Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries
type: Item
instance of: scholarly article
title: Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries
author: Tao Li
author: Jaume Llibre
published in: Journal of Differential Equations
publication date: 2022-11-28T00:00:00Z
review text: In this paper, the authors investigate the number of crossing limit cycles for a class of planar piecewise polynomial Hamiltonian systems of degree \(n\) separated by the curve \(y= x^m\), where \(m\) and \(n\) are positive integers. The system under consideration has the following form: \[ \left(\dot{x}, \dot{y}\right)=\begin{cases} \left(-\partial_y H^+(x, y), \partial_x H^+(x, y)\right), & y> x^m, \\ \left(-\partial_y H^-(x, y), \partial_x H^-(x, y)\right), & y<x^m, \end{cases} \] where \(H^+(x, y)\) and \(H^-(x, y)\) are real polynomials of degree \(n+1\). Let \(\mathcal{H}(m, n)\) be the maximum number of crossing limit cycles of this system. The authors prove that: \[ \mathcal{H}(m, n)\leq \sum_{p=1}^{q}\left[\frac{(mn+m-1)^{2p}}{2p}\right], \] where \(q=\left[(mn+m)/2\right]\), \([\cdot]\) denotes the integer part function. In particular, \(\mathcal{H}(1, 2)\leq 1\). The authors also investigate how to obtain a lower bound of \(\mathcal{H}(m, n)\) by perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this purpose they first present a complete classification of the center conditions of the system with \(n=1\). Then they develop an arbitrary order Melnikov-like function for general \(d\)-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. By using the newly obtained Melnikov-like function, the authors obtain a lower bound and an upper bound for the maximum number of crossing limit cycles bifurcating from the unperturbed periodic orbits up to any order.
reviewed by: Zhengdong Du
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37J46
zbMATH DE Number: 7623989
zbMATH Keywords: bifurcation theory
zbMATH Keywords: limit cycle
zbMATH Keywords: nonlinear switching boundary
zbMATH Keywords: non-smooth center
zbMATH Keywords: piecewise Hamiltonian system
MaRDI profile type: Publication
full work available at URL: j.jde.2022.11.007
OpenAlex ID: W4309683868
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DOI: 10.1016/J.JDE.2022.11.007
description: scientific article; zbMATH DE number 7623989