Showing papers in "Publicacions Matematiques in 1997"
TL;DR: In this paper, the authors associate several residue-type indices to a singular point of a two-dimensional holomorphic vector field and establish some relations between them, especially when the singular point is a generalized curve and the separatrix is the maximal one.
Abstract: One can associate several residue-type indices to a singular point of a two-dimensional holomorphic vector field. Some of these indices depend also on the choice of a separatrix at the singular point. We establish some relations between them, especially when the singular point is a generalized curve and the separatrix is the maximal one. These local results have global consequences, for example concerning the construction of logarithmic forms defining a given holomorphic foliation.
64 citations
TL;DR: In this paper, it was shown that the asymptotic stability of the positive equilibrium of the Holling-Tanner model does not imply that it is globally stable, and the main tool used is the computation of Poincare-Lyapunov constants in case a weak focus occurs.
Abstract: This paper deals with the following question: does the asymptotic stability of the positive equilibrium of the Holling-Tanner model imply it is also globally stable? We will show that the answer to this question is negative. The main tool we use is the computation of Poincare-Lyapunov constants in case a weak focus occurs. In this way we are able to construct an example with two limit cycles.
58 citations
TL;DR: The integrability of two-dimensional autonomous systems in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree was studied in this article.
Abstract: In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.
58 citations
TL;DR: In this article, a continuous planar piecewise linear system with two linear zones is considered, and several bifurcations giving rise to limit cycles are pointed out, and a technique for analysis of such systems is developed.
Abstract: Continuous planar piecewise linear systems with two linear zones are considered. Due to their low differentiability specific techniques of analysis must be developed. Several bifurcations giving rise to limit cycles are pointed out.
35 citations
TL;DR: It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group as mentioned in this paper, which is the subgroup of the smallest simple group.
Abstract: It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.
24 citations
TL;DR: In this paper, a genus formula for the Poincare problem on algebraic surfaces with rational first-integral first integral has been given, and it is shown that the general solution is a non-singular curve.
Abstract: Given a foliation $\Cal F$ in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case $S=\Bbb P^2$ some new counter-examples to the classic formulation of the Poincare problem are presented. If $S$ is a rational surface and $\Cal F$ has singularities of type $(1,1)$ or $(1,-1)$ we prove that the general solution is a non-singular curve.
23 citations
TL;DR: In this article, a singular integral operator with rough kernel was studied, and it was shown that it is a bounded operator in the space Ω(R^n) and this bound is independent of the coefficients of the operator.
Abstract: In this paper we study a singular integral operator $T$ with rough kernel. This operator has singularity along sets of the form $\{x=Q(|y|)y'\}$, where $Q(t)$ is a polynomial satisfying $Q(0)=0$. We prove that $T$ is a bounded operator in the space $L^2(R^n)$, $n\ge 2$, and this bound is independent of the coefficients of $Q(t)$.
We also obtain certain Hardy type inequalities related to this operator.
22 citations
TL;DR: In this paper, the authors resolve the center-focus problem for a specific class of cubic systems and determine the number of limit cycles which can bifurcate from a fine focus.
Abstract: We resolve the centre-focus problem for a specific class of cubic systems and determine the number of limit cycles which can bifurcate from a fine focus. We also describe the methods which we have developed to investigate these questions in general. These involve extensive use of Computer Algebra; we have chosen to use REDUCE.
21 citations
TL;DR: In this article, the authors study phase portraits of quadratic vector fields with a weak focus of third order at the origin and show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields.
Abstract: We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle.
19 citations
TL;DR: In this paper, the first step of the successive derivatives algorithm is described, which gives also the first derivative of the period function, as well as new presentations of formulas obtained by Carmen Chicone and Marc Jacobs.
Abstract: The algorithm of the successive derivatives introduced in \cite{5} was implemented in \cite{7}, \cite{8}. This algorithm is based on the existence of a decomposition of 1-forms associated to the relative cohomology of the Hamiltonian function which is perturbed. We explain here how the first step of this algorithm gives also the first derivative of the period function. This includes, for instance, new presentations of formulas obtained by Carmen Chicone and Marc Jacobs in \cite{3}.
18 citations
TL;DR: In this article, a generalization of the classical Darboux method is presented to generate integrable systems with a given null divergence factor, which plays a fundamental role in the problem of the center and in the determination of the limit cycles.
