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74
Computation at the onset of chaos
 The Santa Fe Institute, Westview
, 1988
"... 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 perioddoubl ..."
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Cited by 86 (14 self)
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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 perioddoubling and bandmerging cascades, we derive expressions for the entropy, the interdependence ofmachine 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 contextfree Chomsky grammar in this case, we give finite descriptions at the higher computational level of contextfree Lindenmayer systems. We construct a restricted indexed contextfree grammar and its associated oneway 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
Dynamical systems, Measures and Fractals via Domain Theory
 Information and Computation
, 1995
"... 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 ar ..."
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Cited by 68 (19 self)
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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 omegacontinuous 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 nondeterministic 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.
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... 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 dist ..."
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Cited by 48 (10 self)
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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...
ALMOST EVERY REAL QUADRATIC MAP IS EITHER REGULAR OR STOCHASTIC
, 1997
"... 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 “ regul ..."
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Cited by 29 (3 self)
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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.
Scale Invariance in Biology: Coincidence Or Footprint of a Universal Mechanism?
, 2001
"... In this article, we present a selfcontained 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}fnoise where f denotes the frequency of a signal (temporal scale i ..."
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Cited by 24 (1 self)
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In this article, we present a selfcontained 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}fnoise 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 scalefree 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 (powerlaw distributions, fractals and 1}fnoise) and of critical phenomena. We then review typical mathematical models exhibiting such properties : edge of chaos, cellular automata and selforganized 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.
The Small Scale Structure of SpaceTime: A Bibliographical Review
, 1995
"... 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 ..."
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Cited by 19 (0 self)
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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
Expanding direction of the period doubling operator
 Commun. Math. Phys
, 1992
"... We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “PerronFrobenius 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 st ..."
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Cited by 10 (1 self)
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We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “PerronFrobenius 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.
Chasing chaos
 in: Congress on Evolutionary Computation
, 2003
"... 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 scenario ..."
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Cited by 9 (6 self)
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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 Bcell 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 Bcell algorithm in tracking multiple, moving targets. A number of experiments are conducted and results reported from the Bcell algorithm and standard hybrid genetic algorithm approaches. These results show the benefit of the Bcell algorithm approach when compared against these heuristic approaches. 1
Functional dynamics I: Articulation process
 Physica D 138
, 2000
"... 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 selfreference term f ◦ f, introduced by recalling that operation in ..."
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Cited by 5 (1 self)
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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 selfreference 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 piecewiseflat 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
Regularity Properties of Critical Invariant Circles of Twist Maps, and Their Universality
, 2008
"... We compute accurately the golden critical invariant circles of several areapreserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics o ..."
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Cited by 3 (0 self)
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We compute accurately the golden critical invariant circles of several areapreserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier analysis method developed by R. de la Llave and one of the authors, we compute numerically the Hölder regularities of these functions. Our computations show that – withing their numerical accuracy – these regularities are the same for the different maps studied. We discuss how our findings are related to some previous results: (a) to the constants giving the scaling behavior of the iterates on the critical invariant circle (discovered by Kadanoff and Shenker); (b) to some characteristics of the singular invariant measures connected with the distribution of iterates. Some of the functions studied have pointwise Hölder regularity that is different at different points. Our results give a convincing numerical support to the fact that the points with different Hölder exponents of these functions are interspersed in the same way for different maps, which is a strong indication that the underlying twist maps belong to the same universality class.