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december 2014 Limit cycles for a class of quintic $\mathbb{Z}_6$-equivariant systems without infinite critical points
M.J. Àlvarez, I.S. Labouriau, A.C. Murza
Bull. Belg. Math. Soc. Simon Stevin 21(5): 841-857 (december 2014). DOI: 10.36045/bbms/1420071857


We analyze the dynamics of a class of $\mathbb{Z}_6$-equivariant systems of the form $\dot{z}=pz^2\bar{z}+sz^3\bar{z}^2-\bar{z}^{5},$ where $z$ is complex, the time $t$ is real, while $p$ and $s$ are complex parameters. This study is the natural continuation of a previous work (M.J. Àlvarez, A. Gasull, R. Prohens, Proc. Am. Math. Soc. \textbf{136}, (2008), 1035--1043) on the normal form of $\mathbb{Z}_4$-equivariant systems. Our study uses the reduction of the equation to an Abel one, and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points.


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M.J. Àlvarez. I.S. Labouriau. A.C. Murza. "Limit cycles for a class of quintic $\mathbb{Z}_6$-equivariant systems without infinite critical points." Bull. Belg. Math. Soc. Simon Stevin 21 (5) 841 - 857, december 2014.


Published: december 2014
First available in Project Euclid: 1 January 2015

zbMATH: 1308.34039
MathSciNet: MR3298481
Digital Object Identifier: 10.36045/bbms/1420071857

Primary: 34C07 , 34C14
Secondary: 34C23 , 37C27

Keywords: limit cycles , Planar autonomous ordinary differential equations , symmetric polynomial systems

Rights: Copyright © 2014 The Belgian Mathematical Society


Vol.21 • No. 5 • december 2014
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