On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields. (English) Zbl 0534.34059

It is shown by the author in (*) ibid. 41, 301-312 (1981; Zbl 0442.34029) that Poincaré-Lyapunov constants are important control variables for the creation of limit cycles. So, an investigation of structure of Poincaré-Lyapunov constants is interesting. The author of this paper gives two explicit formulas about the structure of Poincaré-Lyapunov constants and Lyapunov functions. Also, the order of Poincaré-Lyapunov constants is defined, which has been vague for a long time. Both of this paper and (*) prove the following theorem: If there is a certain number of algebraic independent Poincaré-Lyapunov constants, then there will be at least the same number of limit cycles which can create from the neighborhood of weak focus for polynomial vector fields \((E_ n)\). This theorem stated above can be regarded as the second theorem of any degree of generality regarding the existence of closed orbits. (Poincaré- Bendixson annuli theorem, as Lefschetz said in 1940, is the theorem of any degree of generality regarding the existence of closed orbits.)


34D20 Stability of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations


Zbl 0442.34029
Full Text: DOI


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