Narrow Results

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliσ_τ-shift rules and eight hyper Bernoulliσ_τ-shift rules, the latter including such famous rules and . All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞.

Basin tree diagrams of all ten complex Bernoulli σ_τ-shift rules are exhibited for lengths L = 3, 4, …, 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely,

Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules , and . In particular, the period "T" of all attractors of rules , and , as well as of all isles of Eden of rules and , increases linearly with unit slope, in logarithmic scale, with the length L.

This paper presents the basin tree diagrams of all hyper Bernoulliσ_τ-shift rules for string lengths L = 3, 4, …, 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule has no Isles of Eden for all L, and that local rules and are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules , , , , as well as all invariant local rules, such as rules , , and , subject to no constraints.

This article is a short review of the recent results on properties of nonlinear fractional maps which are maps with power- or asymptotically power-law memory. These maps demonstrate the new type of attractors - cascade of bifurcations type trajectories, power-law convergence/divergence of trajectories, period doubling bifurcations with changes in the memory parameter, intersection of trajectories, and overlapping of attractors. In the limit of small time steps these maps converge to nonlinear fractional differential equations.

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameters values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors as well as periodic and quasi-periodic solutions and solutions that approach fixed points.

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property.

This stage of our journey through the universe of one-dimensional binary Cellular Automata is devoted to period-1 rules, constituting the first of the six groups in which we systematized the 88 globally-independent CA rules.

The first part of this article is mainly dedicated to reviewing the terminology and the empirical results found in the previous papers of our quest. We also introduce the concept of the ω-limit orbit with the purpose of linking our work to the classical theory of nonlinear dynamical systems. Moreover, we present the basin tree diagrams of all period-1 rules — except for rule , which is trivial — along with their Boolean cubes and time-1 characteristic functions.

In the second part, we prove a theorem demonstrating that all rules belonging to group 1 have robust period-1 rules for any finite, and infinite, bit-string length L. This is the first time we give analytical results on the behavior of CA local rules for large values of L and, consequently, for bi-infinite bit strings.

The theoretical treatment is complemented by two remarkable practical results: an explicit formula for generating isomorphic basin trees, and an algorithm for creating new periodic orbits by concatenation. We also provide several examples of both of them, showing how they help to avoid tedious simulations.

Motivation: A grand challenge in the modeling of biological systems is the identification of key variables which can act as targets for intervention. Good intervention targets are the "key players" in a system and have significant influence over other variables; in other words, in the context of diseases such as cancer, targeting these variables with treatments and interventions will provide the greatest effects because of their direct and indirect control over other parts of the system. Boolean networks are among the simplest of models, yet they have been shown to adequately model many of the complex dynamics of biological systems. Often ignored in the Boolean network model, however, are the so called basins of attraction. As the attractor states alone have been shown to correspond to cellular phenotypes, it is logical to ask which variables are most responsible for triggering a path through a basin to a particular attractor.

Results: This work claims that logic minimization (i.e. classical circuit design) of the collections of states in Boolean network basins of attraction reveals key players in the network. Furthermore, we claim that the key players identified by this method are often excellent targets for intervention given a network modeling a biological system, and more importantly, that the key players identified are not apparent from the attractor states alone, from existing Boolean network measures, or from other network measurements. We demonstrate these claims with a well-studied yeast cell cycle network and with a WNT5A network for melanoma, computationally predicted from gene expression data.

We discuss some of the basic features of extremal black holes in four-dimensional extended supergravities. Firstly, all regular solutions display an attractor behavior for the scalar field evolution towards the black hole horizon. Secondly, they can be obtained by solving first order ow equations even when they are not supersymmetric, provided one identifies a suitable superpotential W which also gives the black hole entropy at the horizon and its ADM mass at spatial infinity. We focus on N=8 supergravity and we review the basic role played by U-duality of the underlying supergravity in determining the attractors, their entropies, their masses and in classifying both regular and singular extremal black holes.

