Limit cycle bifurcations by perturbing piecewise Hamiltonian systems with a nonregular switching line via multiple parameters

Abstract

<p style='text-indent:20px;'>In this paper, we investigate limit cycle bifurcations by perturbing planar piecewise Hamiltonian systems with a switching line <inline-formula><tex-math id="M1">\begin{document}$ \left\{(x,y): y = \pm kx, k\right. $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ \left.\in(0,+\infty), x\geqslant0\right\} $\end{document}</tex-math></inline-formula> via multiple parameters. With the help of Han and Xiong [<xref ref-type="bibr" rid="b3">3</xref>], Han and Liu [<xref ref-type="bibr" rid="b5">5</xref>] and Xiong [<xref ref-type="bibr" rid="b18">18</xref>], we obtain the second and third terms in expansions of the first order Melnikov function. As an application, we consider limit cycle bifurcations of a piecewise near-Hamiltonian system and prove that the system has four limit cycles.</p>