Limit cycles in an $ m $-piecewise discontinuous polynomial differential system

Abstract

<abstract><p>In this paper, I study a planar $ m $-piecewise discontinuous polynomial differential system $ \dot{x} = y, \dot{y} = -x-\varepsilon(f(x, y)+g_m(x, y)h(x)) $, which has a linear center in each zone partitioned by those switching lines, where $ f(x, y) = \sum_{i+j = 0}^na_{ij}x^iy^j $, $ h(x) = \sum_{j = 0}^lb_jx^j, a_{ij}, b_j\in\mathbb{R}, n, l\in\mathbb{N} $, and $ g_m(x, y) $ with the positive even number $ m $ as the union of $ m/2 $ different straight lines passing through the origin of coordinates dividing the plane into sectors of angle $ 2\pi/m $. Using the averaging theory, I provide the lower bound $ L_m(n, l) $ for the maximun number of limit cycles, which bifurcates which bifurcating from the annulus of the origin of this system.</p></abstract>