A new result on averaging theory for a class of discontinuous planar differential systems with applications
type: Item
instance of: scholarly article
title: A new result on averaging theory for a class of discontinuous planar differential systems with applications
zbMATH Open document ID: 1386.34081
DOI: 10.4171/RMI/970
author: Jackson Itikawa
author: Jaume Llibre
author: Douglas D. Novaes
published in: Revista Matemática Iberoamericana
publication date: 2018-01-04T00:00:00Z
review text: Summary: We develop the averaging theory at any order for computing the periodic solutions of periodic discontinuous piecewise differential system of the form \[ \frac{dr}{d\theta}=r'=\begin{cases} F^+(\theta,r,\varepsilon) \quad \text{if} \quad 0 \leq \theta \leq \alpha,\\ F^-(\theta,r,\varepsilon) \quad \text{if} \quad \alpha \leq \theta \leq 2\pi,\end{cases} \] where \(F^{\pm}(\theta,r,\epsilon)=\sum_{i=1}^k\epsilon^i F_i^{\pm}(\theta,r)+\epsilon^{k+1} R^{\pm}(\theta,r,\epsilon)\) with \(\theta \in \mathbb S^1\) and \(r\in D\), where D is an open interval of \(\mathbb R^+\), and \(\epsilon\) is a small real parameter. Applying this theory, we provide lower bounds for the maximum number of limit cycles that bifurcate from the origin of quartic polynomial differential systems of the form \(\dot{x}=-y+xp(x,y)\), \(\dot{y}=x+yp(x,y)\), with \(p(x,y)\) a polynomial of degree \(3\) without constant term, when they are perturbed, either inside the class of all continuous quartic polynomial differential systems, or inside the class of all discontinuous piecewise quartic polynomial differential systems with two zones separated by the straight line \(y=0\).
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 6821830
zbMATH Keywords: periodic solution
zbMATH Keywords: averaging method
zbMATH Keywords: non-smooth differential system
zbMATH Keywords: discontinuous differential system
zbMATH Keywords: uniform isochronous center
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6821830