Ergodic properties of triangle partitions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Ergodic crossover in partially self-avoiding stochastic walks

Consider a one-dimensional environment with N randomly distributed sites. An agent explores this random medium moving deterministically with a spatial memory μ. A crossover from local to global exploration occurs in one dimension at a well-defined memory value μ1=log2N. In its stochastic version, the dynamics is ruled by the memory and by temperature T, which affects the hopping displacement. This dynamics also shows a crossover in one dimension, obtained computationally, between exploration schemes, characterized yet by the trajectory size (Np) (aging effect). In this paper we provide an analytical approach considering the modified stochastic version where the parameter T plays the role of a maximum hopping distance. This modification allows us to obtain a general analytical expression for the crossover, as a function of the parameters μ, T, and Np. Differently from what has been proposed by previous studies, we find that the crossover occurs in any dimension d. These results have been validated by numerical experiments and may be of great value for fixing optimal parameters in search algorithms. © 2013 American Physical Society.

Magnetic Field Line Escape: Comparison with Mean Free Path

In the present work, we determine the fraction of magnetic field lines that reach the tokamak wall leaving the plasma surrounded by a chaotic layer created by resonant perturbations at the plasma edge. The chaotic layer arises in a scenario where an integrable magnetic field with reversed magnetic shear is perturbed by an ergodic magnetic limiter. For each considered line, we calculate its connection length, i.e. the number of toroidal turns that the field lines complete before reaching the wall. We represent the results in the poloidal section in which the initial coordinates are chosen. We also estimate the radial profile of the fraction of field lines, for different temperatures, whose connection lengths are smaller than the electron collisional mean free path.

Partial hyperbolicity for symplectic diffeomorphisms

We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic. (c) 2005 Elsevier SAS. All rights reserved.

Non-periodic bifurcations of one-dimensional maps

We prove that a 'positive probability' subset of the boundary of '{uniformly expanding circle transformations}' consists of Kupka-Smale maps. More precisely, we construct an open class of two-parameter families of circle maps (f(alpha,theta))(alpha,theta) such that, for a positive Lebesgue measure subset of values of alpha, the family (f(alpha,theta))(theta) crosses the boundary of the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.

Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes

In this paper, we propose a model for the destruction of three-dimensional horseshoes via heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call generating. © 2008 Cambridge University Press.

Plasma confinement in tokamaks with robust torus

We present a non-linear symplectic map that describes the alterations of the magnetic field lines inside the tokamak plasma due to the presence of a robust torus (RT) at the plasma edge. This RT prevents the magnetic field lines from reaching the tokamak wall and reduces, in its vicinity, the islands and invariant curve destruction due to resonant perturbations. The map describes the equilibrium magnetic field lines perturbed by resonances created by ergodic magnetic limiters (EMLs). We present the results obtained for twist and non-twist mappings derived for monotonic and non-monotonic plasma current density radial profiles, respectively. Our results indicate that the RT implementation would decrease the field line transport at the tokamak plasma edge. © 2010 Elsevier B.V. All rights reserved.

Caos e termalização na teoria de Yang-Mills com quebra espontânea de simetria

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Suspensões de Poisson, ergodicidade e o teorema central do limite

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Aspectos dinâmicos e ergódicos dos intercâmbios de intervalos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Bilhares planares

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Teorema ergódico multiplicativo de oseledets

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sistemas dinâmicos e substituições

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Law of large numbers for certain cylinder flows

We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit a law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a Gaussian distribution.

Chaotic planar piecewise smooth vector fields with non-trivial minimal sets

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.