Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve

Autor: Andrade, Kamila da S., Cespedes, Oscar A. R., Cruz, Dayane R., Novaes, Douglas D.
Rok vydání: 2020
Předmět:
Zdroj: Journal of Differential Equations 287 (2021) 1-36
Druh dokumentu: Working Paper
DOI: 10.1016/j.jde.2021.03.039
Popis: In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)\geq 4,$ $H(3)\geq 8,$ $H(n)\geq7,$ for $n\geq 4$ even, and $H(n)\geq 9,$ for $n\geq 5$ odd. This improves all the previous results for $n\geq2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.
Databáze: arXiv