Uniqueness of limit cycles for polynomial first-order differential equations

Abstract

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE x ′ = ∑ k = 1 m a k sin i k ( t ) cos j k ( t ) x n k in terms of { i k } , { j k } , { n k } . Our main result characterizes, under some additional hypotheses, the exponents { i k } , { j k } , { n k } , such that for any choice of a 1 , … , a m ∈ R the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x ′ = y + x R ( x , y ) , y ′ = − x + y R ( x , y ) , where R ( x , y ) = ∑ k = 1 m a k x i k y j k . Concretely, when the set { i k + j k : k = 1 , … , m } has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents { i k } , { j k } such that the origin of the rigid system is a center for any choice of a 1 , … , a m ∈ R and also when there are no limit cycles surrounding the origin for any choice of a 1 , … , a m ∈ R .