## Journal of Applied Mathematics and Computation

 ISSN Print: 2576-0645 Downloads: 87905 Total View: 1370222 Frequency: quarterly ISSN Online: 2576-0653 CODEN: JAMCEZ Email: jamc@hillpublisher.com

## Volumes & Issues

### Current Issue

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.

## References

[1] D. Hilbert, Mathematical problems, Bulletin of the American Mathematical Society, 8 (1902) 437-479.

[2] Jibin Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, International Journal Bifurcation and Chaos, 13 (2003), 47-106.

[3] Yun Tian, X. Shang, Maoan. Han. Bifurcation of limit cycles in a piecewise smooth near-integrable, J. Math. Anal. Appl., 504 (2021), 125578.

[4] O. Ramirez, A.M. Alves, Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold, Nonlinear Anal.: Real World Appl., 57 (2021) 103188.

[5] Jihua Yang, Erli Zhang, Mei Liu, Limit cycle cifurcations of a piecewise smooth Hamiltonian systems with a generalized heteroclinic loop through a cusp, Commu. on Pure and Appl. Anal., 16 (2017) 2321-2336.

[6] H. Liang, S. Li, X. Zhang, Limit cycles and global dynamics of planar piecewise linear refracting systems of focus-focus type, Nonlinear Analysis: Real World Applications, 58 (2021) 103228.

[7] Maoan Han, Weinian Zhang, On Hopf bifurcation in non-smooth planar systems, Journal of Differential Equations, 248 (2010) 2399-2416.

[8] Xia Liu, Maoan Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, International Journal Bifurcation and Chaos, 20 (2010) 1379-1390.

[9] Feng Liang, Maoan Han, V. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Analysis, 75 (2012) 4355-4374.

[10] Jihua Yang, Liqin Zhao, Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations, Journal of Differential Equations, 264 (2018) 5734-5757.

## 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