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\begin{equation*}I_{k}(h)=\oint_{\Gamma_{h}}f_{k}(x)ydx,\hspace{۰.۵cm} k=۰,۱,\end{equation*}with \Gamma_{h}=\Gamma_{h}^{L}+\Gamma^{R}_{h}, where \Gamma^{L}_{h}=\{(x,y)\in \mathbb{R}^{۲}| \frac{۱}{۲}y^۲+\Psi_{۲}(x)=h, \  x<۰ \} and \Gamma_{h}^{R}=\{(x,y)\in \mathbb{R}^{۲}|\frac{۱}{۲}y^۲+\Psi_۱(x)=h,\  x>۰ \}. We prove that the monotonicity of the presented criterion function implies the monotonicity of the ratio \frac{I_۱(h)}{I_۰(h)} and provide a few examples to explain the application of this criterion.

Keywords:

Piecewise-smooth differential systems , Melnikov function , Monotonicity , Abelian integral , Limit cycle