0000000000144640

AUTHOR

José Martínez-alfaro

showing 4 related works from this author

Plane foliations with a saddle singularity

2012

Abstract We study the set of planar vector fields with a unique singularity of hyperbolic saddle type. We found conditions to assure that a such vector field is topologically equivalent to a linear saddle. Furthermore, we describe the plane foliations associated to these vector fields. Such a foliation can be split in two subfoliations. One without restriction and another one that is topologically characterized by means of trees.

Planar vector fieldsSingular foliationsPlane (geometry)Mathematical analysisPlanar vector fieldsType (model theory)SingularityFoliation (geology)Vector fieldGeometry and TopologyTopological conjugacySaddleMathematicsSaddle singularityTopology and its Applications
researchProduct

Planar maps whose second iterate has a unique fixed point

2007

Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of either (c1) the union of the number 0 with the intervals ]-\infty, -1] and [1,\infty[, or (c2) t…

Discrete mathematics37G10; 37G15; 34K18Algebra and Number TheoryApplied Mathematics37G15Dynamical Systems (math.DS)Fixed point37G10Homothetic transformationPlanar graphSet (abstract data type)symbols.namesakeMathematics - Classical Analysis and ODEsSimple (abstract algebra)Classical Analysis and ODEs (math.CA)FOS: MathematicssymbolsEmbeddingDifferentiable functionMathematics - Dynamical Systems34K18AnalysisEigenvalues and eigenvectorsMathematicsJournal of Difference Equations and Applications
researchProduct

Singular levels and topological invariants of Morse Bott integrable systems on surfaces

2016

Abstract We classify up to homeomorphisms closed curves and eights of saddle points on orientable closed surfaces. This classification is applied to Morse Bott foliations and Morse Bott integrable systems allowing us to define a complete invariant. We state also a realization Theorem based in two transformations and one generator (the foliation of the sphere with two centers).

Pure mathematicsIntegrable systemApplied Mathematics010102 general mathematicsMathematical analysisMorse code01 natural scienceslaw.inventionlawSaddle point0103 physical sciencesFoliation (geology)Topological invariants010307 mathematical physics0101 mathematicsInvariant (mathematics)Mathematics::Symplectic GeometryEQUAÇÕES DIFERENCIAIS ORDINÁRIASAnalysisCircle-valued Morse theoryMorse theoryMathematics
researchProduct

An upper bound of the index of an equilibrium point in the plane

2012

Abstract We give an upper bound of the index of an isolated equilibrium point of a C 1 vector field in the plane. The vector field is decomposed in gradient and Hamiltonian components. This decomposition is related with the Loewner vector field. Associated to this decomposition we consider the set Π where the gradient and Hamiltonian components are linearly dependent. The number of branches of Π starting at the equilibrium point determines the upper bound of the index.

Equilibrium pointApplied MathematicsMathematical analysisGradient systemsUpper and lower boundsIndexsymbols.namesakesymbolsVector fieldLinear independenceHamiltonian systemsHamiltonian (quantum mechanics)AnalysisPlanar differential systemsMathematics
researchProduct