American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, 1982.  Google Scholar

[8]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Eqns., (2007), Art ID 98427, 10 pp.  Google Scholar

[15]

C. Li, Two problems of planar quadratic systems, (Chinese), Science in China Math, 26 (1983), 471-481.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. E. Malkin, Criteria for the center for a certain differential equation, (Russian), Volz. Mat. Sb. Vyp, 2 (1964), 87-91.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Sansone and R. Conti, Non-Linear Differential Equations, $2^{nd}$ edition, Pergamon Press, New York, 1964.  Google Scholar

[25]

D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expo. Math., 8 (1990), 3-25.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.  Google Scholar

[29]

Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems, Shanghai Science $&$ Technology Pub., Shanghai, 1995. Google Scholar

[30]

H. Yoshida, Necessary condition for existence of algebraic first integrals I and II, Celestial Mech., 31 (1983), 363-379. doi: 10.1007/BF01230293.  Google Scholar

[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.  Google Scholar

[32]

H. Zoladek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, 1982.  Google Scholar

[8]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Eqns., (2007), Art ID 98427, 10 pp.  Google Scholar

[15]

C. Li, Two problems of planar quadratic systems, (Chinese), Science in China Math, 26 (1983), 471-481.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. E. Malkin, Criteria for the center for a certain differential equation, (Russian), Volz. Mat. Sb. Vyp, 2 (1964), 87-91.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Sansone and R. Conti, Non-Linear Differential Equations, $2^{nd}$ edition, Pergamon Press, New York, 1964.  Google Scholar

[25]

D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expo. Math., 8 (1990), 3-25.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.  Google Scholar

[29]

Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems, Shanghai Science $&$ Technology Pub., Shanghai, 1995. Google Scholar

[30]

H. Yoshida, Necessary condition for existence of algebraic first integrals I and II, Celestial Mech., 31 (1983), 363-379. doi: 10.1007/BF01230293.  Google Scholar

[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.  Google Scholar

[32]

H. Zoladek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.  Google Scholar

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