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

Zbl 0442.34029
Full Text:

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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