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On the integrability of a tritrophic food chain model
https://ddd.uab.cat/record/226091
In this paper we work with a vastly analyzed tritrophic food chain model. We provide a complete characterization of their Darboux polynomials and of their exponential factors. We also show the non-existence of polynomial first integrals, of rational first integrals, of local analytic first integrals in a neighborhood of the origin, of first integrals that can be described by formal series and of Darboux first integrals. Llibre, JaumeFri, 03 Jul 2020 10:15:45 GMThttps://ddd.uab.cat/record/2260912010Polynomial solutions of equivariant polynomial Abel differential equations
https://ddd.uab.cat/record/221349
Let a(x) be non-constant and let bj(x), for j = 0, 1, 2, 3, be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation a(x)y = b(x)y + b(x)y, with b(x) =/ 0, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a(x)yy = b0(x) + b(x)y, with b(x) =/ 0, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions. Llibre, JaumeWed, 15 Apr 2020 15:23:09 GMThttps://ddd.uab.cat/record/2213492018The cubic polynomial differential systems with two circles as algebraic limit cycles
https://ddd.uab.cat/record/199332
In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles. Giné, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993322018Limit cycles coming from some uniform isochronous centers
https://ddd.uab.cat/record/169459
This article is about the weak 16--th Hilbert problem, i. e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers \[ x= -y x^2 y (x^2 y^2)^n, y= x x y^2 (x^2 y^2)^n, \] of degree 2n 3 and we perturb them inside the class of all polynomial differential systems of degree 2n 3. For n=0,1 we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n=2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center. Liang, HaihuaMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694592016Liouvillian first integrals for generalized Liénard polynomial differential systems
https://ddd.uab.cat/record/150587
We study the Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form x' = y, y' = −g(x) − f(x)y, where g(x) and f(x) are arbitrary polynomials such that 2 ≤ deg g ≤ deg f. Llibre, JaumeFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1505872013Limit cycles bifurcating from a 2-dimensional isochronous torus in R^3
https://ddd.uab.cat/record/150463
In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in ℝ3, when we perturb it inside a class of differential systems. This method is based in the averaging theory. As far as we know all applications of this method have been made perturbing noncompact surfaces, as for instance a plane or a cylinder in ℝ3. Here we consider polynomial perturbations of degree d of an isochronous torus. We prove that, up to first order in the perturbation, at most 2(d+1) limit cycles can bifurcate from a such torus and that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate from such a torus for every ν ∈ {2, 4,. . . ,2(d + 1)}. Llibre, JaumeFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504632011Liouvillian first integrals for generalized Riccati polynomial differential systems
https://ddd.uab.cat/record/145360
We study the existence and non-existence of Liouvillian first integrals for the generalized Riccati polynomial differential systems of the form x'=y, y'= a(x)y^2 b(x)y c(x), where a(x), b(x) and c(x) are polynomials. Llibre, JaumeTue, 12 Jan 2016 16:38:11 GMThttps://ddd.uab.cat/record/1453602015