Dynamical systems described by real and complex variables are currently one of the most popular areas of scientific research. These systems play an important role in several fields of physics, engineering, and computer sciences, for example, laser systems, control (or chaos suppression), secure communications, and information science. Dynamical basic properties, chaos (hyperchaos) synchronization, chaos control, and generating hyperchaotic behavior of these systems are briefly summarized. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. They are also used to describe and simulate the physics of detuned laser and thermal convection of liquid flows, where the electric field and the atomic polarization amplitudes are both complex. Clearly, if the variables of the system are complex the equations involve twice as many variables and control parameters, thus making it that much harder for a hostile agent to intercept and decipher the coded message. Chaotic and hyperchaotic complex systems are stated as examples. Finally there are many open problems in the study of chaotic and hyperchaotic complex nonlinear dynamical systems, which need further investigations. Some of these open problems are given.

Logistic-like first order iterative maps, defined here as X_n+1 = rX_n^λ(1-X_n)^μ, are examined. The parameters r, λ and μ are positive real numbers, while the variable x and its map range from 0 to 1, the latter yielding the upper value of r for which full chaos is obtained. Depending on the values of λ and μ, the resulting x's can have a totally different behavior from those of the logistic map, given by λ = μ =1. The focus here is on fixed points since their existence, for given values of λ and μ, is necessary for obtaining chaotic x's. The purpose of the paper is four-fold: first, to define regions of existence for the fixed point(s) in terms of the parameters r, λ and μ; second, to determine the nature of the fixed points, whether they are attractors (stable), repellors (unstable) or super-stable, according to the values of the parameters; third, to define those maps for which the fixed points can be written in explicit algebraic form; and fourth, for iterative nearby maps, to obtain their fixed points in an approximate algebraic form in terms of the exact fixed points. The approximation is based on Newton's method, one step from the nearest iterative map whose fixed points can be obtained exactly, in explicit form. The validity of the fixed point approximation depends on the stability of the fixed points and is, subsequently, established in respect of well defined surfaces of the parameters r, λ and μ.

A major challenge for neuroscientists is to deduce and explain the neural mechanisms of the rapid transposition between stimulus energy and abstract concept - between the specific and the generic - in both material and conceptual aspects. Brain researchers are attempting three explanations of perception in terms of neural codes. Cellular neurobiologists find rate and frequency codes for stimulus features in trains of action potentials induced by stimuli and carried by topologically organized axons. Cognitivists correlate grouped firings of nerve cell assemblies with generalizations over classes of stimuli (faces, objects, odorants, words, etc.). Dynamicists correlate 2-D spatial patterns of brain waves (EEG) with the meanings of stimuli, which contain the knowledge about the stimuli. The patterns self-organize by trajectories through high-dimensional brain state space. This multivariate code is expressed in landscapes of non-convergent ('chaotic') attractors, which form the memory bank of the brain. Each pattern is formed by the dissipative dynamics of the cortical system. Its formation is preceded by a discontinuity in the oscillation of the EEG when a stimulus directs a search trajectory into a basin of attraction. Convergence to an attractor implements the cognitive process of generalization and abstraction. The attractor constructs the memory of the stimulus, and expresses it in the spatial pattern of amplitude modulation of a carrier wave. I propose that by its attractor dynamics the sensory cortex constructs knowledge from the information that is provided by the senses. Whereas sensory information is expressed in the action potentials of neurons and networks of neurons, knowledge is expressed in the continuous field dynamics that is indirectly observed in the EEG, in which the synapse and the action potential that support mental function are treated not as bits of information but as infinitesimals in differential calculus.

The analysis of industrial clusters has been carried on until now almost prevalently through the traditional statistical approaches used in empirical research. The growing and promising studies of inter-firm relationships stimulated the application of network methodologies, mostly confined to social network analysis. However, though this methodology gives a lot of interesting results, it suffers of the limitation of being static. In order to overcome this failure, this paper applies the methodology of Boolean networks, which allows a dynamic analysis. More specifically, the focus is on the knowledge exchanges flowing through the network of collaborations for innovation, which is indicated by current literature as a fundamental factor of competitiveness of industrial clusters. In particular, it has been studied the inter-firm transfer of managerial knowledge into the aerospace industrial cluster of the Lazio Region (Italy). Both the application of the Boolean network methodology and the content of the managerial knowledge network are traits of originality respect to the literature on industrial clusters and inter-firm relationships. Moreover, for it uses empirical data and introduces some innovative methodological devices this work can be an example replicable in other studies. The main findings are that the number of attractors are very sensitive to the threshold of activation of firms to transfer knowledge, and that even small changes determine the presence of key-players in the attractors (final stable states).

