The center problem on piecewise smooth differential systems with two zones

Abstract

In this paper we shall study $ \Sigma $-center problem on piecewise smooth systems with two zones divided by a straight line in the real plane. We first give some necessary and sufficient conditions for the crossing $ \Sigma $-center by taking advantage of the forms of the first integrals. Then for applications, we completely describe the $ \Sigma $-centers from both the algebra and geometry points of view when the both of two sub-systems are linear.