127 research outputs found
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 matemà tic francès Henri Poincaré recollida per a una sessió de l'assignatura de lliure elecció "La Matemà tica 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
- …