Theory of Limit Cycles

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American Mathematical Soc., 1986 - 435页
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Over the past two decades the theory of limit cycles, especially for quadratic differential systems, has progressed dramatically in China as well as in other countries. This monograph, updating the 1964 first edition, includes these recent developments, as revised by eight of the author's colleagues in their own areas of expertise. The first part of the book deals with limit cycles of general plane stationary systems, including their existence, nonexistence, stability, and uniqueness. The second section discusses the global topological structure of limit cycles and phase-portraits of quadratic systems. Finally, the last section collects important results that could not be included under the subject matter of the previous two sections or that have appeared in the literature very recently. The book as a whole serves as a reference for college seniors, graduate students, and researchers in mathematics and physics.
 

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目录

Fundamental Concepts Concrete Examples and Several Criteria to Establish Existence and Nonexistence of Limit Cycles
5
Multiplicity and Stability of Limit Cycles
25
Limit Cycles in Rotated Vector Fields
41
General Behavior of Limit Cycles Depending on a Parameter
77
Existence of a Limit Cycle
91
Uniqueness of a Limit Cycle
117
Existence of Several Limit Cycles
155
Structural Stability of Differential Systems
175
General Properties and Relative Positions of Limit Cycles in Quadratic Differential Systems
245
Classification of Quadratic Differential Systems Limit Cycles of Equations of Class I
261
Global Structure of Trajectories of Equations of Class II without Limit Cycles
281
Relative Positions of Limit Cycles and Conditions for Having at Most One and Two Limit Cycles in Equations of Class II
305
Some Local and Global Properties of Equations of Class III
335
The Method of Dulac Functions in the Qualitative Study of Quadratic Systems
359
Limit Cycles in Bounded Quadratic Systems
371
Appendix
401

Work of M Frommer and N N Bautin
191
Global Structural Analysis of Some Quadratic Systems without Limit Cycles
219

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