Gustavo Gutierrez

From exactly solvable chaotic maps to stochastic dynamics

Abstract For a class of nonlinear chaotic maps, the exact solution can be written as Xn=P(θkn), where P(t) is a periodic function, θ is a real parameter and k is an integer number. A generalization of these functions: Xn=P(θzn), where z is a real parameter, can be proved to produce truly random sequences. Using different functions P(t) we can obtain different distributions for the random sequences. Similar results can be obtained with functions of type Xn=h[f(n)], where f(n) is a chaotic function and h(t) is a noninvertible function. We show that a dynamical system consisting of a chaotic map coupled to a map with a noninvertible nonlinearity can generate random dynamics. We present physical systems with this kind of behavior. We report the results of real experiments with nonlinear circuits and Josephson junctions. We show that these dynamical systems can produce a type of complexity that cannot be observed in common chaotic systems. We discuss applications of these phenomena in dynamics-based computation.

Chaos-induced true randomness

We investigate functions of type Xn=P(θzn), where P(t) is a periodic function, θ and z are real parameters. We show that these functions produce truly random sequences. We prove that a class of autonomous dynamical systems, containing nonlinear terms described by periodic functions of the variables, can generate random dynamics. We generalize these results to dynamical systems with nonlinearities in the form of noninvertible functions. Several examples are studied in detail. We discuss how the complexity of the dynamics depends on the kind of nonlinearity. We present real physical systems that can produce random time-series. We report the results of real experiments using nonlinear circuits with noninvertible I–V characteristics. In particular, we show that a Josephson junction coupled to a chaotic circuit can generate unpredictable dynamics.

Vertical Granular Transport in a Vibrated U-Tube

We present an experimental study of the collective motion of grains inside a U shaped tube undergoing vertical oscillations, and we develop a very simple quantitative model that captures relevant features of the observed behavior. The height difference between the granular columns grows exponentially with time when the system is shaken at sufficiently low frequencies. In vacuum the exponential growth is suppressed.

Power law for the permeability in a two-dimensional disordered porous medium

We report numerical calculations of the permeability as a function of porosity for a two-dimensional disordered porous medium. This medium is modeled using the well known “Swiss Cheese” model. The fluid is simulated using a cellular automata algorithm. We find that for relatively high porosities the permeability decays exponentially with the density of obstacles. As a consequence of this exponential behavior, a power-law dependence of the permeability as a function of the porosity is obtained, for this model system. We find that the power-law exponent is given by the ratio between two characteristic scales. One scale is given by the inverse of the area of one obstacle, and is approximately equal to the density of obstacles necessary to reach the percolation threshold. The other scale is equal to the average change in the density of obstacles necessary for the permeability to be reduced to about 1/e of its original value.

Silo Collapse: An Experimental Study

The purpose of this experimental work is to develop some basic insight into the pre-buckling behavior and the buckling transition toward plastic collapse of a granular silo by studying different patterns of deformation generated on thin paper cylindrical shells during granular discharge. We study the collapse threshold considering the influence of the bed heights, flow rates and grain sizes. We compare the patterns that appear during the discharge of spherical beads, with those obtained in the axially compressed cylindrical shells. When the height of the granular column is close to the collapse threshold, we observe a ladder like pattern that rises around the cylinder surface in a spiral path of diamond shaped localizations, and develops into a plastic collapsing fold that grows around the collapsing silo.

Characterization of invariant patterns in a slowly rotated granular tumbler

We report experimental results of the pattern developed by a mixture of two types of grains in a triangular rotating tumbler operating in the avalanche regime. At the centroid of the triangular tumbler an invariant zone appears where the grains do not move relative to the tumbler. We characterize this invariant zone by its normalized area, Ai, and its circularity index as a function of the normalized filling area A. We find a critical filling area so that only for A>Ac invariant zones are obtained. These zones scale as Ai∼(A−Ac)2 near Ac. We have obtained a maximum in the circularity index for A≈0.8, for which the shape of the invariant zone is closer to a circular one. The experimental results are reproduced by a simple model which, based on the surface position, accounts for all the possible straight lines within the triangle that satisfy the condition of constant A. We have obtained an analytic expression for the contour of the invariant zone. Experimentally, we obtained a displacement in Ac that we explain in terms of a finite width of the avalanche region. This displacement is needed only to correct the size of the invariant zone, not its shape.

Behavior of grains in contact with the wall of a silo during the initial instants of a discharge-driven collapse

We study experimentally gravity-driven granular discharges of laboratory scale silos, during the initial instants of the discharge. We investigate deformable wall silos around their critical collapse height, as well as rigid wall silos. We propose a criterion to determine a maximum time for the onset of the collapse and find that the onset of collapse always occurs before the grains adjacent to the wall are sliding down. We conclude that the evolution of the static friction towards a state of maximum mobilization plays a crucial role in the collapse of the silo.