2000, 6(4): 809-828. doi: 10.3934/dcds.2000.6.809

Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

2. 

Departamento de Matemáticas, Universidad de Oviedo, Avda. Calvo Sotelo, s/n, 33007, Oviedo, Spain, Spain

Received  June 1999 Revised  June 2000 Published  August 2000

Recently, in [9] we characterized the set of planar homogeneous vector fields that are structurally stable and we obtained the exact number of the topological equivalence classes. Furthermore, we gave a first extension of the Hartman-Grobman Theorem for planar vector fields. In this paper we study the structural stability in the set $H_{m,n}$ of planar semi-homogeneous vector fields $X = (P_m,Q_n)$, where $P_m$ and $Q_n$ are homogeneous polynomial of degree $m$ and $n$ respectively, and $0 < m < n$. Unlike the planar homogeneous vector fields, the semi-homogeneous ones can have limit cycles, which prevents to characterize completely those planar semi-homogeneous vector fields that are structurally stable. Thus, in the first part of this paper we will study the local structural stability at the origin and at infinity for the vector fields in $H_{m,n}$. As a consequence of these local results, we will complete the extension of the Hartman-Grobman Theorem to the nonlinear planar vector fields. In the second half of this paper we define a subset $\Delta_{m,n}$ that is dense in $H_{m,n}$ and whose elements are structurally stable. We prove that there exist vector fields in $\Delta_{m,n}$ that have at least $(m+n)/2$ hyperbolic limit cycles.
Citation: Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[1]

Xiao-Song Yang. Index sums of isolated singular points of positive vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1033-1039. doi: 10.3934/dcds.2009.25.1033

[2]

Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015

[3]

Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177

[4]

Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531

[5]

Isaac A. García, Jaume Giné, Susanna Maza. Linearization of smooth planar vector fields around singular points via commuting flows. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1415-1428. doi: 10.3934/cpaa.2008.7.1415

[6]

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010

[7]

Magdalena Caubergh, Freddy Dumortier, Stijn Luca. Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 963-980. doi: 10.3934/dcds.2010.27.963

[8]

Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387

[9]

M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743

[10]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661

[11]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[12]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019

[13]

M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems & Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219

[14]

Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997

[15]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257

[16]

Davor Dragičević. Admissibility, a general type of Lipschitz shadowing and structural stability. Communications on Pure & Applied Analysis, 2015, 14 (3) : 861-880. doi: 10.3934/cpaa.2015.14.861

[17]

Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763

[18]

Jaume Llibre, Claudia Valls. Centers for polynomial vector fields of arbitrary degree. Communications on Pure & Applied Analysis, 2009, 8 (2) : 725-742. doi: 10.3934/cpaa.2009.8.725

[19]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073

[20]

David Russell. Structural parameter optimization of linear elastic systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1517-1536. doi: 10.3934/cpaa.2011.10.1517

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (3)

[Back to Top]