Limit cycles of Abel equations of the first kind

Abstract

Consider the scalar differential equation x′=∑i=0mai(t)xni, where ai(t) are T-periodic analytic functions, and 1≤ni≤n. For any polynomial Q(x)=xn0−∑i=1mαixni, the equation can be written as x′=a0Q(x)+R(t,x). Let W be the Wronskian of Q and R with respect to x, and Q˜, W˜ the previous polynomials after removing multiplicity of roots and solutions of the differential equation. We prove that if the vector field defined by the differential equation is “transversal” at every point of Q˜(x)=0 or W˜(t,x)=0 then the number of limit cycles (isolated periodic solutions in the set of periodic solutions) of the differential equation is at most 3n−1.