On the Integrability of Quasihomogeneous Systems and Quasidegenerate Infinity Systems

Abstract

The integrability of quasihomogeneous systems is considered, and the properties of the first integrals and the inverse integrating factors of such systems are shown. By solving the systems of ordinary differential equations which are established by using the vector fields of the quasihomogeneous systems, one can obtain an inverse integrating factor of the systems. Moreover, the integrability of a class of systems (quasidegenerate infinity systems) which generalize the so-called degenerate infinity vector fields is considered, and a method how to obtain an inverse integrating factor of the systems from the first integrals of the corresponding quasihomogeneous systems is shown.

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

“…The quasi-homogeneous polynomial differential systems have also gained wide attention since the beginning of the 21st century. The result obtained mainly include the integrability [3,4,16,20,21,27,31], polynomial and rational first integrability [2,9], normal forms [4], centers [1,5,6,35,42,43] and limit cycles [22,23,25,29]. Recently, the authors of [20] establish an algorithm for obtaining all the quasi-homogeneous but non-homogeneous polynomial systems with a given degree.…”

Global dynamics of planar quasi-homogeneous differential systems

“…For a quasi-homogeneous polynomial differential system (1), a weight vector w = ( s 1 , s 2 ,d) is minimal for system (1) if any other weight vector (s 1 , s 2 , d) of system (1) satisfiess 1 ≤ s 1 ,s 2 ≤ s 2 andd ≤ d. Clearly each quasihomogeneous polynomial differential system has a unique minimal weight vector. When s 1 = s 2 = 1, system (1) is a homogeneous one of degree d. Quasi-homogeneous polynomial differential systems have been intensively investigated by many different authors from integrability point of view, see for example [2,14,15,16] and the references therein. It is well known that all planar quasi-homogeneous vector fields are Liouvillian integrable, see e.g.…”

“…Quasi-homogeneous polynomial differential systems have been intensively investigated by many different authors from integrability point of view, see for example [2,14,15,16] and the references therein. It is well known that all planar quasi-homogeneous vector fields are Liouvillian integrable, see e.g.…”

Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

“…Smooth Quasi-homogeneous polynomial differential systems have been intensively studied by a great deal of authors from different views. We refer readers to see for example the integrability [2,17,19,21,29], the centers and limit cycles [1,15,18,24], the algorithm to compute quasi-homogeneous systems with a given degree [14], the characterization of centers or topological phase portraits for quasi-homogeneous equations of degrees 3-5 respectively [5,26,32] and the references therein.…”