Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries
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.
scientific article; zbMATH DE number 7623989
scientific article