Uniqueness of the limit cycles for complex differential equations with two monomials

Abstract

We prove that any complex differential equation with two monomials of the form z˙=azkz¯l+bzmz¯n, with k,l,m,n non-negative integers and a,b∈C, has one limit cycle at most. Moreover, we characterise when such a limit cycle exists and prove that then it is hyperbolic. For an arbitrary equation of the above form, we also solve the centre-focus problem and examine the number, position, and type of its critical points. In particular, we prove a Berlinskiĭ-type result regarding the geometrical distribution of the critical points stabilities. © 2022 The Author(s)