Invariant algebraic curves of large degree for quadratic system

Abstract

In this paper we present for the first time examples of algebraic limit cycles and saddle loops of degree greater than 4 for planar quadratic systems. In particular, we give examples of algebraic limit cycles of degree 5 and 6, and algebraic saddle loops of degree 3 and 5 surrounding a strong focus. We also give an example of an invariant algebraic curve of degree 12 for which the quadratic system has no Darboux integrating factors or first integrals.

On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity

“…The second class was found in [16]. More recently, two new classes have been found and in [9] the authors proved that there are no other algebraic limit cycles of degree 4 for quadratic vector fields. The uniqueness of these limit cycles was proved in [11].…”

“…The uniqueness of these limit cycles was proved in [11]. It is known that there are quadratic polynomial differential systems having algebraic limit cycles of degree 5 and 6, see [9], and that this limit cycle is the unique one for these differential systems. Other results on algebraic limit cycles can be found in [20].…”

On some open problems in planar differential systems and Hilbert’s 16th problem

“…Chavarriga et al [29] have found all the cases of algebraic limit cycles of degree 4 for quadratic polynomial systems and Chavarriga, Giacomini and Llibre [16] have proved their uniqueness. Christopher et al [38] have given examples of quadratic systems with an algebraic limit cycle of degree 5 and 6. The fact that these limit cycles are hyperbolic is proved in [64].…”

Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems

“…It is shown in [9] that there are no algebraic limit cycles of degree 3 for a quadratic system. In [13], two examples of quadratic systems with an algebraic limit cycle of degree 5 and 6 are described. We will show that none of these quadratic systems has a Liouvillian first integral.…”

“…In a work due to C. Christopher, J. Llibre and G.Świrszcz [13] two families of quadratic systems with an algebraic limit cycle of degrees five and six, respectively, are given. These two families are constructed by means of a birrational transformation of system (21).…”