Abstract: Let $(P, Q)$ be a $C^{1}$ vector field defined in a open subset $U \subset R^{2}$. We call a null divergence factor a $C^{1}$ solution $V(x, y)$ of the equation $P \frac{\partial V}{\partial x} + Q \frac{\partial V}{\partial y} = \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right) \, V$. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method to generate integrable systems.
TL;DR: In this paper, it was shown that the interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the $C 1$ interior of all the full-expansions of the manifold.
Abstract: In this paper, we show that the $C^1$ interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the $C^1$ interior of the set of all expansive diffeomorphisms. And the $C^1$ interior of the set of all continuum-wise fully expansive diffeomorphisms on a surface is investigated. Furthermore, we have necessary and sufficient conditions for a diffeomorphism belonging to these open sets to be Anosov.
TL;DR: In this article, an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R n and an affine hyperplane is given.
Abstract: We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R n and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated.
TL;DR: In this article, it was shown that the ring of Gevrey power series is a noetherian ring, and a spectral synthesis theorem for some classes of ultradifferentiable functions.
Abstract: Given a subring of the ring of formal power series defined by the growth of the coefficients, we prove a necessary and sufficient condition for it to be a noetherian ring. As a particular case, we show that the ring of Gevrey power series is a noetherian ring. Then, we get a spectral synthesis theorem for some classes of ultradifferentiable functions.
TL;DR: In this article, the authors answer the question posed by Ian Stewart which Padovan numbers are at the same time as Fibonacci numbers, and give a result on the difference between Padovan and fibonacci number and on the growth of Padovan number with negative indices.
Abstract: We answer the question posed by Ian Stewart which Padovan numbers are at the same time Fibonacci numbers. We give a result on the difference between Padovan and Fibonacci numbers, and on the growth of Padovan numbers with negative indices.
TL;DR: In this article, the authors study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Holder inequalities.
Abstract: We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Holder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1,1).
TL;DR: The Caratheodory's Conjecture is equivalent to the Loewner's conjecture for planar vector fields as mentioned in this paper, which states that there are no umbilics of index bigger than one.
Abstract: Let $r = 3, 4, \ldots, \infty, \omega$. The $C^r$-Caratheodory's Conjecture states that every $C^r$ convex embedding of a 2-sphere into ${\Bbb R}^{3}$ must have at least two umbilics. The $C^r$-Loewner's conjecture (stronger than the one of Caratheodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if $r
e \omega$, $C^r$-Caratheodory's Conjecture is equivalent to the following one:
Let $\rho >0$ and $\beta : U \subset \Bbb R^2 \to \Bbb R$, be of class $C^r$, where $U$ is a neighborhood of the compact disc $D(0, \rho) \subset \Bbb R^2$ of radius $\rho$ centered at $0$. If $\beta$ restricted to a neighborhood of the circle $\partial D(0, \rho)$ has the form $\beta(x, y) = (a x^2 + b y^2)/( x^2 + y^2)$, where $a < b < 0$, then the vector field (defined in $U$) that takes $(x, y)$ to $(\beta_{xx}(x,y) - \beta_{yy}(x,y), 2 \beta_{xy}(x,y) )$ has at least two singularities in $D(0, \rho)$.
TL;DR: In this article, a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus Yamabe Question in dimension $n\ge 3$ is given.
Abstract: In the paper \cite{CEGHM} a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension $n\ge 3$ are given. In the present paper we explain a way for obtaining a family of polynomial counterexamples containing the above ones. Finally we study the global dynamics of the examples given in \cite{CEGHM}.
TL;DR: In this paper, it was shown that the structure group of an oriented 8-dimensional vector bundle can be reduced to a group of 3 linearly independent sections or a 3-dimensional subbundle.
Abstract: Let $\xi$ be an oriented 8-dimensional vector bundle. We prove that the structure group $SO(8)$ of $\xi$ can be reduced to $Sp(2)$ or $Sp(2)\cdot Sp(1)$ if and only if the vector bundle associated to $\xi$ via a certain outer automorphism of the group $Spin(8)$ has 3 linearly independent sections or contains a $3$-dimensional subbundle. Necessary and sufficient conditions for the existence of an $Sp(2)$-structure in $\xi$ over a closed connected spin manifold of dimension~$8$ are also given in terms of characteristic classes.
TL;DR: In this paper, it was shown that the flatness of a line of double limit cycles in a generic 2-parameter family at its end point is a codimension 2 saddle connection bifurcation point.
