International Journal of Nonlinear Analysis and Applications

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

Document Type : Research Paper

Authors

1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 84156-83111.

2 Department of Mathematical Sciences, Kashan University, Kashan, Iran

Abstract

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{0.5cm} k=0,1,
\end{equation*}
with $\Gamma_{h}=\Gamma_{h}^{L}+\Gamma^{R}_{h}$, where $\Gamma^{L}_{h}=\{(x,y)\in \mathbb{R}^{2}| \frac{1}{2}y^2+\Psi_{2}(x)=h, \  x<0 \}$ and $\Gamma_{h}^{R}=\{(x,y)\in \mathbb{R}^{2}|
\frac{1}{2}y^2+\Psi_1(x)=h,\  x>0 \}$. We prove that the monotonicity of the presented criterion function implies the monotonicity of the ratio $\frac{I_1(h)}{I_0(h)}$ and provide a few examples to explain the application of this criterion.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 6
June 2024
Pages 1-17
  • Receive Date: 16 October 2022
  • Accept Date: 10 July 2023