Universality classes of spatiotemporal intermittency

We discuss the spatiotemporal intermittency (STI) seen in the coupled sine circle map lattice. The phase diagram of this system, when updated with random initial conditions, shows very rich behaviour including synchronised solutions, and STI of various kinds. These behaviours are organised around the bifurcation boundary of the synchronised solutions, as well as an infection line which separates the lower part of the phase diagram into a spreading and a non-spreading regime. The STI seen at the bifurcation boundary in the spreading regime belongs convincingly to the directed percolation (DP) universality class. In the non-spreading regime, spatial intermittency (SI) with temporally regular bursts is seen at the bifurcation boundary. The laminar length distribution scales as a power-law with an exponent which is quite distinct from DP behaviour. Therefore, both DP and non-DP universality classes are seen in this system. When the coupled map lattice is mapped to a cellular automaton via coarse graining, a transition from a probabilistic cellular automaton to a deterministic cellular automaton at the infection line signals the transition from spreading to non-spreading behaviour.     

The dynamical origin of the universality classes of spatiotemporal intermittency

Two universality classes of spatiotemporal intermittency are seen in the spreading and non-spreading regimes of the sine circle map lattice, spatiotemporal intermittency of the directed percolation class, and spatial intermittency, not of the DP class, where the temporal behavior is regular. The transition between the two classes maps to a probabilistic to deterministic transition of the equivalent cellular automaton of the model, and is seen to have its dynamic origin in an attractor-widening crisis.     

Directed percolation phase transition at the onset of STI in an inhomogeneous coupled map lattice

A model for inhomogeneously coupled logistic maps is considered to find some critical exponents in the transition from inhomogeneous steady state to spatiotemporal chaos through spatiotemporal intermittency. The laminar state in the model is described by inhomogeneous steady state with spatial period two. We obtain a complete set of static exponents which match with the corresponding directed percolation (DP) values in (1+1) dimension. We also find four nonuniversal spreading exponents in which three exponents are in agreement with DP values. The model in which absorbing state is inhomogeneous steady state, contributes a new example in evidence of Pomeau's [18] conjecture that the onset of STI in a deterministic system belongs to DP universality class.     

Spatiotemporal intermittency on fractal lattices

The transition to turbulence via spatiotemporal intermittency is investigated for coupled maps defined on generalized Sierpinski gaskets, a class of deterministic fractal lattices. Critical exponents that characterize the onset of intermittency are computed as a function of the fractal dimension of the lattice. Windows of spatiotemporal intermittency are found as the coupling parameter is varied for lattices with a fractal dimension greater than two. This phenomenon is associated with a collective chaotic behavior of the fractal array of coupled maps.     

Spatio-temporal intermittency in coupled map lattices

The transition to turbulence in a one-dimensional array of maps coupled by diffusion is shown to display critical properties resembling those of directed percolation. The analogy is supported by the reconstruction of a probabilistic cellular automation with closely similar statistical properties. Numerical results suggest however that spatio-temporal intermittency does not belong to the same universality class as directed percolation.     

Computation by Asynchronously Updating Cellular Automata

A known method to compute on an asynchronously updating cellular automaton is the simulation of a synchronous computing model on it. Such a scheme requires not only an increased number of cell states, but also the simulation of a global synchronization mechanism. Asynchronous systems tend to use synchronization only on a local scale—if they use it at all. Research on cellular automata that are truly asynchronous has been limited mostly to trivial phenomena, leaving issues such as computation unexplored. This paper presents an asynchronously updating cellular automaton that conducts computation without relying on a simulated global synchronization mechanism. The two-dimensional cellular automaton employs a Moore neighborhood and 85 totalistic transition rules describing the asynchronous interactions between the cells. Despite the probabilistic nature of asynchronous updating, the outcome of the dynamics is deterministic. This is achieved by simulating delay-insensitive circuits on it, a type of asynchronous circuit that is known for its robustness to variations in the timing of signals. We implement three primitive operators on the cellular automaton from which any arbitrary delay-insensitive circuit can be constructed and show how to connect the operators such that collisions of crossing signals are avoided.     

