On The Classifications Of DAEs Under GT-Equivalence

Many mathematical models are differential algebraic equations (DAE) in practical applications. It may occur bifurcation phenomenon in the neighborhood of a singular point, thus, the research on singular points and concerning classifications of DAE has certain practical value. By the singularity theory and normal theory, this paper mainly studies the classifications of DAE which contains two real state variables and a bifurcation parameter, under GT—Equivalence. GT—Equivalence not only can maintain the corresponding relationships between the orbits of GT—Equivalent systems, but also can reserve the directions. And the form GT—Equivalent system is simple and easy to grasp.Moreover, using differential topological theory and methods, this paper gets some properties with the solutions of several special two-dimensional DAEs. For instance, in the neighborhood of origin, each connected components exists solution, and the orbit passing every point of each connected component is the whole connected component. For each connected component has and only has two directions, then it can classify DAEs according to the directions of connected components. The results includes: when the first projection of the algebraic part as y1 2k1 +1 +δ1x2l1?1 and the first projection of the algebraic part as DAE is certainly GT—Equivalent to one of (εx ,g), (εx 2 ,g)(ε=±1),and gives several classification theories when the algebraic part has some other forms.There are four chapters in this paper. Chapter 1 briefly introduces the research background and status of DAE, the contents and significances of this paper. Chapter 2 introduces basic concepts. Chapter 3 introduces the classifications of DAE which algebraic codimension is equal or less than 3 under GT—Equivalence, and gives the results of the classifications. Chapter 4 is the main part of this paper, studies on the classifications of several kinds of two-dimensional DAE under GT—Equivalence. When the algebraic part of DAE has several special forms, this chapter gives the results of the classifications and proves the theories.