Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points

Álvarez, M.; Gasull, A.; Prohens, R.

doi:10.1090/S0002-9939-07-09072-7

Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points
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by M. J. Álvarez, A. Gasull and R. Prohens

Proc. Amer. Math. Soc. 136 (2008), 1035-1043

DOI: https://doi.org/10.1090/S0002-9939-07-09072-7

Published electronically: November 30, 2007

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Abstract:

In this paper we study those cubic systems which are invariant under a rotation of $2\pi /4$ radians. They are written as $\dot {z}=\varepsilon z+p z^2\bar {z}-\bar {z}^3,$ where $z$ is complex, the time is real, and $\varepsilon =\varepsilon _1+i\varepsilon _2$, $p=p_1+ip_2$ are complex parameters. When they have some critical points at infinity, i.e. $|p_2|\le 1$, it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. $|p_2|>1,$ there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.

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