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In this chapter, Theorem 1.1 is proved. The theorem about the singular equilibriums and 1-dimensional flows with infinite-equilibriums in crossing and product polynomial systems is proved first. The first integral manifolds are obtained for positive and negative saddle, centers, parabola-saddles and inflection-saddle. The theorem about hybrid networks of singular equilibriums and 1-dimensional flows in crossing and product polynomial systems are proved, and the corresponding infinite-equilibriums are discussed. The theorem about hybrid networks of simple equilibriums and 1-dimensional flows with infinite-equilibriums in crossing and product polynomial systems is proved. The theorem about appearing and switching of simple and singular 1-dimensional flows and equilibriums is proved. Finally, the theorem about the switching of networks of simple and singular equilibriums and flows is proved.

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Proof of Theorem 1.1

  • 章节

  • 发表日期：

    2026年

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  • Albert C. J. Luo

摘要

In this chapter, Theorem 1.1 is proved. The theorem about the singular equilibriums and 1-dimensional flows with infinite-equilibriums in crossing and product polynomial systems is proved first. The first integral manifolds are obtained for positive and negative saddle, centers, parabola-saddles and inflection-saddle. The theorem about hybrid networks of singular equilibriums and 1-dimensional flows in crossing and product polynomial systems are proved, and the corresponding infinite-equilibriums are discussed. The theorem about hybrid networks of simple equilibriums and 1-dimensional flows with infinite-equilibriums in crossing and product polynomial systems is proved. The theorem about appearing and switching of simple and singular 1-dimensional flows and equilibriums is proved. Finally, the theorem about the switching of networks of simple and singular equilibriums and flows is proved.