A criterion for the monotonicity of the ratio of two Abelian integrals in piecewise-smooth differential systems

In this paper, we present a new criterion function for investigating the monotonicity of the ratio of two Abelian integrals in piecewise-smooth differential systems, and then, apply it to deal with some examples. More precisely, we consider the Abelian integrals of the form Ik(h) = ꭍ fk(x)ydx, k = 0, 1, with Γh = ΓL h + ΓR h , where ΓL h = {(x, y) ∈ R 2 | 1 2 y 2 + Ψ2(x) = h, x 0} and ΓR h = {(x, y) ∈ R² | 1/2 y² + Ψ1(x) = h, x 0}. We prove that the monotonicity of the presented criterion function implies the monotonicity of the ratio I1(h)/I0(h) and provide a few examples to explain the application of this criterion.