Qualitative theory of flows on surfaces (a review)
Within the last fifty years the geometric theory of dynamical systems was widely appreciated and nowadays this field is accessible for a broad scientific audience. In fact, the nonlinear dynamics has become a subject with many applications in general sciences. In this paper many concepts and results on dynamical systems defined in 2-dimensional manifolds obtained in the last decades are gathered and discussed. The summary of this work says that: ``The geometric study of flows on manifolds has a long history in mathematics, even though it was not considered as a special field until it appeared in the works of Poincaré, Lyapunov, and Bendixson. After that, the subject developed rapidly, and, at the present time, it is the focus of extensive research. It is impossible to touch upon all sides of the geometric theory of flows in one article. The purpose of this review is to present some aspects of the geometric theory of flows on 2-manifolds (of course, within the frame of our knowledge). Our main interest is the topological structure and the topological equivalence of surface flows, asymptotic properties of semitrajectories and the closing lemma, etc. (see the contents). However, our article was not written for specialists only; we have also attempted to provide an introduction to the subject. We would like to thank the many mathematicians who took part in valuable discussions of problems of the theory of dynamical systems, and especially D. Anosov, I. Bronshtein, V. Grines, G. Levitt, M. Malkin, Shigenori Matsumoto, I. Nikolaev, and A. Zorich.'' The article is organized as follows: Section 1: Introduction of basic concepts, some useful constructions, terminology and notations. Section 2: The structure of limit sets. Section 3: The topological structure of a flow. Section 4: Space of flows (structural stability, Morse-Smale flows and bifurcation). Section 5: Topological classification of flows (more on equivalence of flows). Section 6: On the equivalence between the topological and smooth properties of flows. Section 7: Miscellany. An extensive bibliography is attached to the article.
scientific article; zbMATH DE number 1213213
scientific article