Topological Classification Of Lienard Systems

This paper discuss mainly topological classification of Lienard systems which in-clude two cases: (1) the systems without closed orbit; (2) the systems with at least oneclosed orbit. The paper ^[1-3] gave a complete classification of Lienard systems withoutclosed orbit and obtained 64 possible topological structures. With this understandingwe provide some concrete equations which ensure the existence of every topologicalstructure and thereby yield the feasibility and realization of the classification. Then, wediscuss the topological classification of Lienard systems with at least one closed orbitand give the concrete principle and method of classification. The whole contents is divided into five chapters. Chapter 1, as the beginning of this paper, offers some relative knowledge, such aspreliminary qualitative theory of differential equation, the existence theorem of limitcycles of Lienard equation, and basis of functional analysis about open set constitutionon a line. Chapter 2, as the second part of this thesis, gives the main results obtained by ^[1-3]and corresponding proof, introduces the method of classification and the 64 possibletopological structures. Chapter 3, as one of the main parts of this thesis, focuses on how to realize theclassification in Chapter 2. As we all know, the topological classification of Lienardsystems without closed orbit is important, however, the proof of feasibility and real-ization of the classification is more important. Only the feasibility and realization canilluminate the rationality and signification of the classification. with this understandingutilizing Filippov transformation and method of patching phase plane we provide someconcrete examples about Lienard equations satisfying some conditions and prove theseexamples possess possible topological structures. According 64 different examples,that all of 64 possible topological structures can be realized is prove. Chapter 4, as the other main part of this thesis, resumes to discuss the topologicalclassification of the Lienard systems with at least one closed orbit. As we know thatLienard systems hold an important place about the study of limit cycles in the theory ofplane dynamical systems. Therefore, the study of topological classification of Lienardsystems with at least one closed orbit is more profound and more complicated. Thenwe consider this problem, give the concrete principle and method of classification bythe corresponding relation between the point set of Y+ and the point set of Y-, and atlast prove that the systems have ∞ possible topological structures which can be dividedinto 40 groups. At the end of the paper, it is proposed the remark of this paper and the furtherstudy direction, many related references are listed.