American Institute of Mathematical Sciences

Advanced
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
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show all references

References:
[1]

Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056.  Google Scholar

[2]

Nonlinearity, 22 (2009), 396-420. doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[3]

Nonlinear Anal., 73 (2010), 1318-1327. doi: 10.1016/j.na.2010.04.059.  Google Scholar

[4]

Adv. Math., 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006.  Google Scholar

[5]

Mat. Sb., 30 (1925), 181-196, Amer. Math. Soc. Transl., 1954 (1954), 1-19.  Google Scholar

[6]

J. Math. Anal. Appl., 331 (2007), 1284-1298. doi: 10.1016/j.jmaa.2006.09.066.  Google Scholar

[7]

Springer-Verlag, Berlin, 1982.  Google Scholar

[8]

Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

Internat. J. Bifur. Chaos, 13 (2003), 995-1002. doi: 10.1142/S021812740300700X.  Google Scholar

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J. Diff. Eqns., 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032.  Google Scholar

[11]

J. Diff. Eqns., 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010.  Google Scholar

[12]

Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547. doi: 10.3934/dcds.2013.33.4531.  Google Scholar

[13]

J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484.  Google Scholar

[14]

Adv. Difference Eqns., (2007), Art ID 98427, 10 pp.  Google Scholar

[15]

(Chinese), Science in China Math, 26 (1983), 471-481.  Google Scholar

[16]

J. Dyn. Diff. Eqns., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1.  Google Scholar

[17]

Nonlinear Dyn., 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8.  Google Scholar

[18]

Nonlinearity, 15 (2002), 1269-1280. doi: 10.1088/0951-7715/15/4/313.  Google Scholar

[19]

(Russian), Volz. Mat. Sb. Vyp, 2 (1964), 87-91.  Google Scholar

[20]

J. Diff. Eqns., 121 (1995), 67-108. doi: 10.1006/jdeq.1995.1122.  Google Scholar

[21]

Qual. Theory Dyn. Syst., 13 (2014), 39-72. doi: 10.1007/s12346-013-0105-5.  Google Scholar

[22]

Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4727-8.  Google Scholar

[23]

J. Diff. Eqns., 123 (1995), 388-436. doi: 10.1006/jdeq.1995.1168.  Google Scholar

[24]

$2^{nd}$ edition, Pergamon Press, New York, 1964.  Google Scholar

[25]

Expo. Math., 8 (1990), 3-25.  Google Scholar

[26]

Nonlinearity, 17 (2004), 1407-1426. doi: 10.1088/0951-7715/17/4/015.  Google Scholar

[27]

(Russian), Dokl. Akad. Nauk SSSR, 301 (1988), 1297-1301, translation in: Soviet Math. Dokl., 38 (1989), 198-201.  Google Scholar

[28]

Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.  Google Scholar

[29]

Shanghai Science $&$ Technology Pub., Shanghai, 1995. Google Scholar

[30]

Celestial Mech., 31 (1983), 363-379. doi: 10.1007/BF01230293.  Google Scholar

[31]

Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992.  Google Scholar

[32]

Topol. Methods Nonlinear Anal., 4 (1994), 79-136.  Google Scholar

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