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**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.)

### MSC:

34D20 | Stability of solutions to ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

### Citations:

Zbl 0442.34029
Full Text:
DOI

### References:

[1] | Poincaré, H., Mémoire sur les courbes définies par une équation différentielle, J. Math. Pures Appl., 3, 7, 375-422 (1881) · JFM 13.0591.01 |

[2] | Lyapunov, A. M., Problème Général de la Stabilité du Mouvement (1935), ONTI: ONTI Moscow/Leningrad, [Russian] |

[3] | Shi, S., A method of constructing cycles without contact around a weak focus, J. Differential Equations, 41, 301-312 (1981) · Zbl 0442.34029 |

[4] | Nemytskii, V. V.; Stepanov, V. V., Qualitative Theory of Differential Equations (1960), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 0089.29502 |

[5] | Bautin, N. N., Amer. Math. Soc. Transl., 100 (1954) · Zbl 0059.08201 |

[6] | Zariski, O.; Samuel, P., Commutative Algebra, (Graduate Texts in Mathematics, Vol. 1 (1976), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0121.27901 |

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