Highest weak focus order for trigonometric Liénard equations

Abstract

Given a planar analytic differential equation with a critical point which is a weak focus of order k, it is well known that at most k limit cycles can bifurcate from it. Moreover, in case of analytic Liénard differential equations this order can be computed as one half of the multiplicity of an associated planar analytic map. By using this approach, we can give an upper bound of the maximum order of the weak focus of pure trigonometric Liénard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Liénard case a similar result known for polynomial Liénard equations.

The first author is partially supported by “Agencia Estatal de Investigación” and “Ministerio de Ciencia, Innovación y Universidades”, Grant number MTM2016-77278-P and AGAUR, Generalitat de Catalunya, grant 2017-SGR-1617. The second author is partially supported by a MINECO/FEDER grant number MTM2017-84383-P and an AGAUR grant number 2017SGR-1276. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.