On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems

Abstract

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems.

Keywords

• discontinuous piecewise differential system
• continuous piecewise differential system
• first integral
• non-algebraic limit cycle
• linear system
• quadratic center