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Chaos

Edited by Jon Lawhead (University of Southern California)
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Summary Chaotic systems have two definitive properties: (1) they are deterministic, and (2) their long-term behavior sensitively depends on their initial conditions.  Edward Lorenz, the father of modern chaos theory, summarized chaos as being present when "the past determines the future, but the approximate past doesn't determine the approximate future."  The most familiar depiction of chaotic behavior is the so-called "butterfly effect," in which a very small perturbation of the global weather system (the flapping of a butterfly's wings in Argentina) results in a very large change to that same system (the genesis of a hurricane in Texas three weeks later).  Of course, it is clear that not all butterfly flaps spawn hurricanes (thankfully!), so the central problem of chaos theory is the precise mathematical modeling of this sensitive dependence relationship, and the determination of when very small changes are likely to have very large effects.  Many important natural systems exhibit chaotic dynamics under certain circumstances; in addition to the global weather system, the global climate, social systems (like the economy), and biological systems (like the human brain) can sometimes exhibit chaotic dynamics.  The understanding and modeling of chaos is an important part of understanding complex natural systems.
Key works Lorenz 1963 is the earliest mature articulation of the central ideas of chaos theory, though it builds on concepts from Saltzman 1962.  More recent developments of the formalism include Abarbanel 1992 and Guan unknown.  The concept has also been expanded to define a notion of structural chaos by Mayo-Wilson 2015.
Introductions Strogatz 2001Auyang manuscriptLorenz 1963Lawhead forthcoming
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  1. H. D. I. Abarbanel (1992). Local Lyapunov Exponents Computed From Observed Data. Journal of Nonlinear Science 2 (3):343-365.
    We develop methods for determining local Lyapunov exponents from observations of a scalar data set. Using average mutual information and the method of false neighbors, we reconstruct a multivariate time series, and then use local polynomial neighborhood-to-neighborhood maps to determine the phase space partial derivatives required to compute Lyapunov exponents. In several examples we demonstrate that the methods allow one to accurately reproduce results determined when the dynamics is known beforehand. We present a new recursive QR decomposition method for finding (...)
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  2. A. Abbasi, S. H. Fathi, G. B. Gharehpatian, A. Gholami & H. R. Abbasi (2013). Voltage Transformer Ferroresonance Analysis Using Multiple Scales Method and Chaos Theory. Complexity 18 (6):34-45.
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  3. Ralph Abraham (1994). Chaos, Gaia, Eros a Chaos Pioneer Uncovers the Three Great Streams of History.
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  4. Philip Anderson & Jack Cohen (1999). Reviews: Coping with Uncertainty, Insights From the New Sciences of Chaos, Self-Organization, and Complexity, Uri Merry. [REVIEW] Emergence: Complexity and Organization 1 (2):106-108.
    (1999). Reviews: Coping with Uncertainty, Insights from the New Sciences of Chaos, Self-Organization, and Complexity, Uri Merry. Emergence: Vol. 1, No. 2, pp. 106-108.
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  5. S. M. Anlage (2000). Book Review: Quantum Chaos-An Introduction. [REVIEW] Foundations of Physics 30 (7):1135-1138.
  6. I. Antoniou & Z. Suchanecki (1997). The Fuzzy Logic of Chaos and Probabilistic Inference. Foundations of Physics 27 (3):333-362.
    The logic of a physical system consists of the elementary observables of the system. We show that for chaotic systems the logic is not any more the classical Boolean lattice but a kind of fuzzy logic which we characterize for a class of chaotic maps. Among other interesting properties the fuzzy logic of chaos does not allow for infinite combinations of propositions. This fact reflects the instability of dynamics and it is shared also by quantum systems with diagonal singularity. We (...)
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  7. Fatihcan M. Atay, Sarika Jalan & Jürgen Jost (2009). Randomness, Chaos, and Structure. Complexity 15 (1):29-35.
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  8. Harald Atmanspacher, Characterizing Spontaneous Irregular Behavior in Coupled Map Lattices.
    Two-dimensional coupled map lattices display, in a specific parameter range, a stable phase (quasi-) periodic in both space and time. With small changes to the model parameters, this stable phase develops spontaneous eruptions of nonperiodic behavior. Although this behavior itself appears irregular, it can be characterized in a systematic fashion. In particular, parameter-independent features of the spontaneous eruptions may allow useful empirical characterizations of other phenomena that are intrinsically hard to predict and reproduce. Specific features of the distributions of lifetimes (...)
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  9. Harald Atmanspacher, Ontic and Epistemic Descriptions of Chaotic Systems.
