On the center criterion of planar quasi-homogeneous polynomial differential systems

Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems

“…In this paper we are interested in the centers of the discontinuous piecewise smooth quasi-homogeneous polynomial differential systems. It is known that the quasi-homogeneous polynomial differential systems of even degree have no centers, see [3,19,31,34]. However, as we shall see the discontinuous piecewise smooth quasi-homogeneous polynomial differential systems of even degree can have centers.…”

“…The centers of the quasi-homogeneous polynomial differential systems have been classified, and all are global centers see [15] and [19]. The normal forms of the quasi-homogeneous polynomial differential systems of degree n having a center at the origin were obtained in [34].…”

“…where D = (p − 1)ς + (n − 1)κ and s = gcd(ς, κ). Furthermore, the integers (ς + κ)/s and κ/s are coprime and at least one of them is odd, see [32,34]. From the algorithm described in subsection 3.1 of [13] the quasi-homogeneous but non-homogeneous polynomial differential system (1) of degree n with weight vector (s 1 , s 2 , m) can be written as…”