Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x′=y, y′=−fm(x)y−gn(x), where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated.

The method of Puiseux series and invariant algebraic curves

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems [Formula: see text], [Formula: see text] with [Formula: see text]. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with [Formula: see text] can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family [Formula: see text] of the Chinese classification, that is, of differential system [Formula: see text] with [Formula: see text]. In addition, we provide a complete and correct proof of the nonexistence of algebraic limit cycles for these equations.

On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

We present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

Limit Cycles and Invariant Curves in a Class of Switching Systems with Degree Four

In this paper, a class of switching systems which have an invariant conic x2+cy2=1,c∈R, is investigated. Half attracting invariant conic x2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved.

Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers

In the integrability of polynomial differential systems it is well known that the invariant algebraic curves play a relevant role. Here we will see that they can also play an important role with respect to limit cycles.In this paper, we study quadratic polynomial systems with an algebraic periodic orbit of degree [Formula: see text] surrounding a center. We show that there exists only one family of such systems satisfying that an algebraic limit cycle of degree [Formula: see text] can bifurcate from the period annulus of the mentioned center under quadratic perturbations.