The Monotonicity of the Ratio of Two Line Integrals in Piecewise Smooth Differential Systems

Abstract

In this work, we study the monotonicity of the ratio of two line integrals

$$\begin{aligned} I_{i}(h)=\int _{\Gamma _{h}^{+}}f_{i}(x)ydx+\int _{\Gamma _{h}^{-}}f_{i}(x)ydx, i=0,1, \end{aligned}$$

with $\Gamma ^{+}_{h}=\lbrace (x,y)\in {\mathbb {R}}^{2}~\vert ~ H^{+}(x,y)=h,~x> 0\rbrace$ and $\Gamma _{h}^{-}=\lbrace (x,y)\in {\mathbb {R}}^{2}~\vert ~ H^{-}(x,y)={\tilde{h}},~x\le 0\rbrace$, where $H^{+}(x,y)$ and $H^{-}(x,y)$ have the form $\frac{1}{2}y^{2}+\Psi _{1}(x)$ and $\frac{1}{2}y^{2}+\Psi _{2}(x)$, respectively. We first present a criterion function defined directly by the functions which appear in the above integrals and prove the monotonicity of this criterion function implying the monotonicity of the ratio of these line integrals which is very important to find the number of limit cycles. Then we give several examples to illustrate the application of this criterion.