Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {ik}, {jk}, {nk}. Our main result characterizes, under some additional hypotheses, the exponents {ik}, {jk}, {nk}, such that for any choice of a1,…,am∈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+xR(x,y), y′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {ik}, {jk} such that the origin of the rigid system is a center for any choice of a1,…,am∈R and also when there are no limit cycles surrounding the origin for any choice of a1,…,am∈R.