Giovani L. Vasconcelos

Long-range correlations and nonstationarity in the Brazilian stock market

We report an empirical study of the Ibovespa index of the Sao Paulo Stock Exchange in which we detect the existence of long-range correlations. To analyze our data we introduce a rescaled variant of the usual Detrended Fluctuation Analysis that allows us to obtain the Hurst exponent through a one-parameter fitting. We also compute a time-dependent Hurst exponent H(t) using three-year moving time windows. In particular, we find that before the launch of the Collor Plan in 1990 the curve H(t) remains, in general, well above 1/2, while afterwards it stays close to 1/2. We thus argue that the structural reforms set off by the Collor Plan has lead to a more efficient stock market in Brazil. We also suggest that the time dependence of the Ibovespa Hurst exponent could be described in terms of a multifractional Brownian motion.

Scaling Relations for Diversity of Languages

The distribution of living languages is investigated and scaling relations are found for the diversity of languages as a function of the country area and population. These results are compared with data from Ecology and from computer simulations of fragmentation dynamics where similar scalings appear. The language size distribution is also studied and shown to display two scaling regions: (i) one for the largest (in population) languages and (ii) another one for intermediate-size languages. It is then argued that these two classes of languages may have distinct growth dynamics, being distributed on the sets of different fractal dimensions.

Phase transitions in a spring-block model of earthquakes

We investigate the nature of the phase transitions that occur in one version of the Burridge-Knopoff spring-block model, which has been used to simulate tectonic motion along a fault. We find that by adjusting the parameters of the model, a variety of qualitatively different behaviors can be observed. As we vary one of the parameters, the system goes from a region where all the blocks are continuously moving to a region where the motion occurs in short, but violent, ruptures. We investigate the properties of the dynamics in these two regimes and the nature of the transition that occurs when the system is tuned from one regime into the other.

Exact solutions for steady bubbles in a Hele-Shaw cell with rectangular geometry

Exact solutions are presented for an arbitrary number of steadily moving bubbles in a Hele-Shaw channel when surface tension is neglected. According to the symmetry displayed by the bubbles, the solutions are classified into two groups: (i) solutions where the bubbles are symmetrical about the channel centreline and (ii) solutions in which the bubbles have fore-and-aft symmetry. The general solutions are expressed in integral form but in some special cases analytic solutions in terms of elliptic integrals are found. The possible relevance of these exact solutions to experiments is also briefly discussed

Power law periodicity in the tangent bifurcations of the logistic map

Numerical studies were carried out for the average of the logistic map on the tangent bifurcations from chaos into periodic windows. A critical exponent of 12 is found on the average amplitude as one approaches the transition. Additionally, the averages oscillate with a period that decreases with the same exponent. This Power Law Periodicity is related to the reinjection mechanism of the map. The undulations appear at control parameter values much earlier than the values where the critical exponent of the bifurcation shows significant changes in the average amplitude.

Geometrical Model for a Particle on a Rough Inclined Surface

A simple geometrical model is presented for the gravity-driven motion of a single particle on a rough inclined surface. Adopting a simple restitution law for the collisions between the particle and the surface, we arrive at a model in which the dynamics is described by a one-dimensional map. This map is studied in detail and it is shown to exhibit several dynamical regimes (steady state, chaotic behavior, and accelerated motion) as the model parameters vary. A phase diagram showing the corresponding domain of existence for these regimes is presented. The model is also found to be in good qualitative agreement with recent experiments on a ball moving on a rough inclined line.

Single-particle model for a granular ratchet

A simple model for a granular ratchet corresponding to a single grain bouncing off a vertically vibrating sawtooth-shaped base is studied. Depending on the model parameters, horizontal transport is observed in both the preferred and unfavoured directions. A phase diagram is presented indicating the regions in parameter space where the different regimes (no current, normal current, and current reversal) occur.

Selection of the Taylor-Saffman bubble does not require surface tension.

A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex plane. It is then demonstrated that the only stable fixed point (attractor) of the nonsingular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established theory, bubble selection in a Hele-Shaw cell does not require surface tension. The solutions reported here significantly extend previous results for a simply connected geometry (finger) to a doubly connected one (bubble). We conjecture that the same selection rule without surface tension holds for Hele-Shaw flows of arbitrary connectivity.

Doubly periodic array of bubbles in a Hele-Shaw cell

Exact solutions are presented for a doubly periodic array of steadily moving bubbles in a Hele-Shaw cell when surface tension is neglected. It is assumed that the bubbles are either symmetrical with respect to the channel centreline or have fore-and-aft symmetry, or both, so that the relevant flow domain can be reduced to a simply connected region. By using conformal-mapping techniques, a general solution with any number of bubbles per unit cell is obtained in integral form. Several examples are given, including solutions for multi-file arrays of bubbles in the channel geometry and doubly periodic solutions in an unbounded cell.

Fingering in a channel and tripolar Loewner evolutions.

A class of Laplacian growth models in the channel geometry is studied using the formalism of tripolar Loewner evolutions, in which three points, namely, the channel corners and infinity, are kept fixed. Initially, the problem of fingered growth, where growth takes place only at the tips of slit-like fingers, is revisited and a class of exact exact solutions of the corresponding Loewner equation is presented for the case of stationary driving functions. A model for interface growth is then formulated in terms of a generalized tripolar Loewner equation and several examples are presented, including interfaces with multiple tips as well as multiple growing interfaces. The model exhibits interesting dynamical features, such as tip and finger competition.