L. I. Reyes

Exact solutions to chaotic and stochastic systems

We investigate functions that are exact solutions to chaotic dynamical systems. A generalization of these functions can produce truly random numbers. For the first time, we present solutions to random maps. This allows us to check, analytically, some recent results about the complexity of random dynamical systems. We confirm the result that a negative Lyapunov exponent does not imply predictability in random systems. We test the effectiveness of forecasting methods in distinguishing between chaotic and random time series. Using the explicit random functions, we can give explicit analytical formulas for the output signal in some systems with stochastic resonance. We study the influence of chaos on the stochastic resonance. We show, theoretically, the existence of a new type of solitonic stochastic resonance, where the shape of the kink is crucial. Using our models we can predict specific patterns in the output signal of stochastic resonance systems. (c) 2001 American Institute of Physics.

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.

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.

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.