On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields
type: Item
instance of: scholarly article
title: On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields
zbMATH Open document ID: 0534.34059
DOI: 10.1016/0022-0396(84)90133-5
author: Songling Shi
published in: Journal of Differential Equations
publication date: 1984-01-01T00:00:00Z
review text: 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.)
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 3847831
zbMATH Keywords: weak focus
zbMATH Keywords: Poincaré-Lyapunov constants
zbMATH Keywords: limit cycles
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: closed orbits
MaRDI profile type: Publication
full work available at URL: 0022-0396(84)90133-5
OpenAlex ID: W1976624180
cites work: A method of constructing cycles without contact around a weak focus
cites work: Q3264374
cites work: Q5832271
cites work: Q5732722
description: scientific article; zbMATH DE number 3847831