Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

Abstract

Abstract. This paper deals with the existence of Darboux first integrals for the planar polynomial differential systemsẋ = λx − y + P n+1 (x, y) + xF 2n (x, y),ẏ = x + λy + Q n+1 (x, y)+yF 2n (x, y), where P i (x, y), Q i (x, y) and F i (x, y) are homogeneous polynomials of degree i. Inside this class we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 we present necessary and sufficient conditions for being Darboux integrable.

Bifurcation analysis and integrability in the segmented disc dynamo with mechanical friction

“…In the theory of polynomial differential equations, Darboux and Liouvillian integrals act a crucial role. Here we only study the Darboux integral of equations (1.1) by applying the theory of algebraic invariant surfaces, exponential factors, and Darboux polynomials (see [17][18][19][20][21][22][23][24][25][26][27][28]). We make further efforts to learn the behavior and geometry structure of equations (1.1) by studying their integrability.…”

“…for some polynomial L h , which is called the cofactor of h(x, y, z) [17][18][19][20][21][22][23][24][25][26][27][28]. If h(x, y, z) is a Darboux polynomial, then the surface h(x, y, z) = 0 is an invariant algebraic surface [18-21, 25, 28], which means that if an orbit of system (1.1) has a point on the surface h(x, y, z) = 0, then the whole orbit is contained in it.…”