The 11th part of our tour through one-dimensional binary Cellular Automata concerns period-2 rules, which form the second group in our classification of the 88 globally-independent CA rules according to the properties of their periodic orbits. In this article, we display the basin tree diagrams of all period-2 rules along with their time-2 characteristic functions, and then we prove that all rules belonging to group 2 have robust period-2 ω-limit orbits for any finite, and infinite, bit string length. This rigorous result, which pairs with the one about period-1 rules given in the tenth installment of our chronicle, confirms what we stated about period-2 rules on the basis of empirical evidence. In the second part of this tutorial, we introduce the notion of quasi global-equivalence and prove that there are only 82 quasi globally-independent CA rules. For the first time, we show that the space-time patterns of globally-independent local rules can depend on each other, and we present an example of quasi-global transformation. We also define the super string, and its unique decimal representation , dubbed the super decimal, which provides a completely transparent yet rigorous proof that rule 170 is chaotic when L → ∞. Moreover, we present the basin tree generation formulas, which uncover the analytical relationships between basin trees of globally-equivalent rules. Last but not least, for pedagogical and epistemological reasons, we conclude this paper with the selection of rule137, instead of rule 110, as the prototypic universal Turing machine for our future discourse.

This 12th part of our Nonlinear Dynamics Perspective of Cellular Automata concludes a series of three articles devoted to CA local rules having robust periodic ω-limit orbits. Here, we consider only the two rules, 131 and 133, constituting the third of the six groups in which we classified the 1D binary Cellular Automata. Among the numerous theoretical results contained in this article, we emphasize the complete characterization of the ω-limit orbits, both robust and nonrobust, of these two rules and the proof that period-3 and period-6 ω-limit orbits are dense for 131 and 133, respectively. Furthermore, we will also introduce the fundamental concepts of perfect period-T orbitsets and riddled basins, and see how they emerge in rule 131.

As stated in the title, we also focus on permutive rules, which have been introduced in a previous installment of our series but never thoroughly studied. Indeed, we will review some of the well-known properties of such rules, like the surjectivity, examining their implications for finite and bi-infinite Cellular Automata.

Finally, we propose a new list of the 88 globally-independent local rules, which is slightly different from the one we have used so far but has the great advantage of being selected via a rigorous methodology and not an arbitrary choice. For the sake of completeness, we display in the appendix the basin tree diagrams and the portraits of the ω-limit orbits of the rules from this refined table which have not yet been reported in our previous articles.

More than one third of the 88 globally-independent Cellular Automata rules exhibit robust simple Bernoulli-shift dynamics. Among them we find rule 170 , which we proved to be chaotic in the previous episodes of our chronicle, and rule 184 , the famous global majority rule. Therefore, we cannot overstate the importance of the Bernoulli σ_τ-shift rules which we will present in two parts of our continuing odyssey on the Nonlinear Dynamics Perspective of Cellular Automata. This paper covers the first 15 of the 30 Bernoulli σ_τ-shift rules. In this paper, after recalling the main concepts of Bernoulli rules — such as the role of the three Bernoulli parameters σ, τ and β — we will display the basin tree diagrams of these rules together with a convenient summary of the results extracted from them. Then, we will show that the superstring is an excellent testing signal to find the robust behavior of a given rule. Finally, we will conclude this paper with a discussion about the difference between robust and nonrobust ω-limit orbits of the Bernoulli σ_τ-shift rules.

Over the past eight years, we have studied one of the simplest, yet extremely interesting, dynamical systems; namely, the one-dimensional binary Cellular Automata. The most remarkable results have been presented in a series of papers which is concluded by the present article. The final stop of our odyssey is devoted to the analysis of the second half of the 30 Bernoulli σ_τ-shift rules, which constitute the largest among the six groups in which we classified the 256 local rules. For all these 15 rules, we present the basin-tree diagrams obtained by using each bit string with L ≤ 8 as initial state, a summary of the characteristics of their ω-limit orbits, and the space-time patterns generated from the superstring. Also, in the last section we summarize the main results we obtained by means of our “nonlinear dynamics perspective”.