The Morse Equation Existence And Its Applications

Morse theory since 20th century 20's by H. Morse has proposed, had the considerable development. The classical Morse theory has produced the smooth manifold topological property and its on between the Morse function non- degenerated singular point reciprocity. The 20th century 50's, Smale and so on its promotion hyperbolic fixed points and the manifold topological property relations which flows for the smooth manifold on gradient, thus obtained the Morse-Smale equation. At the end of the 70's, human and so on Conley has promoted the Morse index, obtained the Conley index which flows, thus has established the Morse equality in the general foundation which flows. Afterwards human and so on Szymzzak obtained about the mapping Conley index, this conclusion has promoted the Morse index. And Zhang Yuguang used this Conley index to obtain about the mapping Morse equation, this equation might regard as flows or the homographism Morse equation promotion. In this article, we discuss the Morse equation first which maps under certain condition existence. Also applies this equality to prove the Brouwer fixed points principle. Finally, we produced this equation to be separated half dynamic system branch some applications.