Diego Maza

The control of chaos: Theory and applications

Abstract Control of chaos refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior. We review the major ideas involved in the control of chaos, and present in detail two methods: the Ott–Grebogi–Yorke (OGY) method and the adaptive method. We also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, we describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, we give a review of relevant experimental applications of these ideas and techniques.

Jamming during the discharge of granular matter from a silo.

In this work, we present an experimental study of the jamming that stops the free flow of grains from a silo discharging by gravity. When the outlet size is not much bigger than the beads, granular material jams the outlet of the container due to the formation of an arch. Statistical data from the number of grains fallen between consecutive jams are presented. The information that they provide can help one to understand the jamming phenomenon. As the ratio between the size of the orifice and the size of the beads is increased, the probability that an arch blocks the outlet decreases. We show here that there is a power-law divergence of the mean avalanche size for a finite critical radius. Beyond this critical radius, no jamming can occur and the flow is never stopped. The dependence of the arch formation on the shape and the material of the grains has been explored. It has been found that the material properties of the grains do not affect the arch formation probability. On the contrary, the shape of the grains deeply influences it. A simple model to interpret the results is also discussed.

Clogging transition of many-particle systems flowing through bottlenecks

When a large set of discrete bodies passes through a bottleneck, the flow may become intermittent due to the development of clogs that obstruct the constriction. Clogging is observed, for instance, in colloidal suspensions, granular materials and crowd swarming, where consequences may be dramatic. Despite its ubiquity, a general framework embracing research in such a wide variety of scenarios is still lacking. We show that in systems of very different nature and scale -including sheep herds, pedestrian crowds, assemblies of grains, and colloids- the probability distribution of time lapses between the passages of consecutive bodies exhibits a power-law tail with an exponent that depends on the system condition. Consequently, we identify the transition to clogging in terms of the divergence of the average time lapse. Such a unified description allows us to put forward a qualitative clogging state diagram whose most conspicuous feature is the presence of a length scale qualitatively related to the presence of a finite size orifice. This approach helps to understand paradoxical phenomena, such as the faster-is-slower effect predicted for pedestrians evacuating a room and might become a starting point for researchers working in a wide variety of situations where clogging represents a hindrance.

Pattern formation without heating in an evaporative convection experiment

We present an evaporation experiment in a single fluid layer reproducing conditions of volatile fluids in nature. When latent heat associated to the evaporation is large enough, the heat flow through the free surface of the layer generates temperature gradients that can destabilize the conductive motionless state giving rise to convective cellular structures without any external heating. Convective cells can be then observed in the transient range of evaporation from an initial depth value to a minimum threshold depth, after which a conductive motionless state appears until the evaporation finishes with an unwetting sequence. The sequence of convective patterns obtained here without heating is similar to that obtained in Benard-Marangoni convection. This work presents the sequence of spatial bifurcations as a function of the layer depth. The transition between square-to-hexagonal pattern, known from non-evaporative experiments, is obtained here with a similar change in wavelength.

Jamming and critical outlet size in the discharge of a two-dimensional silo

We present an experimental study of jamming in the discharge of grains through an opening in a two-dimensional silo. For a wide range of outlet sizes, we obtain the size distribution of avalanche defined as the number of grains that fall between two consecutive jams. From these distributions, we obtain the probability that the silo jams before N particles pass through the orifice. Then a simple model of arch formation is proposed that predicts the shape of the jamming probability function and reveals that it does not exist a critical size of the orifice above which there is not jamming.

Unjamming a granular hopper by vibration

We present an experimental study of the outflow of a hopper continuously vibrated by a piezoelectric device. Outpouring of grains can be achieved for apertures much below the usual jamming limit observed for non-vibrated hoppers. Granular flow persists down to the physical limit of one grain diameter, a limit reached for a finite vibration amplitude. For the smaller orifices, we observe an intermittent regime characterized by alternated periods of flow and blockage. Vibrations do not significantly modify the flow rates both in the continuous and the intermittent regime. The analysis of the statistical features of the flowing regime shows that the flow time significantly increases with the vibration amplitude. However, at low vibration amplitude and small orifice sizes, the jamming time distribution displays an anomalous statistics.

Identification of arches in two-dimensional granular packings.

We identify arches in a bed of granular disks generated by a molecular dynamic-type simulation. We use the history of the deposition of the particles to identify the supporting contacts of each particle. Then, arches are defined as sets of mutually stable disks. Different packings generated through tapping are analyzed. The possibility of identifying arches from the static structure of a deposited bed, without any information on the history of the deposition, is discussed.

Role of vibrations in the jamming and unjamming of grains discharging from a silo.

We present experimental results of the jamming of noncohesive particles discharged from a flat bottomed silo subjected to vertical vibration. When the exit orifice is only a few grain diameters wide, the flow can be arrested due to the formation of blocking arches. Hence, an external excitation is needed to resume the flow. The use of a continuous gentle vibration is a usual technique to ease the flow in such situations. Even though jamming is less frequent, it is still an issue in vibrated silos. There are, in principle, two possible mechanisms through which vibrations may facilitate the flow: (i) a decrease in the probability of the formation of blocking arches and (ii) the breakage of blocking arches once they have been formed. By measuring the time intervals inside an avalanche during which no particles flow through the outlet, we are able to estimate the probability of breaking a blocking arch by vibrations. The result agrees with the prediction of a bivariate probabilistic model in which the formation of blocking arches is equally probable in vibrated and nonvibrated silos. This indicates that the second aforementioned mechanism is mainly responsible for improving the flowability in gently vibrated silos.

Towards a relevant set of state variables to describe static granular packings

We analyze, experimentally and numerically, the steady states, obtained by tapping, of a two-dimensional granular layer. Contrary to the usual assumption, we show that the reversible (steady state branch) of the density-acceleration curve is nonmonotonous. Accordingly, steady states with the same mean volume can be reached by tapping the system with very different intensities. Simulations of dissipative frictional disks show that equal volume steady states have different values of the force moment tensor. Additionally, we find that steady states of equal stress can be obtained by changing the duration of the taps; however, these states present distinct mean volumes. These results confirm previous speculations that the volume and the force moment tensor are both needed to describe univocally equilibrium states in static granular assemblies.

Stress distribution of faceted particles in a silo after its partial discharge

We present experimental and numerical results of the effect that a partial discharge has on the morphological and micro-mechanical properties of non-spherical, convex particles in a silo. The comparison of the particle orientation after filling the silo and its subsequent partial discharge reveals important shear-induced orientation, which affects stress propagation. For elongated particles, the flow induces an increase in the packing disorder which leads to a reduction of the vertical stress propagation developed during the deposit generated prior to the partial discharge. For square particles, the flow favors particle alignment with the lateral walls promoting a behavior opposite to the one of the elongated particles: vertical force transmission, parallel to gravity, is induced. Hence, for elongated particles the flow developed during the partial discharge of the silo leads to force saturation with depth whereas for squares the flow induces hindering of the force saturation observed during the silo filling.