Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems

被引:24
作者
Algaba, A. [2 ]
Freire, E. [1 ]
Gamero, E. [1 ]
Garcia, C. [2 ]
机构
[1] Univ Seville, Dept Appl Math 2, ESI, Seville, Spain
[2] Univ Huelva, Dept Math, Fac Ciencias, Huelva, Spain
关键词
Polynomial systems; Monodromy; Center-focus problem; Integrability; 1ST INTEGRALS;
D O I
10.1016/j.na.2009.09.012
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
This paper deals with planar quasi-homogeneous polynomial vector fields, and addresses three major questions: the monodromy, the center-focus and the integrability problems. We characterize the monodromic planar quasi-homogeneous polynomial vector fields, and we give a condition to distinguish between a center and a focus in this case. Also, we provide conditions which characterize the integrability of quasi-homogeneous polynomial systems under non-resonance conditions. The results obtained allow us to analyse two monodromic planar systems with degenerate linear part: one of them with nilpotent linearization, and another one with null linear part. (C) 2009 Elsevier Ltd. All rights reserved.
引用
收藏
页码:1726 / 1736
页数:11
相关论文
共 12 条
[1]   Quasi-homogeneous normal forms [J].
Algaba, A ;
Freire, E ;
Gamero, E ;
García, C .
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2003, 150 (01) :193-216
[2]   TOPOLOGICAL EQUIVALENCE OF A PLANE VECTOR FIELD WITH ITS PRINCIPAL PART DEFINED THROUGH NEWTON POLYHEDRA [J].
BRUNELLA, M ;
MIARI, M .
JOURNAL OF DIFFERENTIAL EQUATIONS, 1990, 85 (02) :338-366
[3]  
Bruno A.D., 1989, Local Methods in Nonlinear Differential Equations
[4]   Local analytic integrability for nilpotent centers [J].
Chavarriga, J ;
Giacomin, H ;
Giné, J ;
Llibre, J .
ERGODIC THEORY AND DYNAMICAL SYSTEMS, 2003, 23 :417-428
[5]   ALGEBRAIC CONDITIONS FOR A CENTER OR A FOCUS IN SOME SIMPLE SYSTEMS OF ARBITRARY DEGREE [J].
COLLINS, CB .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1995, 195 (03) :719-735
[6]  
Dumortier F., 1991, Structures in Dynamics, Finite Dimensional Deterministic Studies, Vvol 2, P161
[7]   On the differentiability of first integrals of two dimensional flows [J].
Li, WG ;
Llibre, J ;
Nicolau, M ;
Zhang, X .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2002, 130 (07) :2079-2088
[8]   On the center problem for degenerate singular points of planar vector fields [J].
Mañosa, V .
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2002, 12 (04) :687-707
[9]   A CHARACTERIZATION OF CENTERS VIA 1ST INTEGRALS [J].
MAZZI, L ;
SABATINI, M .
JOURNAL OF DIFFERENTIAL EQUATIONS, 1988, 76 (02) :222-237
[10]  
MEDVEDEVA NB, 2002, ST PETERBOURG MATH J, V13, P253