Limit cycles for a class of quintic \(\mathbb{Z}_6\)-equivariant systems without infinite critical points. (English) Zbl 1308.34039

Summary: We analyze the dynamics of a class of \(\mathbb{Z}_6\)-equivariant systems of the form \(\dot z= pz^2\overline z+ sz^3\overline z^2-\overline z^5\), 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 [the first author et al., Proc. Am. Math. Soc. 136, No. 3, 1035–1043 (2008; Zbl 1142.34017)] 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.


34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations


Zbl 1142.34017
Full Text: arXiv Euclid