    Traditional philosophical discourse draws a distinction between ontology and epistemology and generally enforces this distinction by keeping the two subject areas separated and unrelated. In addition, the relationship between the two areas is of central importance to physics and philosophy of physics. For instance, all kinds of measurement-related problems force us to consider both our knowledge of the states and observables of a system (epistemic perspective) and its states and observables independent of such knowledge (ontic perspective). This applies to quantum (...)
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  10. Harald Atmanspacher & Robert C. Bishop (2007). Stability Conditions in Contextual Emergence. Chaos and Complexity Letters 2:139-150.
    The concept of contextual emergence is proposed as a non-reductive, yet welldefined relation between different levels of description of physical and other systems. It is illustrated for the transition from statistical mechanics to thermodynamical properties such as temperature. Stability conditions are crucial for a rigorous implementation of contingent contexts that are required to understand temperature as an emergent property. It is proposed that such stability conditions are meaningful for contextual emergence beyond physics as well.
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  11. David Aubin (1998). A Cultural History of Catastrophes and Chaos: Around the Institut des Hautes Études Scientifiques. Princeton.
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  12. David Aubin & Amy Dalmedico (2002). Writing the History of Dynamics Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures. Historia Mathematica 29:1–67.
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  13. Sunny Auyang, How Science Comprehends Chaos.
    Behaviors of chaotic systems are unpredictable. Chaotic systems are deterministic, their evolutions being governed by dynamical equations. Are the two statements contradictory? They are not, because the theory of chaos encompasses two levels of description. On a higher level, unpredictability appears as an emergent property of systems that are predictable on a lower level. In this talk, we examine the structure of dynamical theories to see how they employ multiple descriptive levels to explain chaos, bifurcation, and other complexities of nonlinear (...)
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  14. R. Badii (1997). Complexity: Hierarchical Structures and Scaling in Physics. Cambridge University Press.
    This is a comprehensive discussion of complexity as it arises in physical, chemical, and biological systems, as well as in mathematical models of nature. Common features of these apparently unrelated fields are emphasised and incorporated into a uniform mathematical description, with the support of a large number of detailed examples and illustrations. The quantitative study of complexity is a rapidly developing subject with special impact in the fields of physics, mathematics, information science, and biology. Because of the variety of the (...)
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  15. Arek Bagiânski & Agnieszka Wierzchucka (1999). Chaos.
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  16. Gerold Baier (1995). A Strategy for Higher Chaos. In R. J. Russell, N. Murphy & A. R. Peacocke (eds.), Chaos and Complexity. Vatican Observatory Publications 189.
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  17. Riccardo Baldissone (2013). Chaos Beyond Order: Overcoming the Quest for Certainty and Conservation in Modern Western Sciences. Cosmos and History: The Journal of Natural and Social Philosophy 9 (1):35-49.
    Chaos theory not only stretched the concept of chaos well beyond its traditional semantic boundaries, but it also challenged fundamental tenets of physics and science in general. Hence, its present and potential impact on the Western worldview cannot be underestimated. I will illustrate the relevance of chaos theory in regard to modern Western thought by tracing the concept of order, which modern thinkers emphasised as chaos’ dichotomic counterpart. In particular, I will underline how the concern of seventeenth-century natural philosophers with (...)
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  18. Robert W. Batterman (1993). Defining Chaos. Philosophy of Science 60 (1):43-66.
    This paper considers definitions of classical dynamical chaos that focus primarily on notions of predictability and computability, sometimes called algorithmic complexity definitions of chaos. I argue that accounts of this type are seriously flawed. They focus on a likely consequence of chaos, namely, randomness in behavior which gets characterized in terms of the unpredictability or uncomputability of final given initial states. In doing so, however, they can overlook the definitive feature of dynamical chaos--the fact that the underlying motion generating the (...)
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  19. Robert W. Batterman (1991). Chaos, Quantization, and the Correspondence Principle. Synthese 89 (2):189 - 227.
  20. Robert W. Batterman & Homer White (1996). Chaos and Algorithmic Complexity. Foundations of Physics 26 (3):307-336.
    Our aim is to discover whether the notion of algorithmic orbit-complexity can serve to define “chaos” in a dynamical system. We begin with a mostly expository discussion of algorithmic complexity and certain results of Brudno, Pesin, and Ruelle (BRP theorems) which relate the degree of exponential instability of a dynamical system to the average algorithmic complexity of its orbits. When one speaks of predicting the behavior of a dynamical system, one usually has in mind one or more variables in the (...)
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  21. Roger A. Beaumont (1994). War, Chaos, and History. Monograph Collection (Matt - Pseudo).
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  22. Christopher Belanger (2013). On Two Mathematical Definitions of Observational Equivalence: Manifest Isomorphism and Epsilon-Congruence Reconsidered. Studies in History and Philosophy of Science Part B 44 (2):69-76.
