Cyclicity of a Class of Hamiltonian Systems under Perturbations of Piecewise Smooth Polynomials

In the study of the weakened Hilbert’s 16th problem, there usually appears period annulus surrounding more than one singular point, implying that there are often more than three generate functions in the Abelian integrals (or the first order Melnikov functions). It is also known that there are usually more than three generate functions in the study of the bifurcation of limit cycles in piecewise smooth Hamiltonian systems. This paper aims to explore a method so that we can estimate the upper bound of limit cycles bifurcating from the period annulus if the unperturbed system has some good properties such as Picard–Fuchs equations. This method is based on [Gavrilov & Iliev, 2009] and is useful for smooth and piecewise smooth polynomial vector fields of a general degree. Finally, we present an example to illustrate an application of the theoretical results.