On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields
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.)
scientific article; zbMATH DE number 3847831