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

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.