Computation at levels beyond storage and transmission of information appears in physical systems at phase transitions. We investigate this phenomenon using minimal computational models of dynamical systems that undergo a transition to chaos as a function of a nonlinearity parameter. For period-doubling and band-merging cascades, we derive expressions for the entropy, the interdependence of-machine complexity and entropy, and the latent complexity of the transition to chaos. At the transition deterministic finite automaton models diverge in size. Although there is no regular or context-free Chomsky grammar in this case, we give finite descriptions at the higher computational level of context-free Lindenmayer systems. We construct a restricted indexed context-free grammar and its associated one-way nondeterministic nested stack automaton for the cascade limit language. This analysis of a family of dynamical systems suggests a complexity theoretic description of phase transitions based on the informational diversity and computational complexity of observed data that is independent of particular system control parameters. The approach gives a much more refined picture of the architecture of critical states than is available via

We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X- X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L are respectively the Vietoris hyperspace, the upper hyperspace and the lower hyperspace functors. We show that if (X, f) is chaotic, then so is (UX, U f). When X is locally compact UX, is a continuous bounded complete dcpo. If X is second countable as well, then UX will be omega-continuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of non-deterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of finite Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...

We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc: x ↦ → x² + c has zero measure. This yields the statement in the title (where “ regular ” means to have an attracting cycle and “stochastic” means to have an absolutely continuous invariant measure). An application to the MLC problem is given.

In this article, we present a self-contained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}f-noise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scale-free phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (power-law distributions, fractals and 1}fnoise) and of critical phenomena. We then review typical mathematical models exhibiting such properties : edge of chaos, cellular automata and self-organized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.

This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “Perron-Frobenius type operator”, to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof 1 is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point. Contents §1 Introduction. §2 The Period Doubling Operator and the Induced Operator. §2.1 From the period doubling operator to the induced operator. §2.2 The induced operator Lϕ.

Both simple and hybrid genetic algorithms encounter difficulties when presented with a function which has multiple values. Similarly, changing environments or functions which change rapidly present other problems. This paper presents an algorithm that is capable of coping with both of these scenarios: it can accommodate multiple solutions simultaneously and can track changes in optima efficiently. The proposed B-cell algorithm is inspired by the natural immune system, which itself displays similar capabilities of tracking multiple, moving targets in the form of infectious agents. This paper employs two nonlinear mappings which display chaotic behaviour to demonstrate the effectiveness of the B-cell algorithm in tracking multiple, moving targets. A number of experiments are conducted and results reported from the B-cell algorithm and standard hybrid genetic algorithm approaches. These results show the benefit of the B-cell algorithm approach when compared against these heuristic approaches. 1

The articulation process of dynamical networks is studied with a functional map, a minimal model for the dynamic change of relationships through iteration. The model is a dynamical system of a function f, not of variables, having a self-reference term f ◦ f, introduced by recalling that operation in a biological system is often applied to itself, as is typically seen in rules in the natural language or genes. Starting from an inarticulate network, two types of fixed points are formed as an invariant structure with iterations. The function is folded with time, until it has finite or infinite piecewise-flat segments of fixed points, regarded as articulation. For an initial logistic map, attracted functions are classified into step, folded step, fractal, and random phases, according to the degree of folding. Oscillatory dynamics are also found, where function values are mapped to several fixed points periodically. The significance of our results to prototype categorization in language is discussed. 1

As a first step toward realizing a dynamical system that evolves while spontaneously determining its own rule for time evolution, function dynamics (FD) is analyzed. FD consists of a functional equation with a self-referential term, given as a dynamical system of a one-dimensional map. Through the time evolution of this system, a dynamical graph (a network) emerges. This graph has three interesting properties: (i) vertices appear as stable elements, (ii) the terminals of directed edges change in time, and (iii) some vertices determine the dynamics of edges, and edges determine the stability of the vertices, complementarily. Two aspects of FD are studied, the generation of a graph (network) structure and the dynamics of this graph (network) in the system.