On the limit cycles of the piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center.

While the limit cycles of the discontinuous piecewise differential systems formed by two linear differential systems separated by one straight line have been studied intensively, and up to now there are examples of these systems with at most 3 limit cycles. There are almost no works studying the limit cycles of the discontinuous piecewise differential systems formed by one linear differential system and a quadratic polynomial differential system separated by one straight line. In this paper using the averaging theory up to seven order we prove that the discontinuous piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center separated by one straight line can have 8 limit cycles. More precisely, at every order of the averaging theory from order one to order seven we provide the maximum number of limit cycles that can be obtained using the averaging theory. Primary 34C05, 34A34. • The purpose of this paper is the study of a piecewise differential system formed by a linear focus or center and a quadratic weak focus separated by the straight-line x = 0. • Study the maximum number of limit cycles of this discontinuous piecewise differential system. • Using averaging theory we prove that this piecewise differential system can have at most eight limit cycles. [ABSTRACT FROM AUTHOR]