# A new result on averaging theory for a class of discontinuous planar differential systems with applications

### Jackson Itikawa

Universidade de São Paulo, São Carlos, Brazil### Jaume Llibre

Universitat Autònoma de Barcelona, Bellaterra, Spain### Douglas Duarte Novaes

Universidade Estadual de Campinas, Brazil

## Abstract

We develop the averaging theory at any order for computing the periodic solutions of periodic discontinuous piecewise differential system of the form

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 $\dot{x}=-y+xp(x,y),\quad\dot{y}=x+yp(x,y),$ 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 $y=0$.

## Cite this article

Jackson Itikawa, Jaume Llibre, Douglas Duarte Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications. Rev. Mat. Iberoam. 33 (2017), no. 4, pp. 1247–1265

DOI 10.4171/RMI/970