Statistical analysis of the transition to turbulence in plane Couette flow

We argue on general grounds that the transition to turbulence in plane Couette flow is best studied experimentally at a statistical level. We present such a statistical analysis of experimental data guided by a parallel investigation of a simple coupled map lattice model for spatiotemporal intermittency. We confirm that this generic type of spatiotemporal chaos is relevant in the context of plane Couette flow, where the linear stability of the laminar regime at all Reynolds numbers insures the necessary local subcriticality. Using large ensembles of similar experiments, we show the existence of a well-defined threshold Reynolds number above which a unique, turbulent, intermittent attractor coexists with the laminar flow. Furthermore, our data reveals that this transition to spatiotemporal intermittency is discontinuous, i.e. akin to a first-order phase transition. Received: 10 April 1998 / Revised: 22 June 1998 / Accepted: 24 June 1998     

New insights into traffic dynamics: a weighted probabilistic cellular automaton model

From the macroscopic viewpoint for describing the acceleration behaviour of drivers, a weighted probabilistic cellular automaton model （the WP model, for short） is proposed by introducing a kind of random acceleration probabilistic distribution function. The fundamental diagrams, the spatiotemporal patterns, are analysed in detail. It is shown that the presented model leads to the results consistent with the empiricaZ data rather well, nonlinear flow-density relationship existing in lower density regions, and a new kind of traific phenomenon called neo-synchronized flow. Furthermore, we give the criterion for distinguishing the high-speed and low-speed neo-synchronized flows and clarify the mechanism of this kind of traffic phenomenon. In addition, the result that the time evolution of distribution of headways is displayed as a normal distribution further validates the reasonability of the neo-synchronized flow. These findings suggest that the diversity and the randomicity of drivers and vehicles have indeed a remarkable effect on traffic dynamics.     

Two-spin-majority cellular automaton as a model of 2D cluster and interface growth

A new probabilistic cellular automaton model is introduced to simulate cluster and interface growth in two dimensions. The dynamics of this model is an extension to higher dimensions of the compact directed percolation studied by Essam. Numerical results indicate that the two-dimensional cluster coarsening and growth can be described only approximately by the conventional cluster size scaling due to a crossover in the growth mode. The spreading of the initially flat interface follows a purely diffusional,t ^1/2, law.     

Coupled map lattices as cellular automata

We approach the problem of the complex dynamics of coupled map lattices (CML) by proposing a reduction to deterministic cellular automata (CA) with more than two states per site. The reduction scheme replaces the local map by an approximation in terms of a step function based on a straightforward analysis of the local dynamics. The variation of the spatial coupling in the CML then translates itself as a path in the spaces of rules for the equivalent deterministic CA. The transition to turbulence via spatiotemporal intermittency in the CML is then interpreted as a transition in the space of rules. The observed nonuniversality of this transition can be traced back to the nature of the rules involved on both sides of the transition region and to the character of the escape process from the turbulent state, either strongly deterministic or quasiprobabilistic. The relation between CML, deterministic, and probabilistic CA and the possibility of a mean-field treatment of the dynamics of CML are discussed at a more formal level.     

Chaotic transition in a three-coupled phase-locked loop system

The chaotic transition is observed in a three-coupled phase-locked loop (PLL) system in both experiments and numerical simulations. In this system, three PLL oscillators are connected with the periodic boundary condition. Intermittency is found in partially synchronized phase, in which two of three oscillators synchronize with each other and form a pair, and the chaotic transition occurs due to the recombination of synchronized pairs so that different pair is re-formed. In this phase, on-off intermittency is also observed and statistical analyses are carried out for on-off intermittent time series. This intermittency is considered as a hybrid type of intermittency with both on-off intermittency and intermittency due to the recombination of synchronized pairs present in the same time series. We also show the chaotic transition phenomena in a three-coupled logistic map system. (c) 2001 American Institute of Physics.     

Traffic flow characteristics in a mixed traffic system consisting of ACC vehicles and manual vehicles: A hybrid modelling approach

In this paper, we have investigated traffic flow characteristics in a traffic system consisting of a mixture of adaptive cruise control (ACC) vehicles and manual-controlled (manual) vehicles, by using a hybrid modelling approach. In the hybrid approach, (i) the manual vehicles are described by a cellular automaton (CA) model, which can reproduce different traffic states (i.e., free flow, synchronised flow, and jam) as well as probabilistic traffic breakdown phenomena; (ii) the ACC vehicles are simulated by using a car-following model, which removes artificial velocity fluctuations due to intrinsic randomisation in the CA model. We have studied the traffic breakdown probability from free flow to congested flow, the phase transition probability from synchronised flow to jam in the mixed traffic system. The results are compared with that, where both ACC vehicles and manual vehicles are simulated by CA models. The qualitative and quantitative differences are indicated.     

