Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries

This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems of degree nwith the switching boundary y= x(m), where mand nare positive integers. We answer a version of the 16th Hilbert problem for such systems, providing an upper bound for the maximum number of limit cycles in the function of mand n. We also are devoted to giving a lower bound via perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this, a complete classification of the center conditions is established. In pursuit of a better lower bound, we require that this global center is nonlinear induced by the piecewise linearity, instead of the normal linear differential center considered in most of the existing articles. This renders that the traveling time of the unperturbed periodic orbits in each smooth zone is not calculable explicitly. Thus it is difficult to use the known Melnikov functions and averaged functions to study the bifurcating limit cycles. To overcome this difficulty, we develop an arbitrary order Melnikov-like function, which does not depend on the traveling time, for general d-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. Finally, employing the new Melnikovlike function to the considered perturbation problem, we obtain a lower bound and an upper bound for the maximum number of limit cycles bifurcating from the unperturbed periodic orbits up to any order. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems with switching boundaries. It provides an upper bound for the maximum number of limit cycles in terms of positive integers m and n. The paper also establishes a lower bound by perturbing piecewise linear Hamiltonian systems and classifies the center conditions. The importance of this paper is rated 9 out of 10.