Abstract: Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to sub-analytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with the line of saddle connections. It is possible to prove that the flatness is smooth and to compute its asymptotic properties.
TL;DR: In this article, the authors introduce the well-behaved uniformizing averages, which are useful in resummation theory and sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields.
Abstract: The aim of this expository paper is to introduce the well-behaved uniformizing averages, which are useful in resummation theory. These averages associate three essential, but often antithetic, properties: respecting convolution; preserving realness; reproducing lateral growth. These new objects are serviceable in real resummation and we sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields.
TL;DR: In this article, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given for symmetric piecewise linear bi-dimensional systems.
Abstract: Symmetric piecewise linear bi-dimensional systems are very common in control engineering. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable. For such systems, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given.
TL;DR: In this article, the authors show that basic algebro-geometric concepts such as the concept of intersection multiplicity of projective curves at a point in the complex projective plane, are needed in the study of planar polynomial vector fields and in particular in summing up the information supplied by bifurcation diagrams of global families of Polynomial systems.
Abstract: In this work we show that basic algebro-geometric concepts such as the concept of intersection multiplicity of projective curves at a point in the complex projective plane, are needed in the study of planar polynomial vector fields and in particular in summing up the information supplied by bifurcation diagrams of global families of polynomial systems. Algebro-geometric concepts are helpful in organizing and unifying in more intrinsic ways this information.
TL;DR: In this article, it was shown that the nuclear case of the Frechet-Hilbert space can be solved in polynomial time if and only if $[\Cal H(O_E)]'\in(\Omega)$ (resp.
Abstract: Let $E$ be a Frechet (resp. Frechet-Hilbert) space. It is shown that $E\in (\Omega)$ (resp. $E\in (DN)$) if and only if $[\Cal H(O_E)]'\in(\Omega)$ (resp. $[\Cal H(O_E)]'\in (DN)$). Moreover it is also shown that $E\in (DN)$ if and only if $\Cal H_b(E')\in (DN)$. In the nuclear case these results were proved by Meise and Vogt \cite{2}.
TL;DR: In this article, the existence and convergence of logarithmic series for quadratic vector fields on the plane is studied, and an algorithm for the location of singularities is developed.
Abstract: Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that quadratic systems with psi-series that are not Laurent series do not have an algebraic first integral. Besides, a criterion about non-existence of an analytic first integral is given.
TL;DR: In this paper, sufficient conditions are formulated for the perturbed oscillator to have an invariant torus in the phase cylinder, and some perturbed solutions are in the basin of attraction of this torus and are thus entrained to the dynamical behavior of the perturbation system on the torus.
Abstract: The response of an oscillator to a small amplitude periodic excitation is discussed. In particular, sufficient conditions are formulated for the perturbed oscillator to have an invariant torus in the phase cylinder. When such an invariant torus exists, some perturbed solutions are in the basin of attraction of this torus and are thus entrained to the dynamical behavior of the perturbed system on the torus. In particular, the perturbed solutions in the basin of attraction of the invariant torus are entrained to a subharmonic or to a quasi periodic motion.
TL;DR: In this article, it was shown that an ideal if and only if every non-zero element of the ideal is a molecule is linearly ordered, and connections among the classes of ideals that have various combinations of the properties of being implicative, essential, weakly essential, maximal or prime.
Abstract: We show that an ideal $I$ of an $MV$-algebra $A$ is linearly ordered if and only if every non-zero element of $I$ is a molecule. The set of molecules of $A$ is contained in $\operatorname{Inf}(A)\cup B_2(A)$ where $B_2(A)$ is the set of all elements $x\in A$ such that $2x$ is idempotent. It is shown that $I
e \{0\}$ is weakly essential if and only if $B^\perp \subset B(A).$ Connections are shown among the classes of ideals that have various combinations of the properties of being implicative, essential, weakly essential, maximal or prime.
TL;DR: In this paper, a harmonic functional calculus is introduced in order to generalize some extended versions of theorems of von Neumann, Heinz, and Fan, and the purpose of this paper is to introduce a new functional calculus.
Abstract: The purpose of this paper is to introduce a harmonic functional calculus in order to generalize some extended versions of theorems of von Neumann, Heinz and Ky Fan.
TL;DR: In this paper, the Alexander ideals of classical knots are characterised, a result which extends to certain higher dimensional knots, and a result that extends to higher-dimensional knots as well.
Abstract: The Alexander ideals of classical knots are characterised, a result which extends to certain higher dimensional knots.