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Qualitative theory of differential equations and singularity theory

Abstract

Singularity theory has wide applications to various areas in Mathematics and in particular to differential Geometry and to the qualitative study of ordinary and partial differential equations. These branches of Mathematics feed back, in turn, into and enrich singularity theory. The following project, with well defined objectives, aims at consolidating research activities in the state of São Paulo in geometric aspects of dynamical systems and singularity theory. The interaction will promote development in the following areas: 1) qualitative theory of differential equations and applications; 2) generic properties of submanifolds in Euclidean spaces; 3) classification of singularities, the study of their topology as their invariants. In the research proposal on "qualitative theory of differential equations and applications" we shall study problems related to the Global Asymptotic Stability Conjecture and the injectivity of maps from Rn to Rn which are local diffeomorphisms. Flows in dimensional manifolds and transformations of the interval will also be studied M well M the singularities of certain classes of differential equations in Rn using singularity theory. The aim of the proposal on "generic properties of submanifolds in Euclidean and hyperbolic spaces" is to study the geometric properties of smooth and singular submanifolds in Euclidean spaces which are defined by applications whose singularities are finitely determined. The problems to be considered are those concerning embedded surfaces in R4 and singular surfaces in R3. Geometric properties of submanifolds in hyperbolic spaces will also be studied. The project "classification of singularities, the study of their topology as their invariants" is, in some parts, motivated by the problems highlighted above and is a fundamental part of our proposal. Some of the achievements of our group in this area are obtained using geometric methods and Newton polyhedron. The development of this project requires new results in singularity theory and the expertise of the researchers involved is very relevant for achieving the set objectives. The proposal is designed to allow maximum collaboration between the participants. We observe that all the members involved in the project have valuable experience in research related to the proposal, and collaboration between the members of the group has already given fruitful results. (AU)

Scientific publications (18)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
SMANIA, DANIEL. Solenoidal attractors with bounded combinatorics are shy. ANNALS OF MATHEMATICS, v. 191, n. 1, p. 1-79, JAN 2020. Web of Science Citations: 0.
SMANIA, DANIEL. Shy shadows of infinite-dimensional partially hyperbolic invariant sets. Ergodic Theory and Dynamical Systems, v. 39, n. 5, p. 1361-1400, MAY 2019. Web of Science Citations: 1.
DE LIMA, AMANDA; SMANIA, DANIEL. On infinitely cohomologous to zero observables. Ergodic Theory and Dynamical Systems, v. 33, n. 2, p. 375-399, APR 2013. Web of Science Citations: 0.
GUTIERREZ, CARLOS; MARTINEZ-ALFARO, JOSE; VENATO-SANTOS, JEAN. Plane foliations with a saddle singularity. Topology and its Applications, v. 159, n. 2, SI, p. 484-491, FEB 1 2012. Web of Science Citations: 1.
MESSAOUDI, A.; SMANIA, D. EIGENVALUES OF FIBONACCI STOCHASTIC ADDING MACHINE. Stochastics and Dynamics, v. 10, n. 2, p. 291-313, JUN 2010. Web of Science Citations: 6.
BALADI, VIVIANE; SMANIA, DANIEL. Alternative proofs of linear response for piecewise expanding unimodal maps. Ergodic Theory and Dynamical Systems, v. 30, n. 1, p. 1-20, FEB 2010. Web of Science Citations: 11.
GUTIERREZ, CARLOS; LLOYD, SIMON; MEDVEDEV, VLADISLAV; PIRES, BENITO; ZHUZHOMA, EVGENY. TRANSITIVE CIRCLE EXCHANGE TRANSFORMATIONS WITH FLIPS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 26, n. 1, p. 251-263, JAN 2010. Web of Science Citations: 4.
GUTIERREZ, CARLOS; PIRES, BENITO. On C-r-closing for flows on orientable and non-orientable 2-manifolds. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 40, n. 4, p. 553-576, DEC 2009. Web of Science Citations: 0.
GUTIERREZ, CARLOS; MAQUERA, CARLOS. Foliations and polynomial diffeomorphisms of R-3. MATHEMATISCHE ZEITSCHRIFT, v. 262, n. 3, p. 613-626, JUL 2009. Web of Science Citations: 2.
LLIBRE, JAUME; OLIVEIRA, REGILENE D. S. Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 70, n. 10, p. 3549-3560, MAY 15 2009. Web of Science Citations: 7.
BALADI, VIVIANE; SMANIA, DANIEL. SMOOTH DEFORMATIONS OF PIECEWISE EXPANDING UNIMODAL MAPS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 23, n. 3, p. 685-703, MAR 2009. Web of Science Citations: 8.
BALADI, VIVIANE; SMANIA, DANIEL. ANALYTICITY OF THE SRB MEASURE FOR HOLOMORPHIC FAMILIES OF QUADRATIC-LIKE COLLET-ECKMANN MAPS. Proceedings of the American Mathematical Society, v. 137, n. 4, p. 1431-1437, 2009. Web of Science Citations: 7.
GUTIERREZ, C.; LLOYD, S.; PIRES, B. AFFINE INTERVAL EXCHANGE TRANSFORMATIONS WITH FLIPS AND WANDERING INTERVALS. Proceedings of the American Mathematical Society, v. 137, n. 4, p. 1439-1445, 2009. Web of Science Citations: 1.
GUTIERREZ, C.; LLOYD, S.; PIRES, B.; ZHUZHOMA, E. V.; MEDVEDEV, V. S. Exchange transformations reversing orientation. DOKLADY MATHEMATICS, v. 78, n. 1, p. 500-502, AUG 2008. Web of Science Citations: 0.
MARTINS‚ LF; OLIVEIRA‚ R.D.S.; TARI‚ F. On pairs of regular foliations in and singularities of map-germs. Geometriae Dedicata, v. 135, n. 1, p. 103-118, 2008.
MANCINI, SOLANGE; MANOEL, MIRIAM; TEIXEIRA, MARCO ANTONIO. Divergent diagrams of folds and simultaneous conjugacy of involutions. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 12, n. 4, p. 657-674, Apr. 2005.
ARAUJO‚ V.; TAHZIBI‚ A. Stochastic stability at the boundary of expanding maps. Nonlinearity, v. 18, n. 3, p. 939, 2005.
RUAS‚ M.A.S.; DOS SANTOS‚ R.N.A. Real Milnor fibrations and (c)-regularity. MANUSCRIPTA MATHEMATICA, v. 117, n. 2, p. 207-218, 2005.

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