Journal of Applied Mathematics and Computation

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Article http://dx.doi.org/10.26855/jamc.2023.03.015

A Simple Calculation of the First Order Melnikov Function for a Non-smooth Hamilton System

Erli Zhang*, Huimin Li, Jing Huang

School of Statistics and Big Data, Zhengzhou College of Finance and Economics, Zhengzhou, Henan, China.

*Corresponding author: Erli Zhang

Published: May 6,2023

Abstract

One of the important problems in differential systems is the study of the number of limit cycles, which is closely related to the famous Hilbert’s 16th problem. The first order Melnikov function plays a key role in determining the lower and upper bounds of the number of limit cycles for smooth and non-smooth Hamilton systems. But the calculation of some Melnikov functions is very complicated, especially for non-smooth differential systems. In the present article, we use a simple approach to obtain the first order Melnikov function of a non-smooth differential system with a generalized heteroclinic loop through a cusp. We first prove that the first order Melnikov function can be expressed as combinations of some generator functions with polynomial coefficients, which can save a large amount of calculation. Then the upper bound of the number of limit cycles of the perturbed differential system can be obtained by using the existing methods. This method can be applied to other discontinuous differential systems.

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How to cite this paper

A Simple Calculation of the First Order Melnikov Function for a Non-smooth Hamilton System

How to cite this paper:  Erli Zhang, Huimin Li, Jing Huang. (2023) A Simple Calculation of the First Order Melnikov Function for a Non-smooth Hamilton System. Journal of Applied Mathematics and Computation7(1), 142-146.

DOI: http://dx.doi.org/10.26855/jamc.2023.03.015