Electron. J. Differential Equations, Vol. 2020 (2020), No. 90, pp. 1-15.

Finite cyclicity of the contact point in slow-fast integrable systems of Darboux type

Renato Huzak

Using singular perturbation theory and family blow-up we prove that nilpotent contact points in deformations of slow-fast Darboux integrable systems have finite cyclicity. The deformations are smooth or analytic depending on the region in the parameter space. This article is a natural continuation of [1,3], where one studies limit cycles in polynomial deformations of slow-fast Darboux integrable systems, around the "integrable" direction in the parameter space. We extend the existing finite cyclicity result of the contact point to analytic deformations, and under some assumptions we prove that the contact point has finite cyclicity around the "slow-fast" direction in the parameter space.

Submitted December 1, 2019. Published September 6, 2020.
Math Subject Classifications: 34C26, 34E15, 34E17.
Key Words: Blow-up; cyclicity; Darboux systems; singular perturbation theory; slow-fast systems.

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Renato Huzak
Hasselt University, Campus Diepenbeek
Agoralaan Gebouw D
3590 Diepenbeek, Belgium
email: renato.huzak@uhasselt.be

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