Limit cycles for 3-monomial differential equations

Abstract

We study planar polynomial differential equations that in complex coordinates write as z˙=Az+Bzkz¯l+Czmz¯n. We prove that for each p∈N there are differential equations of this type having at least p limit cycles. Moreover, for the particular case z˙=Az+Bz¯+Czmz¯n, which has homogeneous nonlinearities, we show examples with several limit cycles and give a condition that ensures uniqueness and hyperbolicity of the limit cycle.