    In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely mathematical definition of observational equivalence (...)
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  23. Gordon Belot & John Earman (1997). Chaos Out of Order: Quantum Mechanics, the Correspondence Principle and Chaos. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 28 (2):147-182.
    A vast amount of ink has been spilled in both the physics and the philosophy literature on the measurement problem in quantum mechanics. Important as it is, this problem is but one aspect of the more general issue of how, if at all, classical properties can emerge from the quantum descriptions of physical systems. In this paper we will study another aspect of the more general issue-the emergence of classical chaos-which has been receiving increasing attention from physicists but which has (...)
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  24. Gordon Belot & Lina Jansson (2010). Alisa Bokulich, Reexamining the Quantum-Classical Relation: Beyond Reductionism and Pluralism , Cambridge University Press, Cambridge (2008) ISBN 978-0-521-85720-8 Pp. X+195. [REVIEW] Studies in History and Philosophy of Science Part B 41 (1):81-83.
  25. Andrew Belsey (1994). Chaos and Order, Environment and Anarchy. Royal Institute of Philosophy Supplement 36:157-167.
    The distinction between chaos and order has been central to western philosophy, both in metaphysics and politics. At the beginning, it was intrinsic to presocratic natural philosophy, and shortly after that to the cosmology and social philosophy of Plato. Even in the pre-presocratic period there were important intimations of it. Thus Hesiod tells us that ‘first of all did Chaos come into being’ —although exactly what is meant by ‘chaos’ in this context is not clear. between earth and sky . (...)
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  26. Melvyn S. Berger (1995). Order Beyond Periodicity: Fighting Chaos for Quasiperiodic Motion of Nonlinear Hamiltonian Systems. In R. J. Russell, N. Murphy & A. R. Peacocke (eds.), Chaos and Complexity. Vatican Observatory Publications 185.
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  27. K.-F. Berggren & T. Ouchterlony (2001). Chaos in a Quantum Dot with Spin-Orbit Coupling. Foundations of Physics 31 (2):233-242.
    Level statistics and nodal point distribution in a rectangular semiconductor quantum dot are studied for different degrees of spin-orbit coupling. The chaotic features occurring from the spin-orbit coupling have no classical counterpart. Using experimental values for GaSb/InAs/GaSb semiconductor quantum wells we find that level repulsion can lead to the semi-Poisson distribution for nearest level separations. Nodal lines and nodal points are also investigated. Comparison is made with nodal point distributions for fully chaotic states.
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  28. Robert Bishop, Chaos. Stanford Encyclopedia of Philosophy.
    The big news about chaos is supposed to be that the smallest of changes in a system can result in very large differences in that system's behavior. The so-called butterfly effect has become one of the most popular images of chaos. The idea is that the flapping of a butterfly's wings in Argentina could cause a tornado in Texas three weeks later. By contrast, in an identical copy of the world sans the Argentinian butterfly, no such storm would have arisen (...)
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  29. Robert C. Bishop (2004). Nonequilibrium Statistical Mechanics Brussels–Austin Style. Studies in History and Philosophy of Science Part B 35 (1):1-30.
    The fundamental problem on which Ilya Prigogine and the Brussels–Austin Group have focused can be stated briefly as follows. Our observations indicate that there is an arrow of time in our experience of the world (e.g., decay of unstable radioactive atoms like uranium, or the mixing of cream in coffee). Most of the fundamental equations of physics are time reversible, however, presenting an apparent conflict between our theoretical descriptions and experimental observations. Many have thought that the observed arrow of time (...)
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  30. Robert C. Bishop & Frederick M. Kronz (1999). Is Chaos Indeterministic? In Maria Luisa Dalla Chiara (ed.), Language, Quantum, Music. 129--141.
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  31. Marcel Bodea (2005). Chaos and Determinism: Prediction and Anticipation – a Conceptual Distinction. Studia Philosophica 1.
    Today, it is difficult to find much unanimity in what is “the prediction”. New mathematical theories offer the support for an epistemological investigation of predictability. Chaos breaks across the lines that separate the scientific predictions. Chaos poses new conceptual problems in philosophy. Prediction and Anticipation is a conceptual distinction between numerical predictions and geometrical `predictions`. Anticipation means to see what kind of theoretical picture one could develop. Conceptual analysis on a philosophical level is an operational way to clarify the so (...)
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  32. Marcel Bodea (2002). Chaos and Determinism From the Standpoint of the Sensibility to Initial Conditions: An Epistemological Approach. Studia Philosophica 2.
