arXiv subjects
Publications and source records attributed to Raul Rechtman.
Probabilistic Cellular Automata are extended stochastic systems, widely used for modelling phenomena in many disciplines. The possibility of controlling their behaviour is therefore an important topic. We shall present here an approach to the problem of controlling such systems by acting only on the boundary of a target region.
We investigate the phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations, by changing the couplings or the connectivity. The spatial model shows bifurcations in the average magnetization, similar to what seen in the mean-field approximation, induced by the change of the topology, after rewiring short-range to long-range connection, as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution (degree of asynchronism) of the updating. We also examined the effects of inhomogeneity, mixing ferromagnetic and antiferromagnetic coupling, which induces an unexpected bifurcation diagram with a "bubbling" behaviour, as also happens for dilution.
An important question to be addressed regarding system control on a time interval $[0, T]$ is whether some particular target state in the configuration space is reachable from a given initial state. When the target of interest refers only to a portion of the spatial domain, we speak about regional analysis. Cellular Automata (CA) approach have been recently promoted for the study of control problems on spatially extended systems for which the classical approaches cannot be used. An interesting problem concerns the situation where the subregion of interest is not interior to the domain but a portion of its boundary. In this paper we address the problem of regional controllability of cellular automata via boundary actions, i.e., we investigate the characteristics of a cellular automaton so that it can be controlled inside a given region only acting on the value of sites at its boundaries.
We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every site that can be in one of two states that force the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices, B. Z. Webb and E. G. D. Cohen, proved that given an initial position and velocity of the walker and an environment of one type of scatterers, mirrrors or rotators, there is an environment of the other type such that the walks on both environments are equivalent, meaning they visit the same sites at the same time steps. We prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices. The proofs are based both on the geometry of the lattice and the structure of the scattering rule.
We study models of a society composed of a mixture of conformist and reasonable contrarian agents that at any instant hold one of two opinions. Conformists tend to agree with the average opinion of their neighbors and reasonable contrarians to disagree, but revert to a conformist behavior in the presence of an overwhelming majority, in line with psychological experiments. The model is studied in the mean field approximation and on small-world and scale-free networks. In the mean field approximation, a large fraction of conformists triggers a polarization of the opinions, a pitchfork bifurcation, while a majority of reasonable contrarians leads to coherent oscillations, with an alternation of period-doubling and pitchfork bifurcations up to chaos. Similar scenarios are obtained by changing the fraction of long-range rewiring and the parameter of scale-free networks related to the average connectivity.
We explore some aspects of phase transitions in cellular automata. We start recalling the standard formulation of statistical mechanics of discrete systems (Ising model), illustrating the Monte Carlo approach as Markov chains and stochastic processes. We then formulate the cellular automaton problem using simple models, and illustrate different types of possible phase transitions: density phase transitions of first and second order, damage spreading, dilution of deterministic rules, asynchronism-induced transitions, synchronization phenomena, chaotic phase transitions and the influence of the topology. We illustrate the improved mean-field techniques and the phenomenological renormalization group approach.
People are often divided into conformists and contrarians, the former tending to align to the majority opinion in their neighborhood and the latter tending to disagree with that majority. In practice, however, the contrarian tendency is rarely followed when there is an overwhelming majority with a given opinion, which denotes a social norm. Such reasonable contrarian behavior is often considered a mark of independent thought, and can be a useful strategy in financial markets. We present the opinion dynamics of a society of reasonable contrarian agents. The model is a cellular automaton of Ising type, with antiferromagnetic pair interactions modeling contrarianism and plaquette terms modeling social norms. We introduce the entropy of the collective variable as a way of comparing deterministic (mean-field) and probabilistic (simulations) bifurcation diagrams. In the mean field approximation the model exhibits bifurcations and a chaotic phase, interpreted as coherent oscillations of the whole society. However, in a one-dimensional spatial arrangement one observes incoherent oscillations and a constant average. In simulations on Watts-Strogatz networks with a small-world effect the mean field behavior is recovered, with a bifurcation diagram that resembles the mean-field one, but using the rewiring probability as the control parameter. Similar bifurcation diagrams are found for scale free networks, and we are able to compute an effective connectivity for such networks.
We study the problem of master-slave synchronization and control of totalistic cellular automata (CA) by putting a fraction of sites of the slave equal to those of the master and finding the distance between both as a function of this fraction. We present three control strategies that exploit local information about the CA, mainly, the number of nonzero Boolean derivatives. When no local information is used, we speak of synchronization. We find the critical properties of control and discuss the best control strategy compared with synchronization.
