A note on "Relaxation Oscillators with Exact Limit Cycles"

In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work of M.A. Abdelkader entitled ``Relaxation Oscillators with Exact Limit Cycles'', which appeared in J. Math. Anal. Appl. 218 (1998), 308--312. The result given in this note may be used to construct models of Li\'enard differential equations exhibiting a desired limit cycle.Comment: 8 pages, no figure

Transversal conics and the existence of limit cycles

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\'e--Bendixson regions by using transversal conics. We present several examples of known systems in the literature showing different features about limit cycles: hyperbolicity, Hopf bifurcation, sky-blue bifurcation, rotated vector fields, \ldots for which the obtained Poincar\'e--Bendixson region allows to locate the limit cycles. Our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles.Comment: 28 pages; 20 figure

Tres apunts sobre Henri Poincaré

Aquest article es basa en la informació sobre el matemàtic francès Henri Poincaré recollida per a una sessió de l'assignatura de lliure elecció "La Matemàtica i els seus personatges", impartida durant el primer quadrimestre del curs 2008-09 a la Universitat de Lleida. Els tres temes que descriurem i en els que Henri Poincaré va fer aportacions remarcables són: un model de la geometria no Euclidiana, el descobriment del caos, la conjectura de Poincaré

Integrability of planar polynomial differential systems through linear differential equations

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.Comment: 24 pages, no figure

The role of algebraic solutions in planar polynomial differential systems

We study a planar polynomial differential system, given by \dot{x}=P(x,y), \dot{y}=Q(x,y). We consider a function I(x,y)=\exp \{h_2(x) A_1(x,y) \diagup A_0(x,y) \} h_1(x) \prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}, where g_i(x) are algebraic functions, A_1(x,y)=\prod_{k=1}^r (y-a_k(x)), A_0(x,y)=\prod_{j=1}^s (y-\tilde{g}_j(x)) with a_k(x) and \tilde{g}_j(x) algebraic functions, A_0 and A_1 do not share any common factor, h_2(x) is a rational function, h(x) and h_1(x) are functions with a rational logarithmic derivative and \alpha_i are complex numbers. We show that if I(x,y) is a first integral or an integrating factor, then I(x,y) is a Darboux function. In order to prove this result, we show that if g(x) is such that there exists an irreducible polynomial f(x,y) with f(x,g(x)) \equiv 0, then f(x,y)=0 is an invariant algebraic curve of the system. In relation with this fact, we give some characteristics related to particular solutions and functions of the form I(x,y) such as the structure of their cofactor. Moreover, we consider a function of the form \Phi(x,y):= \exp \{h_2(x) A_1(x,y) / A_0 (x,y) \}. We show that if the derivative of \Phi(x,y) with respect to the flow is well defined over A_0(x,y)=0 then \Phi(x,y) gives rise to an exponential factor.Comment: 28 pages, no figure

On the cyclicity of weight-homogeneous centers

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.Comment: 13 pages, no figure