3-Dimensional hopf bifurcation via averaging theory

We consider the Lorenz system ẋ = σ(y - x), ẏ = rx - y - xz and ż = -bz + xy; and the Rössler system ẋ = -(y + z), ẏ = x + ay and ż = b - cz + xz. Here, we study the Hopf bifurcation which takes place at q± = (±√br - b,±√br - b, r - 1), in the Lorenz case, and at s± = (c+√c2-4ab/2, -c+√c2-4ab/2a, c±√c2-4ab/2a) in the Rössler case. As usual this Hopf bifurcation is in the sense that an one-parameter family in ε of limit cycles bifurcates from the singular point when ε = 0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two 2-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other 3- or n-dimensional differential systems.

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group. © 2012 Springer Science+Business Media B.V.

Reduction of infinite dimensional systems to finite dimensions: Compact convergence approach

We consider parameter dependent semilinear evolution problems for which, at the limit value of the parameter, the problem is finite dimensional. We introduce an abstract functional analytic framework that applies to many problems in the existing literature for which the study of asymptotic dynamics can be reduced to finite dimensions via the invariant manifolds technique. Some practical models are considered to show wide applicability of the theory. © 2013 Society for Industrial and Applied Mathematics.

Dynamical properties for a mixed Fermi accelerator model

The behavior of the decay of velocity in a semi-dissipative one-dimensional Fermi accelerator model is considered. Two different kinds of dissipative forces were considered: (i) F-v and; (ii) F-v2. We prove the decay of velocity is linear for (i) and exponential for (ii). During the decay, the particles move along specific corridors which are constructed by the borders of the stable manifolds of saddle points. These corridors organize themselves in a very complicated way in the phase space leading the basin of attraction of the sinks to be seemingly of fractal type. © 2013 Elsevier B.V. All rights reserved.

Variedade riemannianas e imersão do tipo Nash: um ensaio e aplicações

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Intersecções homoclínicas

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Análise da dinâmica de um sistema vibrante não ideal de dois graus de liberdade

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Introdução matemática aos modelos cosmológicos

Pós-graduação em Matemática Universitária - IGCE

Uma introdução ao estudo de fibrados vetoriais

Pós-graduação em Matemática Universitária - IGCE

Dualidade de Poincaré e invariantes cohomológicos

Pós-graduação em Matemática - IBILCE

Forma simplética para ondas não-lineares da corda clássica no espaço Ad'S IND.5'

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Conjuntos minimais de sistemas lineares por partes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Multifractality, stickiness, and recurrence-time statistics

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Cartan geometry of spacetimes with a nonconstant cosmological function Lambda

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Piecewise Linear Systems with Closed Sliding Poly-Trajectories

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)