Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems

被引：25
作者：
Algaba, A. [1 ]
Garcia, C. [1 ]
Reyes, M. [1 ]
机构：
[1] Fac Expt Sci, Dept Math, Huelva 21071, Spain
来源：
CHAOS SOLITONS & FRACTALS | 2012年 / 45卷 / 06期
关键词：
LIMIT-CYCLES; DIFFERENTIAL-EQUATIONS; 1ST INTEGRALS; HAMILTONIAN CENTERS; PLANAR SYSTEMS; VECTOR-FIELDS; BIFURCATION; SHAPE;
D O I：
10.1016/j.chaos.2012.02.016
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We characterize the nilpotent systems whose lowest degree quasi-homogeneous term is (y, sigma x(n))(T), sigma = +/- 1, having a formal inverse integrating factor. We prove that, for n even, the systems with formal inverse integrating factor are formally orbital equivalent to ((x) over dot, (y) over dot)(T) = (y, x(n))(T). In the case n odd, we give a formal normal form that characterizes them. As a consequence, we give the link among the existence of formal inverse integrating factor, center problem and integrability of the considered systems. (C) 2012 Elsevier Ltd. All rights reserved.
引用
收藏
页码：869 / 878
页数：10
相关论文
共 28 条
[1]   An algorithm for computing quasi-homogeneous formal normal forms under equivalence [J].
Algaba, A ;
Freire, E ;
Gamero, E ;
García, C .
ACTA APPLICANDAE MATHEMATICAE, 2004, 80 (03) :335-359
[2]   Reversibility and quasi-homogeneous normal forms of vector fields [J].
Algaba, A. ;
Garcia, C. ;
Teixeira, M. A. .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2010, 73 (02) :510-525
[3]   Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems [J].
Algaba, A. ;
Freire, E. ;
Gamero, E. ;
Garcia, C. .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2010, 72 (3-4) :1726-1736
[4]   Like-linearizations of vector fields [J].
Algaba, A. ;
Garcia, C. ;
Reyes, M. .
BULLETIN DES SCIENCES MATHEMATIQUES, 2009, 133 (08) :806-816
[5]   Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations [J].
Algaba, A. ;
Garcia, C. ;
Reyes, M. .
APPLIED MATHEMATICS AND COMPUTATION, 2009, 215 (01) :314-323
[6]   The integrability problem for a class of planar systems [J].
Algaba, A. ;
Gamero, E. ;
Garcia, C. .
NONLINEARITY, 2009, 22 (02) :395-420
[7]  
Algaba A., 2000, QUAL THEORY DYNAM SY, V1, P133
[8]   REVERSIBILITY AND CLASSIFICATION OF NILPOTENT CENTERS [J].
BERTHIER, M ;
MOUSSU, R .
ANNALES DE L INSTITUT FOURIER, 1994, 44 (02) :465-494
[9]   On a new type of bifurcation of limit cycles for a planar cubic system [J].
Chavarriga, J ;
Giacomini, H ;
Giné, J .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1999, 36 (02) :139-149
[10]   Darboux integrability and the inverse integrating factor [J].
Chavarriga, J ;
Giacomini, H ;
Giné, J ;
Llibre, J .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2003, 194 (01) :116-139
123