Nonlinear Sciences

We study the asymptotic behaviour of symbolic computing systems, notably one-dimensional cellular automata (CA), in order to ascertain whether and at what rate the number of complex versus simple rules dominate the rule space for increasing neighbourhood range and number of symbols (or colours), and how different behaviour is distributed in the spaces of different cellular automata formalisms. Using two different measures, Shannon's block entropy and Kolmogorov complexity, the latter approximated by two different methods (lossless compressibility and block decomposition), we arrive at the same trend of larger complex behavioural fractions. We also advance a notion of asymptotic and limit behaviour for individual rules, both over initial conditions and runtimes, and we provide a formalisation of Wolfram's classification as a limit function in terms of Kolmogorov complexity.

Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using Monte-Carlo simulations. We show that this phenomenon is a second-order phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics.

This paper designs an efficient two-class pattern classifier utilizing asynchronous cellular automata (ACAs). The two-state three-neighborhood one-dimensional ACAs that converge to fixed points from arbitrary seeds are used here for pattern classification. To design the classifier, we first identify a set of ACAs that always converge to fixed points from any seeds with following properties - (1) each ACA should have at least two but not huge number of fixed point attractors, and (2) the convergence time of these ACAs are not to be exponential. In order to address the first issue, we propose a graph, coined as fixed point graph of an ACA that facilitates in counting the fixed points. We further perform an experimental study to estimate the convergence time of ACAs, and find that there are some convergent ACAs which demand exponential convergence time. Finally, we find that there are 71 (out of 256) ACAs which can be effective candidates as pattern classifier. We use each of the candidate ACAs on some standard data sets, and observe the effectiveness of each ACAs as pattern classifier. It is observed that the proposed classifier is very competitive and performs reliably better than many standard existing algorithms.

This paper proposes a method of auto-generation of a centerline graph from a geometrically complex roadmap of real-world traffic systems by using a hierarchical quadtree for cellular automata simulations. Our method is summarized as follows. First, we store the binary values of the monochrome image of target roadmap (one and zero represent the road and the other areas, respectively) in the two-dimensional square map. Second, we recursively divide the square map into sub-leafs by a quadtree until the summed-up value of pixels included inside the leaf becomes equal to or less than one. Third, we gradually remove the distal leaves that are adjacent to the leaves whose depths are shallower than the distal leaf. After that, we trace the remaining distal leaves of the tree using Morton's space-filling curve, while selecting the leaves that keep a certain distance among the previously selected leaves as the nodes of the graph. Finally, each selected node searches the neighboring nodes and stores them as the edges of the graph. We demonstrate our method by generating a centerline graph from a complex roadmap of a real-world airport and by carrying out a typical network analysis using Dijkstra's method.

Most current methods for identifying coherent structures in spatially-extended systems rely on prior information about the form which those structures take. Here we present two new approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures. One, local sensitivity filtering, is a modification of the local Lyapunov exponent approach suitable to cellular automata and other discrete spatial systems. The other, local statistical complexity filtering, calculates the amount of information needed for optimal prediction of the system's behavior in the vicinity of a given point. By examining the changing spatiotemporal distributions of these quantities, we can find the coherent structures in a variety of pattern-forming cellular automata, without needing to guess or postulate the form of that structure. We apply both filters to elementary and cyclical cellular automata (ECA and CCA) and find that they readily identify particles, domains and other more complicated structures. We compare the results from ECA with earlier ones based upon the theory of formal languages, and the results from CCA with a more traditional approach based on an order parameter and free energy. While sensitivity and statistical complexity are equally adept at uncovering structure, they are based on different system properties (dynamical and probabilistic, respectively), and provide complementary information.

Bidirectional transport in (quasi) one-dimensional systems generically leads to cluster-formation and small particle currents. This kind of transport can be described by the asymmetric simple exclusion process (ASEP) with two species of particles. In this work, we consider the effect of non-Markovian site exchange times between particles. Different realizations of the exchange process can be considered: The exchange times can be assigned to the lattice bonds or each particle. In the latter case we specify additionally which of the two exchange times is executed, the earlier one (minimum rule) or the later one (maximum rule). In a combined numerical and analytical approach we find evidence that we recover the same asymptotic behavior as for unidirectional transport for most realizations of the exchange process. Differences in the asymptotic behavior of the system have been found for the minimum rule which is more efficient for fast decaying exchange time distributions.

In this paper, we introduce an ASEP-like transport model for bidirectional motion of particles on a multi-lane lattice. The model is motivated by {\em in vivo} experiments on organelle motility along a microtubule (MT), consisting of thirteen protofilaments, where particles are propelled by molecular motors (dynein and kinesin). In the model, organelles (particles) can switch directions of motion due to "tug-of-war" events between counteracting motors. Collisions of particles on the same lane can be cleared by switching to adjacent protofilaments (lane changes). We analyze transport properties of the model with no-flux boundary conditions at one end of a MT ("plus-end" or tip). We show that the ability of lane changes can affect the transport efficiency and the particle-direction change rate obtained from experiments is close to optimal in order to achieve efficient motor and organelle transport in a living cell. In particular, we find a nonlinear scaling of the mean {\em tip size} (the number of particles accumulated at the tip) with injection rate and an associated phase transition leading to {\em pulsing states} characterized by periodic filling and emptying of the system.

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 address the problem of reconstructing a multi-band signal from its sub-Nyquist point-wise samples. To date, all reconstruction methods proposed for this class of signals assumed knowledge of the band locations. In this paper, we develop a non-linear blind perfect reconstruction scheme for multi-band signals which does not require the band locations. Our approach assumes an existing blind multi-coset sampling method. The sparse structure of multi-band signals in the continuous frequency domain is used to replace the continuous reconstruction with a single finite dimensional problem without the need for discretization. The resulting problem can be formulated within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.