Bulletin of the Belgian Mathematical Society - Simon Stevin

Limit cycles for a class of quintic $\mathbb{Z}_6$-equivariant systems without infinite critical points

M.J. Àlvarez, I.S. Labouriau, and A.C. Murza

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin Volume 21, Number 5 (2014), 841-857.

First available in Project Euclid: 1 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C14: Symmetries, invariants
Secondary: 34C23: Bifurcation [See also 37Gxx] 37C27: Periodic orbits of vector fields and flows

Planar autonomous ordinary differential equations symmetric polynomial systems limit cycles


Àlvarez, M.J.; Labouriau, I.S.; Murza, A.C. Limit cycles for a class of quintic $\mathbb{Z}_6$-equivariant systems without infinite critical points. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 5, 841--857.https://projecteuclid.org/euclid.bbms/1420071857

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