## Theory of Limit CyclesOver 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

5 | |

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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### 常见术语和短语

algebraic curve annular region arc length assume Chen Chinese coefficients completely proved corresponding cycle of system Darboux integral differential equations Dulac function dy/dt easy elementary singular point equations dx/dt equations of class existence of limit family of rotated focus formula global phase-portrait global structure Green's formula Hence homeomorphic i-axis infinite singular point integral curve integral line interior internally stable Lemma Lienard line segment Lyapunov stable method Moreover Nanjing negative neighborhood nodal point number of limit obtain origin parameter point of intersection problem quadratic differential system quadratic system respect right side rotated vector fields saddle point satisfies semistable cycle separatrix cycle shown in Figure singular closed trajectory stable cycle stable limit cycle stable unstable structurally stable sufficient conditions sufficiently small Suppose system dx/dt system of equations topological structures trajectory of system unique limit cycle vicinity y-axis zero