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.

Limit Cycles on Piecewise Smooth Vector Fields with Coupled Rigid Centers

We provide an upper bound for the maximum number of limit cycles of the class of discontinuous piecewise differential systems formed by two differential systems separated by a straight line presenting rigid centers. These two rigid centers are polynomial differential systems with a linear part and a nonlinear homogeneous part. We study the maximum number of limit cycles that such a class of piecewise differential systems can exhibit.

Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

Integrability and Limit Cycles via First Integrals

In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

Existence of Limit Cycles for a Class of Quartic Polynomial Differential System Depending on Parameters

We studied the existence of limit cycles for the quartic polynomial differential systems depending on parameters. To prove that, first, we used the formal series method based on Poincare’ ideas to determine the center-focus. Then, by the Hopf bifurcation theory, we obtained the sufficient condition for the existence of the limit cycles. Finally, we provided some numerical examples for illustration.

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1, 1)SN - (B)

This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, [Formula: see text] phase portraits possessing a finite saddle-node as the only finite singularity and [Formula: see text] phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family [Formula: see text] was reported in [Artés et al., 2020b] where the authors obtained [Formula: see text] topologically distinct phase portraits for systems in the closure [Formula: see text]. In this paper, we provide the complete study of the geometry of family [Formula: see text]. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.