Nonlinear Sciences

We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We prove coalescence for two elementary rules and show that there exists infinitely many coalescing CA. We then conduct an experimental study on all elementary CA and show that some rules exhibit a phase transition, which belongs to the universality class of directed percolation.

We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We prove coalescence for two elementary rules, non coalescence for two other, and show that there exists infinitely many coalescing CA. We then conduct an experimental study on all elementary CA and show that some rules exhibit a phase transition, which belongs to the universality class of directed percolation.

Partitioned cellular automata are known to be an useful tool to simulate linear and nonlinear problems in physics, specially because they allow for a straightforward way to define conserved quantities and reversible dynamics. Here we show how to construct a local coarse graining description of partitioned cellular automata. By making use of this tool we investigate the effective dynamics in this model of computation. All examples explored are in the scenario of lattice gases, so that the information lost after the coarse graining is related to the number of particles. It becomes apparent how difficult it is to remain with a deterministic dynamics after coarse graining. Several examples are shown where an effective stochastic dynamics is obtained after a deterministic dynamics is coarse grained. These results suggest why random processes are so common in nature.

In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".

We study the predictability of emergent phenomena in complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show how to construct local coarse-grained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics.

The advantage of Cellular Potts Model (CPM) is due to its ability for introducing cell-cell interaction based on the well known statistical model i.e. the Potts model. On the other hand, Lattice gas Cellular Automata (LGCA) can simulate movement of cell in a simple and correct physical way. These characters of CPM and LGCA have been combined in a reaction-diffusion frame to simulate the dynamic of avascular cancer growth on a more physical basis.The cellular automaton is evolved on a square lattice on which in the diffusion step tumor cells (C) and necrotic cells (N) propagate in two dimensions and in the reaction step every cell can proliferate, be quiescent or die due to the apoptosis and the necrosis depending on its environment. The transition probabilities in the reaction step have been calculated by the Glauber algorithm and depend on the KCC, KNC, and KNN (cancer-cancer, necrotic-cancer, and necrotic-necrotic couplings respectively). It is shown the main feature of the cancer growth depends on the choice of magnitude of couplings and the advantage of this method compared to other methods is due to the fact that it needs only three parameters KCC, KNC and KNN which are based on the well known physical ground i.e. the Potts model.

We explore a community-detection cellular automata algorithm inspired by human heuristics, based on information diffusion and a non-linear processing phase with a dynamics inspired by human heuristics. The main point of the methods is that of furnishing different "views" of the clustering levels from an individual point of view. We apply the method to networks with local connectivity and long-range rewiring.

We develop a modified version of the totally asymmetric simple exclusion process (TASEP) and use it to reproduce flow on an escalator with two distinct lanes of pedestrian traffic. The model is used to compare strategies with two standing lanes and a standing lane with a walking lane, using theoretical analysis and numerical simulations. The results show that two standing lanes are better for smoother overall transportation, while a mixture of standing and walking is advantageous only in limited cases that have a small number of pedestrians. In contrast, with many pedestrians, the individual travel time of the first several entering particles is always shorter with distinct standing and walking lanes than it is with two standing lanes.

We present results of cellular automata based investigations of rotating spiral autowaves in a nonequilibrium excitable medium which models three-level paramagnetic microwave phonon laser (phaser). The computational model is described in arXiv:cond-mat/0410460v2 and arXiv:cond-mat/0602345v1 . We have observed several new scenarios of self-organization, competition and dynamical stabilization of rotating spiral autowaves under conditions of cross-relaxation between three-level active centers. In particular, phenomena of inversion of topological charge, as well as processes of regeneration and replication of rotating spiral autowaves in various excitable media were revealed and visualized for mesoscopic-scale areas of phaser-type active systems, which model real phaser devices.

The dynamics of rule 54 one-dimensional two-state cellular automaton (CA) are a discrete analog of a space-time dynamics of excitations in nonlinear active medium with mutual inhibition. A cell switches its state 0 to state 1 if one of its two neighbors is in state 1 (propagation of a perturbation) and a cell remains in state 1 only if its two neighbors are in state 0. A lateral inhibition is because a 1-state neighbor causes a 1-state cell to switch to state 0. The rule produces a rich spectrum of space-time dynamics, including gliders and glider guns just from four primitive gliders. We construct a catalogue of gliders and describe them by tiles. We calculate a subset of regular expressions Ψ R54 to encode gliders. The regular expressions are derived from de Bruijn diagrams, tile-based representation of gliders, and cycle diagrams sometimes. We construct an abstract machine that recognizes regular expressions of gliders in rule 54 and validate Ψ R54 . We also propose a way to code initial configurations of gliders to depict any type of collision between the gliders and explore self-organization of gliders, formation of larger tiles, and soliton-like interactions of gliders and computable devices.