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

Center of planar quintic quasi--homogeneous polynomial differential systems

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

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.
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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