Center of planar quintic quasi--homogeneous polynomial differential systems

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May 2015, 35(5): 2177-2191. doi: 10.3934/dcds.2015.35.2177

Center of planar quintic quasi--homogeneous polynomial differential systems

Yilei Tang ^1, , Long Wang ^2, and Xiang Zhang ^3,

1.	Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
2.	Department of Mathematics, Southern Polytechnic State University, Marietta, GA 30060, United States
3.	Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

Received May 2014 Revised July 2014 Published December 2014

Abstract
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In this paper we first characterize all quasi--homogeneous but non--homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.

Keywords: generalized normal sectors., Quasi--homogeneous quintic polynomial systems, blow up, centers, global phase portrait.

Mathematics Subject Classification: Primary: 37G05; Secondary: 37G10, 34C23, 34C2.

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

References:

[1]	A. Algaba, N. Fuentes and C. García, Center of quasi-homogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056.
[2]	A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 396-420. doi: 10.1088/0951-7715/22/2/009.
[3]	A. Algaba, C. García and M. Reyes, Rational integrability of two dimensional quasi-homogeneous polynomial differential systems, Nonlinear Anal., 73 (2010), 1318-1327. doi: 10.1016/j.na.2010.04.059.
[4]	W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006.
[5]	N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1925), 181-196, Amer. Math. Soc. Transl., 1954 (1954), 1-19.
[6]	L. Cairó and J. Llibre, Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3, J. Math. Anal. Appl., 331 (2007), 1284-1298. doi: 10.1016/j.jmaa.2006.09.066.
[7]	S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, 1982.
[8]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.
[9]	I. García, On the integrability of quasihomogeneous and related planar vector fields, Internat. J. Bifur. Chaos, 13 (2003), 995-1002. doi: 10.1142/S021812740300700X.
[10]	B. García, J. Llibre and J. S. Pérez del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Diff. Eqns., 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032.
[11]	L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Diff. Eqns., 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010.
[12]	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.
[13]	A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484.
[14]	Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Eqns., (2007), Art ID 98427, 10 pp.
[15]	C. Li, Two problems of planar quadratic systems, (Chinese), Science in China Math, 26 (1983), 471-481.
[16]	W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homegeneous centers, J. Dyn. Diff. Eqns., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1.
[17]	H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dyn., 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8.
[18]	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.
[19]	K. E. Malkin, Criteria for the center for a certain differential equation, (Russian), Volz. Mat. Sb. Vyp, 2 (1964), 87-91.
[20]	P. Mardesic, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Diff. Eqns., 121 (1995), 67-108. doi: 10.1006/jdeq.1995.1122.
[21]	R. Oliveira and Y. Zhao, Structural stability of planar quasihomogeneous vector fields, Qual. Theory Dyn. Syst., 13 (2014), 39-72. doi: 10.1007/s12346-013-0105-5.
[22]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebraic Approach, Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4727-8.
[23]	C. Rousseau and D. Schlomiuk, Cubic vector fields symmetric with respect to a center, J. Diff. Eqns., 123 (1995), 388-436. doi: 10.1006/jdeq.1995.1168.
[24]	G. Sansone and R. Conti, Non-Linear Differential Equations, $2^{nd}$ edition, Pergamon Press, New York, 1964.
[25]	D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expo. Math., 8 (1990), 3-25.
[26]	Y. Tang and W. Zhang, Generalized normal sectors and orbits in expceptional directions, Nonlinearity, 17 (2004), 1407-1426. doi: 10.1088/0951-7715/17/4/015.
[27]	N. I. Vulpe and K. S. Sibirski, Centro -affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, (Russian), Dokl. Akad. Nauk SSSR, 301 (1988), 1297-1301, translation in: Soviet Math. Dokl., 38 (1989), 198-201.
[28]	Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.
[29]	Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems, Shanghai Science $&$ Technology Pub., Shanghai, 1995.
[30]	H. Yoshida, Necessary condition for existence of algebraic first integrals I and II, Celestial Mech., 31 (1983), 363-379. doi: 10.1007/BF01230293.
[31]	Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992.
[32]	H. Zoladek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.

show all references

References:

[1]	A. Algaba, N. Fuentes and C. García, Center of quasi-homogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056.
[2]	A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 396-420. doi: 10.1088/0951-7715/22/2/009.
[3]	A. Algaba, C. García and M. Reyes, Rational integrability of two dimensional quasi-homogeneous polynomial differential systems, Nonlinear Anal., 73 (2010), 1318-1327. doi: 10.1016/j.na.2010.04.059.
[4]	W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006.
[5]	N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1925), 181-196, Amer. Math. Soc. Transl., 1954 (1954), 1-19.
[6]	L. Cairó and J. Llibre, Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3, J. Math. Anal. Appl., 331 (2007), 1284-1298. doi: 10.1016/j.jmaa.2006.09.066.
[7]	S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, 1982.
[8]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.
[9]	I. García, On the integrability of quasihomogeneous and related planar vector fields, Internat. J. Bifur. Chaos, 13 (2003), 995-1002. doi: 10.1142/S021812740300700X.
[10]	B. García, J. Llibre and J. S. Pérez del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Diff. Eqns., 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032.
[11]	L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Diff. Eqns., 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010.
[12]	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.
[13]	A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484.
[14]	Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Eqns., (2007), Art ID 98427, 10 pp.
[15]	C. Li, Two problems of planar quadratic systems, (Chinese), Science in China Math, 26 (1983), 471-481.
[16]	W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homegeneous centers, J. Dyn. Diff. Eqns., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1.
[17]	H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dyn., 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8.
[18]	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.
[19]	K. E. Malkin, Criteria for the center for a certain differential equation, (Russian), Volz. Mat. Sb. Vyp, 2 (1964), 87-91.
[20]	P. Mardesic, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Diff. Eqns., 121 (1995), 67-108. doi: 10.1006/jdeq.1995.1122.
[21]	R. Oliveira and Y. Zhao, Structural stability of planar quasihomogeneous vector fields, Qual. Theory Dyn. Syst., 13 (2014), 39-72. doi: 10.1007/s12346-013-0105-5.
[22]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebraic Approach, Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4727-8.
[23]	C. Rousseau and D. Schlomiuk, Cubic vector fields symmetric with respect to a center, J. Diff. Eqns., 123 (1995), 388-436. doi: 10.1006/jdeq.1995.1168.
[24]	G. Sansone and R. Conti, Non-Linear Differential Equations, $2^{nd}$ edition, Pergamon Press, New York, 1964.
[25]	D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expo. Math., 8 (1990), 3-25.
[26]	Y. Tang and W. Zhang, Generalized normal sectors and orbits in expceptional directions, Nonlinearity, 17 (2004), 1407-1426. doi: 10.1088/0951-7715/17/4/015.
[27]	N. I. Vulpe and K. S. Sibirski, Centro -affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, (Russian), Dokl. Akad. Nauk SSSR, 301 (1988), 1297-1301, translation in: Soviet Math. Dokl., 38 (1989), 198-201.
[28]	Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.
[29]	Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems, Shanghai Science $&$ Technology Pub., Shanghai, 1995.
[30]	H. Yoshida, Necessary condition for existence of algebraic first integrals I and II, Celestial Mech., 31 (1983), 363-379. doi: 10.1007/BF01230293.
[31]	Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992.
[32]	H. Zoladek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.

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