Antoni Ferragut

A survey on the blow up technique

The blow up technique is widely used in desingularization of degenerate singular points of planar vector fields. In this survey, we give an overview of the different types of blow up and we illustrate them with many examples for better understanding. Moreover, we introduce a new generalization of the classical blow up.

PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE INTEGRATING FACTOR

We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse integrating factor.

POLYNOMIAL VECTOR FIELDS IN R 3 WITH INFINITELY MANY LIMIT CYCLES

We provide a constructive method to obtain polynomial vector fields in ℝ3 having infinitely many limit cycles starting from polynomial vector fields in ℝ2 with a period annulus. We present two exam...

HYPERBOLIC PERIODIC ORBITS FROM THE BIFURCATION OF A FOUR-DIMENSIONAL NONLINEAR CENTER

We study the bifurcation of hyperbolic periodic orbits from a four-dimensional nonlinear center in a class of differential systems. The tool for proving these results is the averaging theory.

Phase portraits of Abel quadratic differential systems of the second kind

ABSTRACT We provide normal forms and the global phase portraits on the Poincaré disk of some Abel quadratic differential equations of the second kind. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

Analytic integrability of the Bianchi Class A cosmological models with 0 k < 1

Many works study the integrability of the Bianchi class A cosmologies with k = 1, where k is the ratio between the pressure and the energy density of the matter. Here we characterize the analytic integrability of the Bianchi class A cosmological models when 0 ⩽ k < 1. We conclude that Bianchi types VI0, VII0, VIII and IX can exhibit chaos whereas Bianchi type I is not chaotic and Bianchi type II is at most partially chaotic.

Phase portraits of Abel quadratic differential systems of second kind with symmetries

ABSTRACT We provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

Conservation Laws in Biochemical Reaction Networks

We study the existence of linear and nonlinear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The nonlinear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using the Darboux theory of integrability, we provide necessary structural (i.e., parameter-independent) conditions on a reaction network to guarantee the existence of nonlinear conservation laws of a certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that nonlinear first integrals only exist under the same structural condition (as in the case of the Lotka--Volterra system). We finally show that the existence of such a first integral generally implies that the ...

Cofactors and equilibria for polynomial vector fields

We provide a relationship between the existence of equilibrium points of dif- ferential systems and the cofactors of invariant algebraic curves and exponential factors of the system.

On the absence of analytic integrability of the Bianchi Class B cosmological models

We follow Bogoyavlenskys approach to deal with Bianchi class B cosmological models. We characterize the analytic integrability of such systems.