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In this chapter, the homoclinic networks of positive and negative saddles with clockwise and counter-clockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks.

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Homoclinic Networks with Centers

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  • 发表日期：

    2025年

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  • Albert C. J. Luo

摘要

In this chapter, the homoclinic networks of positive and negative saddles with clockwise and counter-clockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks.