Planar Semi-quasi Homogeneous Polynomial Differential Systems with a Given Degree

Abstract

This paper study the planar semi-quasi homogeneous polynomial differential systems (short for PSQHPDS), which can be regard as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. Afterward, we apply this algorithm to research the center problem of quadratic and cubic PSQHPDS. It is proved that the quadratic one hasn't center, and, that the cubic one has center if and only if it can be written asẋ = x 2 − y 3 ,ẏ = x after a linear transformation of coordinate and a rescaling of time.

Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems

“…The canonical forms of planar quasi-homogeneous systems of degree 2, 3, 4, 5 have been obtained by several authors, see [15] and the references therein. Very recently, the authors in [16] provided the canonical forms of CSQH.…”

“…Lemma 2.6. ( [16]) After appropriate invertible linear transformations, any planar cubic semi-quasi-homogeneous but non-semi-homogeneous systems with the minimal weight vector w m , can be written as one of the following forms where…”

“…In Lemma 2.6, the terminology of minimal weight vector is used, which was first introduced for the quasi-homogeneous systems in [10] and then generalized to the semi-quasi-homogeneous systems in [16]. A weight vector w…”

“…(iii) If s 1 = s 2 and d 1 = d 2 , system (1) is a quasi-homogeneous one (or weight homogeneous one) with the weight vector (s 1 , s 2 , d 1 ) [1,2,5,10,13,14]. (iv) If s 1 = s 2 and d 1 = d 2 , system (1) is a semi-quasi-homogeneous one with the weight vector (s 1 , s 2 , d 1 , d 2 ) [16,18].…”

“…But as far as we know, there is no a published paper completing the classification of global dynamics of CQH, although this problem was solved for planar quasihomogeneous polynomial systems of degree four and five in [12] and [14], respectively. CSQH: To the best of our knowledge, there is no a published paper concerning the global dynamics and the limit cycles of CSQH, except the one in [16] on its canonical forms.…”