We study a cellular automaton opinion formation model of Ising type, with antiferromagnetic pair interactions modeling anticonformism, and ferromagnetic plaquette terms modeling the social norm constraints. For a sufficiently large connectivity, the mean-field equation for the average magnetization (opinion density) is chaotic. This "chaoticity" would imply irregular coherent oscillations of the whole society, that may eventually lead to a sudden jump into an absorbing state, if present. However, simulations on regular one-dimensional lattices show a different scenario: local patches may oscillate following the mean-field description, but these oscillations are not correlated spatially, so the average magnetization fluctuates around zero (average opinion near one half). The system is chaotic, but in a microscopic sense where local fluctuations tend to compensate each other. By varying the long-range rewiring of links, we trigger a small-world effect. We observe a bifurcation diagram for the magnetization, with period doubling cascades ending in a chaotic phase. As far as we know, this is the first observation of a small-world induced bifurcation diagram. The social implications of this transition are also interesting. In the presence of strong "anticonformistic" (or "antinorm") behavior, efforts for promoting social homogenization may trigger violent oscillations.
The control of chaotic systems implies inducing an unpredictable system to follow a desired trajectory using the smallest "force". In low-dimensional continuous systems, one method is that of reconstructing the tangent space, so that the control may be concentrated on the most expanding directions. This scheme is hard to follow for high-dimensional (extended) systems. This is particularly true for extended systems that exhibit stable chaos, that is, systems which are not chaotic in the usual sense, but are unpredictable for finite perturbations. Prototypical of this class are cellular automata, aka completely discrete dynamical systems. Although usual indicators of chaoticity such as the maximum Lyapunov exponent may be defined for such systems, we show that the usual approach may lead to counter intuitive results, and that it is possible to exploit the characteristics of the system in order to reduce the distance between two replicas with less control.
We show that the thermodynamic entropy density is proportional to the largest Lyapunov ex- ponent (LLE) of a discrete hydrodynamical system, a deterministic two-dimensional lattice gas automaton. The definition of the LLE for cellular automata is based on the concept of Boolean derivatives and is formally equivalent to that of continuous dynamical systems. This relation is jus- tified using a Markovian model. In an irreversible process with an initial density difference between both halves of the system, we find that Boltzmann's H function is linearly related to the expansion factor of the LLE, although the latter is more sensitive to the presence of traveling waves.
We study two problems related to spatially extended systems: the dynamical stability and the universality classes of the replica synchronization transition. We use a simple model of one dimensional coupled map lattices and show that chaotic behavior implies that the synchronization transition belongs to the multiplicative noise universality class, while stable chaos implies that the synchronization transition belongs to the directed percolation universality class.
We study systems with periodically oscillating parameters that can give way to complex periodic or non periodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of $ε$-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative non-trivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.
We study the behavior of a point particle incident from the left on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.
We present a probabilistic cellular automaton with two absorbing states, which can be considered a natural extension of the Domany-Kinzel model. Despite its simplicity, it shows a very rich phase diagram, with two second-order and one first-order transition lines that meet at a tricritical point. We study the phase transitions and the critical behavior of the model using mean field approximations, direct numerical simulations and field theory. A closed form for the dynamics of the kinks between the two absorbing phases near the tricritical point is obtained, providing an exact correspondence between the presence of conserved quantities and the symmetry of absorbing states. The second-order critical curves and the kink critical dynamics are found to be in the directed percolation and parity conservation universality classes, respectively. The first order phase transition is put in evidence by examining the hysteresis cycle. We also study the "chaotic" phase, in which two replicas evolving with the same noise diverge, using mean field and numerical techniques. Finally, we show how the shape of the potential of the field-theoretic formulation of the problem can be obtained by direct numerical simulations.
We study the synchronization of totalistic one dimensional cellular automata (CA). The CA with a non zero synchronization threshold exhibit complex non periodic space time patterns and conversely. This synchronization transition is related to directed percolation. We study also the maximum Lyapunov exponent for CA, defined in analogy with continuous dynamical systems as the exponential rate of expansion of the linear map induced by the evolution rule of CA, constructed with the aid of the Boolean derivatives. The synchronization threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities. The value of this threshold can be used to parametrize the space time complexity of CA.
We perform numerical scattering experiments on a Lorentz array of disks centered on a triangular lattice with L columns and study its transmission and reflection properties. In the finite horizon case, the motion of the particles may be modeled as simple one dimensional random walks with absorbing walls for which the scaling of the transmission and reflection coefficients are known, and agree with those found numerically. In the infinite horizon case the analogy with a simple diffusive process is no longer valid. In this case we compare our results with those expected for a one dimensional Levy walk, again with absorbing walls, for which logarithmic corrections to the scaling relations appear. The scaling with L and the symmetry properties of the forward and the backward are also studied, and some of their salient features are discussed.
Probabilistic cellular automata are prototypes of non equilibrium critical phenomena. This class of models includes among others the directed percolation problem (Domany Kinzel model) and the dynamical Ising model. The critical properties of these models are usually obtained by fine-tuning one or more control parameters, as for instance the temperature. We present a method for the parallel evolution of the model for all the values of the control parameter, although its implementation is in general limited to a fixed number of values. This algorithm facilitates the sketching of phase diagrams and can be useful in deriving the critical properties of the model. Since the criticality here emerges from the asymptotic distribution of some quantities, without tuning of parameters, our method is a mapping from a probabilistic cellular automaton with critical behavior to a self organized critical model with the same critical properties.