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 Lienard 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 Lienard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Lienard case a similar result known for polynomial Lienard equations.