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Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard
Planar quasi-homogeneous polynomial systems with a given weight degree
1. | School of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China |
2. | Department of Mathematics, Shanghai Normal University, Shanghai 200234 |
References:
[1] |
W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Advances in Mathematics, 254 (2014), 233-250.
doi: 10.1016/j.aim.2013.12.006. |
[2] |
A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic systems in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.
doi: 10.1016/0022-247X(90)90359-N. |
[3] |
T. Date and M. Lai, Canonical forms of real homogeneous quadratic transformations, J. Math. Anal. Appl., 56 (1976), 650-682.
doi: 10.1016/0022-247X(76)90031-7. |
[4] |
T. Date, Classification and analysis of two-dimensional homogeneous quadratic differential equations systems, J. Differential Equations, 32 (1979), 311-334.
doi: 10.1016/0022-0396(79)90037-8. |
[5] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theorey of Planar Polynomial Systems, Springer, 2006. |
[6] |
B. García, J. Llibre and J. S. Pérea del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204.
doi: 10.1016/j.jde.2013.07.032. |
[7] |
L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135.
doi: 10.1016/j.jde.2009.02.010. |
[8] |
J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547.
doi: 10.3934/dcds.2013.33.4531. |
[9] |
J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[10] |
J. Llibre and C. Pessoa, On the centers of the weight-homogeneous polynomial vector fields on the plane, J. Math. Anal. Appl., 359 (2009), 722-730.
doi: 10.1016/j.jmaa.2009.06.036. |
[11] |
W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dyn. Diff. Equat., 21 (2009), 133-152.
doi: 10.1007/s10884-008-9126-1. |
[12] |
H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynamics, 78 (2014), 1659-1681.
doi: 10.1007/s11071-014-1541-8. |
[13] |
P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108.
doi: 10.1006/jdeq.1995.1122. |
[14] |
Y. Zhao, Limit cycles for planar semi-quasi-homogeneous polynomial vector fields, J. Math. Anal. Appl., 397 (2013), 276-284.
doi: 10.1016/j.jmaa.2012.07.060. |
show all references
References:
[1] |
W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Advances in Mathematics, 254 (2014), 233-250.
doi: 10.1016/j.aim.2013.12.006. |
[2] |
A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic systems in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.
doi: 10.1016/0022-247X(90)90359-N. |
[3] |
T. Date and M. Lai, Canonical forms of real homogeneous quadratic transformations, J. Math. Anal. Appl., 56 (1976), 650-682.
doi: 10.1016/0022-247X(76)90031-7. |
[4] |
T. Date, Classification and analysis of two-dimensional homogeneous quadratic differential equations systems, J. Differential Equations, 32 (1979), 311-334.
doi: 10.1016/0022-0396(79)90037-8. |
[5] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theorey of Planar Polynomial Systems, Springer, 2006. |
[6] |
B. García, J. Llibre and J. S. Pérea del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204.
doi: 10.1016/j.jde.2013.07.032. |
[7] |
L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135.
doi: 10.1016/j.jde.2009.02.010. |
[8] |
J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547.
doi: 10.3934/dcds.2013.33.4531. |
[9] |
J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[10] |
J. Llibre and C. Pessoa, On the centers of the weight-homogeneous polynomial vector fields on the plane, J. Math. Anal. Appl., 359 (2009), 722-730.
doi: 10.1016/j.jmaa.2009.06.036. |
[11] |
W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dyn. Diff. Equat., 21 (2009), 133-152.
doi: 10.1007/s10884-008-9126-1. |
[12] |
H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynamics, 78 (2014), 1659-1681.
doi: 10.1007/s11071-014-1541-8. |
[13] |
P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108.
doi: 10.1006/jdeq.1995.1122. |
[14] |
Y. Zhao, Limit cycles for planar semi-quasi-homogeneous polynomial vector fields, J. Math. Anal. Appl., 397 (2013), 276-284.
doi: 10.1016/j.jmaa.2012.07.060. |
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