Birth of limit cycles for a class of continuous and discontinuous differential systems in (d + 2)–dimension

ABSTRACT

The orbits of the reversible differential system , , , with and , are periodic with the exception of the equilibrium points (0, 0, z₁,… , z_d). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system , , , using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case, this maximum number is n^d(n − 1)/2, and in the second, it is n^{d + 1}.

KEYWORDS:

• Limit cycle
• averaging method
• periodic orbit
• polynomial differential system
• discontinuous polynomial differential systems

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 34A36
• 34C25
• 34C29
• 37G15

Acknowledgements

We thank the reviewers for their comments and suggestions which help us to improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is partially supported by a MINECO [grant number MTM2013-40998-P]; an AGAUR [grant number 2014SGR-568]; FP7-PEOPLE-2012-IRSES [grant number 318999], [grant number 316338]; CAPES [grant number 88881.030454/2013-01] from the program CSF-PVE. The second author is partially supported by an FAPESP–BRAZIL [grant number 2012/18780–0]. The third author is partially supported by an FAPESP-BRAZIL [grant number 2012/23591–1], [grant number 2013/21078–8].