three parts are demonstrated: the first part is the singularity of the system's Energy-Casimir mapping hierarchy in which the external boundary protons correspond to the system's stability.The second part demonstrates that the internal boundary protons correspond to the system's unstable singularity,stable singularity,etc.The third part demonstrates that the main level protons correspond to the system's dynamic behavior.The third part proves the periodic solving of the main level of the atomic correspondence system,etc.The importance of this paper is to provide a simple way to directly use the results of this paper to obtain a rough overview of the dynamical behavior of the system when analyzing a regenerable system.The paper is structured as follows:Chapter 1 is the introduction,introducing the main work on the Hamilton system,the results of the Energy-Casimir mapping hierarchy method and the background of the research.The second chapter is the preparatory knowledge,which briefly introduces the relevant knowledge of Poisson geometry,the basic concepts of the dynamical system,and the important primers.The third chapter is the main work to obtain general results and prove the correspondence between the Energy-Casimir mapping hierarchy and the dynamical behavior of the system.