Toom’s Model with Glauber Rates: An Exact Solution Using Elementary Methods

A simple spatially two-dimensional stochastic cellular automaton with asymmetric coupling and synchronous updating according to Glauber rates is considered. While detailed balance is violated it is still possible to compute analytically the stationary probability distribution by elementary means. The stationary distribution can be written as a canonical equilibrium distribution of a spin system on a triangular lattice with nearest neighbour coupling. Thus, the cellular automaton shows a nonequilibrium phase transition with Ising critical behaviour.     

Dynamical phase transitions in one-dimensional stochastic cellular automata

We study the influence of dynamic noise and disorder on the evolution of a chaotic cellular automaton model. Three distinct phases are identified corresponding to ordered, random and damage spreading evolution. The time evolution of the associated order parameters is investigated and the critical exponents are calculated close to the phase transition.     

Transition to spatiotemporal chaos in a two-dimensional hydrodynamic system

We study the transition to spatiotemporal chaos in a two-dimensional hydrodynamic experiment where liquid columns take place in the gravity induced instability of a liquid film. The film is formed below a plane grid which is used as a porous media and is continuously supplied with a controlled flow rate. This system can be either ordered (on a hexagonal structure) or disordered depending on the flow rate. We observe, for the first time in an initially structured state, a subcritical transition to spatiotemporal disorder which arises through spatiotemporal intermittency. Statistics of numbers, creations, and fusions of columns are investigated. We exhibit a critical behavior close to the directed percolation one.     

Effects of abnormal excitation on the dynamics of spiral waves

The effect of physiological and pathological abnormal excitation of a myocyte on the spiral waves is investigated based on the cellular automaton model. When the excitability of the medium is high enough, the physiological abnormal excitation causes the spiral wave to meander irregularly and slowly. When the excitability of the medium is low enough,the physiological abnormal excitation leads to a new stable spiral wave. On the other hand, the pathological abnormal excitation destroys the spiral wave and results in the spatiotemporal chaos, which agrees with the clinical conclusion that the early after depolarization is the pro-arrhythmic mechanism of some anti-arrhythmic drugs. The mechanisms underlying these phenomena are analyzed.     

Application of cellular automata to N-body systems

A two-dimensional cellular automaton model is introduced to deal with the dynamics of a finite system of particles whose interactions are simulated by two-body step potentials. The method is illustrated for a potential approximating the standard Lennard-Jones potential, representative for the problem of heavy ion collisions in nuclear physics. From the cellular automaton dynamics thermodynamic equilibrium state variables are introduced in the usual way. The numerical experiments indicate the occurrence of a phase transition. Macroscopically the transition is marked by a singularity in the equation of state; microscopically it manifests itself by the formation of clusters of particles of all sizes, obeying a mass distribution in the form of a power law of exponent 1.35.     

Dynamic critical properties of a one-dimensional probabilistic cellular automaton

Dynamic properties of a one-dimensional probabilistic cellular automaton are studied by Monte Carlo simulation near a critical point which marks a second-order phase transition from an active state to an effectively unique absorbing state. Values obtained for the dynamic critical exponents indicate that the transition belongs to the universality class of directed percolation. Finally the model is compared with a previously studied one to show that a difference in the nature of the absorbing states places them in different universality classes. Received: 6 February 1998 / Revised and Accepted: 17 February 1998     

Phase Transitions for the Cavity Approach to the Clique Problem on Random Graphs

We give a rigorous proof of two phase transitions for a disordered statistical mechanics system used to define an algorithm to find large cliques inside Erdös random graphs. Such a system is a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.     

An Algorithm-Independent Definition of Damage Spreading—Application to Directed Percolation

We present a general definition of damage spreading in a pair of models. Using this general framework, one can define damage spreading in an objective manner that does not depend on the particular dynamic procedure that is being used. The formalism can be used for any spin-model or cellular automaton, with sequential or parallel update rules. At this point we present its application to the Domany–Kinzel cellular automaton in one dimension, this being the simplest model in which damage spreading has been found and studied extensively. We show that the active phase of this model consists of three subphases characterized by different damage-spreading properties.