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Book DetailsTranslations of Mathematical MonographsVolume: 66; 1986; 435 ppMSC: Primary 58; Secondary 34;
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. -
Table of Contents
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Chapters
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Introduction
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Fundamental Concepts, Concrete Examples, and Several Criteria to Establish Existence and Nonexistence of Limit Cycles
-
Multiplicity and Stability of Limit Cycles
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Limit Cycles in Rotated Vector Fields
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General Behavior of Limit Cycles Depending on a Parameter
-
Existence of a Limit Cycle
-
Uniqueness of a Limit Cycle
-
Existence of Several Limit Cycles
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Structural Stability of Differential Systems
-
Work of M. Frommer and N. N. Bautin
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Global Structural Analysis of Some Quadratic Systems without Limit Cycles
-
General Properties and Relative Positions of Limit Cycles in Quadratic Differential Systems
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Classification of Quadratic Differential Systems. Limit Cycles of Equations of Class I
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Global Structure of Trajectories of Equations of Class II without Limit Cycles
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Relative Positions of Limit Cycles and Conditions for Having at Most One and Two Limit Cycles in Equations of Class II
-
Some Local and Global Properties of Equations of Class III
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The Method of Dulac Functions in the Qualitative Study of Quadratic Systems
-
Limit Cycles in Bounded Quadratic Systems
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Appendix
-
-
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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.
-
Chapters
-
Introduction
-
Fundamental Concepts, Concrete Examples, and Several Criteria to Establish Existence and Nonexistence of Limit Cycles
-
Multiplicity and Stability of Limit Cycles
-
Limit Cycles in Rotated Vector Fields
-
General Behavior of Limit Cycles Depending on a Parameter
-
Existence of a Limit Cycle
-
Uniqueness of a Limit Cycle
-
Existence of Several Limit Cycles
-
Structural Stability of Differential Systems
-
Work of M. Frommer and N. N. Bautin
-
Global Structural Analysis of Some Quadratic Systems without Limit Cycles
-
General Properties and Relative Positions of Limit Cycles in Quadratic Differential Systems
-
Classification of Quadratic Differential Systems. Limit Cycles of Equations of Class I
-
Global Structure of Trajectories of Equations of Class II without Limit Cycles
-
Relative Positions of Limit Cycles and Conditions for Having at Most One and Two Limit Cycles in Equations of Class II
-
Some Local and Global Properties of Equations of Class III
-
The Method of Dulac Functions in the Qualitative Study of Quadratic Systems
-
Limit Cycles in Bounded Quadratic Systems
-
Appendix
