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A function I(x, y, t) constant on the solutions of a differential system in R 2 is called an invariant. We classify all planar quadratic differential systems having invariants of the form I (x , y , t) = a x 2 + b x y + c y 2 + d x + e y + c 1 t with c 1 ≠ 0 . There are 13 different families of quadratic systems having invariants of this form. As far as we know this is the first time that quadratic differential systems having an invariant different from a Darboux invariant have been classified [ABSTRACT FROM AUTHOR]

In this paper, we classify the phase portraits of an SIR epidemic dynamics model. Depending on the values of the parameters, this model can exhibit seven different phase portraits. In particular, from a biological point of view we prove that the unique attractors of this model are one or two equilibrium points depending on the values of the parameters and from the phase portraits follow the basins of attraction of these equilibria. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and non-existence of limit cycles for the class of polynomial differential systems of the form x ̇ = λ x + P n (x , y) , y ̇ = μ y + Q n (x , y) , where P n and Q n are homogeneous polynomials of degree n. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the integrability and the non-existence of periodic orbits for the 2-dimensional Kolmogorov systems of the form where n and m are positive integers and and are homogeneous polynomials of degree and m, respectively. [Copyright &y& Elsevier]

We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differential systems separated by the straight-line x = 0 having in x > 0 the general quadratic isochronous center ẋ = − y + x 2 , ẏ = x (1 + y) after an affine transformation, and in x < 0 an arbitrary quadratic isochronous center except for the quadratic isochronous center ẋ = − y + x 2 − y 2 , ẏ = x (1 + 2 y) which has been studied in [Ghermoul et al., 2021]. For these piecewise-continuous differential systems the upper bound of crossing limit cycles is 2, and there are specific examples having one crossing limit cycle. [ABSTRACT FROM AUTHOR]

First, we study the planar continuous piecewise differential systems separated by the straight line x = 0 formed by a linear isochronous center in x > 0 and an isochronous quadratic center in x < 0. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line x = 0 having in x > 0 the general quadratic isochronous center ẋ = − y + x 2 − y 2 , ẏ = x (1 + 2 y) after an affine transformation, and in x < 0 an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise differential systems we have solved the extension of the 16th Hilbert problem. [ABSTRACT FROM AUTHOR]

Abstract: We consider the class of polynomial differential equations , , in where and are homogeneous polynomials of degree and , i.e. the class of polynomial differential systems with homogeneous nonlinearities with a star node at the origin. We prove that these systems are Darboux integrable. Moreover, for these systems we study the existence and non-existence of limit cycles surrounding the equilibrium point located at the origin. [Copyright &y& Elsevier]