# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177
##### References:
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Llibre, Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3,, J. Math. Anal. Appl., 331 (2007), 1284. doi: 10.1016/j.jmaa.2006.09.066. Google Scholar [7] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer-Verlag, (1982). Google Scholar [8] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems,, Springer-Verlag, (2006). Google Scholar [9] I. García, On the integrability of quasihomogeneous and related planar vector fields,, Internat. J. Bifur. Chaos, 13 (2003), 995. 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. 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. 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. 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. doi: 10.1063/1.531484. Google Scholar [14] Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems,, Adv. Difference Eqns., (2007). Google Scholar [15] C. Li, Two problems of planar quadratic systems,, (Chinese), 26 (1983), 471. 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. 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. 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. doi: 10.1088/0951-7715/15/4/313. Google Scholar [19] K. E. Malkin, Criteria for the center for a certain differential equation,, (Russian), 2 (1964), 87. Google Scholar [20] P. Mardesic, C. Rousseau and B. Toni, Linearization of isochronous centers,, J. Diff. Eqns., 121 (1995), 67. 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. 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, (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. doi: 10.1006/jdeq.1995.1168. Google Scholar [24] G. Sansone and R. Conti, Non-Linear Differential Equations,, $2^{nd}$ edition, (1964). Google Scholar [25] D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields,, Expo. Math., 8 (1990), 3. Google Scholar [26] Y. Tang and W. Zhang, Generalized normal sectors and orbits in expceptional directions,, Nonlinearity, 17 (2004), 1407. 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), 301 (1988), 1297. Google Scholar [28] Y. Q. Ye, Theory of Limit Cycles,, Trans. Math. Monographs 66, 66 (1986). Google Scholar [29] Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems,, Shanghai Science $&$ Technology Pub., (1995). Google Scholar [30] H. Yoshida, Necessary condition for existence of algebraic first integrals I and II,, Celestial Mech., 31 (1983), 363. 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 (1992). Google Scholar [32] H. Zoladek, The classification of reversible cubic systems with center,, Topol. Methods Nonlinear Anal., 4 (1994), 79. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2006.09.066. Google Scholar [7] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer-Verlag, (1982). Google Scholar [8] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems,, Springer-Verlag, (2006). Google Scholar [9] I. García, On the integrability of quasihomogeneous and related planar vector fields,, Internat. J. Bifur. Chaos, 13 (2003), 995. 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. 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. 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. 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. doi: 10.1063/1.531484. Google Scholar [14] Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems,, Adv. Difference Eqns., (2007). Google Scholar [15] C. Li, Two problems of planar quadratic systems,, (Chinese), 26 (1983), 471. 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. 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. 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. doi: 10.1088/0951-7715/15/4/313. Google Scholar [19] K. E. Malkin, Criteria for the center for a certain differential equation,, (Russian), 2 (1964), 87. Google Scholar [20] P. Mardesic, C. Rousseau and B. Toni, Linearization of isochronous centers,, J. Diff. Eqns., 121 (1995), 67. 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. 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, (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. doi: 10.1006/jdeq.1995.1168. Google Scholar [24] G. Sansone and R. Conti, Non-Linear Differential Equations,, $2^{nd}$ edition, (1964). Google Scholar [25] D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields,, Expo. Math., 8 (1990), 3. Google Scholar [26] Y. Tang and W. Zhang, Generalized normal sectors and orbits in expceptional directions,, Nonlinearity, 17 (2004), 1407. 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), 301 (1988), 1297. Google Scholar [28] Y. Q. Ye, Theory of Limit Cycles,, Trans. Math. Monographs 66, 66 (1986). Google Scholar [29] Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems,, Shanghai Science $&$ Technology Pub., (1995). Google Scholar [30] H. Yoshida, Necessary condition for existence of algebraic first integrals I and II,, Celestial Mech., 31 (1983), 363. 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 (1992). Google Scholar [32] H. Zoladek, The classification of reversible cubic systems with center,, Topol. Methods Nonlinear Anal., 4 (1994), 79. Google Scholar
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