    The classical determinism is the view for which the only barrier to prediction is our lack of knowledge, due to a lack of observational data or to the lack of knowledge of the relevant laws of nature. A new mathematical theory, called CHAOS, offers a way of understanding order, order masquerading as randomness. My purpose here is an epistemological investigation: to examine a new area of scientific and philosophic inquiry, called ‘deterministic chaos’; to analyse the determinism in relation to unpredictability (...)
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  33. O. Bohigas, P. Lebœuf & M. J. Sánchez (2001). Spectral Spacing Correlations for Chaotic and Disordered Systems. Foundations of Physics 31 (3):489-517.
    New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to diffusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron–Frobenius operator, is derived. (...)
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  34. Oriol Bohigas, Patricio Leboeuf & M. J. Sanchez (2001). Spectral Spacing Correlations for Chaotic and Disordered Systems. Foundations of Physics 31 (3):489-517.
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  35. Norbert W. Bolz (1992). Chaos Und Simulation. Monograph Collection (Matt - Pseudo).
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  36. D. M. Borchert (ed.) (2006). Encyclopedia of Philosophy, Second Edition.
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  37. Moses Boudourides (1996). Chaos and Critical Theory. Neusis 5:115-121.
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  38. Alain Boutot (1991). La philosophie du chaos. Revue Philosophique de la France Et de l'Etranger 181 (2):145 - 178.
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  39. Seamus Bradley, Scientific Uncertainty: A User's Guide. Grantham Institute on Climate Change Discussion Paper.
    There are different kinds of uncertainty. I outline some of the various ways that uncertainty enters science, focusing on uncertainty in climate science and weather prediction. I then show how we cope with some of these sources of error through sophisticated modelling techniques. I show how we maintain confidence in the face of error.
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  40. Robert E. Brooks (1998). Creativity and the Cathartic Moment: Chaos Theory and the Art of Theatre. Dissertation, Louisiana State University and Agricultural & Mechanical College
    This dissertation investigates the potential applications of the scientific paradigm known as "chaos theory" in the examination of dramatic theory. By illuminating the limitations of traditional Newtonian physics and Euclidean geometry, chaos theory conveys philosophical implications that transcend the scientific and provide suitable tools for describing cultural and artistic phenomena. These implications include emphasis on unpredictability, interaction and feedback, qualitative rather that quantitative analyses, and a nonlinear, continuous, even holistic perspective of systems traditionally viewed as dischotomous . ;This study examines (...)
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  41. L. Brown & J. Brown, Out of Chaos and Into a New Identity.
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  42. Stephen Brush (1986). Order Out of Chaos: Man's New Dialogue with Nature. [REVIEW] British Journal for the History of Science 19 (3):371-372.
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  43. Jean E. Burns (2007). Vacuum Radiation, Entropy, and Molecular Chaos. Foundations of Physics 37 (12):1727-1737.
    Vacuum radiation causes a particle to make a random walk about its dynamical trajectory. In this random walk the root mean square change in spatial coordinate is proportional to t 1/2, and the fractional changes in momentum and energy are proportional to t −1/2, where t is time. Thus the exchange of energy and momentum between a particle and the vacuum tends to zero over time. At the end of a mean free path the fractional change in momentum of a (...)
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  44. John L. Casti (1996). Chaos Data Analyzer. Complexity 2 (2):46-47.
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  45. C. M. Caves (1994). Quantum Theory: Concepts and Methods. Foundations of Physics 24:1583-1583.
  46. Carlton M. Caves & R.�Diger Schack (1997). Unpredictability, Information, and Chaos. Complexity 3 (1):46-57.
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  47. Hugues Chat (1995). Towards a Thermodynamic Approach of Spatiotemporal Chaos. In R. J. Russell, N. Murphy & A. R. Peacocke (eds.), Chaos and Complexity. Vatican Observatory Publications 31.
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  48. Boris V. Chirikov (1986). Transient Chaos in Quantum and Classical Mechanics. Foundations of Physics 16 (1):39-49.
    Bogolubov's classical example of statistical relaxation in a many-dimensional linear oscillator is discussed. The relation of the discovered relaxation mechanism to quantum dynamics as well as to some new problems in classical mechanics is considered.
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  49. Kamila Chodarcewicz (2008). Podmiot, chaos i ironia. Friedricha Schlegla pytania o kształtowanie jednostki. Estetyka I Krytyka 1 (1):83-92.
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  50. John Cleave & Ian J. Thompson (1988). Chaos and Order: An Interview with Professor Michael Berry F.R.S. Cogito 2 (1):1-5.
    Michael Berry, Professor of Physics at Bristol University, discusses the philosophical ideas underlying his research to the theories of catastrophes and chaotic systems. He is one of England's leading scientists, and has been instrumental in the growth of interest in qualitative phenomena.
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