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close this bookThe Impact of Chaos on Science and Society (UNU, 1997, 415 pages)
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View the documentPreface
close this folder1. Chaotic dynamics
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View the documentAbstract
View the documentI. Introduction
View the documentII. Measuring Chaos
View the documentIII. Routes to chaos
View the documentIV. Chaos in physical systems
View the documentV. Noise and computer round-off errors
View the documentVI. Hamiltonian systems
View the documentVII. Symbolic dynamics
View the documentVIII. Concluding remarks
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close this folder2. Chaos and politics: applications of nonlinear dynamics to socio-political issues
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View the documentI. Introduction
View the documentII. The evolution of simple models for population dynamics
View the documentIII. Predicting the weather: An intuitive example of chaotic dynamics
View the documentIV. Chaotic dynamics and arms-race models
View the documentV. Future outlook
View the documentVI. Discussion and conclusions: The lessons of nonlinearity
View the documentNotes
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close this folder3. Is the EEG a strange attractor? Brain stem neuronal discharge patterns and electroencephalographic rhythms
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View the documentI. Introduction
View the documentII. The EEG as a global nonlinear oscillator: Quasiperiodic, (
View the documentIII. The neocortical source of the EEG signal
View the documentIV. Hierarchical noise driving of the hierarchical modes of the EEG by brain stem neurons
View the documentV. Deterministic and random models of hierarchical neuronal discharge patterns
View the documentVI. Stochastic resonance and quasiperiodicity in single neuron-neocortical dynamics
View the documentVII. Single neuron dynamics and the EEG: Two clinical examples
View the documentVIII. Summary
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close this folder4. The impact of chaos on mathematics
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View the documentI. Introduction: A historical view
View the documentII. The Lorenz attractor
View the documentIII. The Feigenbaum bifurcation
View the documentIV. Hydrodynamic turbulence
View the documentV. Ergodic theory of differentiable dynamical systems: Axiom A systems
View the documentVI. Ergodic theory of differentiable dynamical systems: General systems
View the documentVII. Quadratic maps of the interval and the Hénon attractor
View the documentVIII. Zeta functions
View the documentIX. Conclusion
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close this folder5. Chaos in neural networks
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View the documentI. Introduction
View the documentII. Chaotic dynamics in nerve membranes
View the documentIII. Chaos in biological neural networks
View the documentIV. Chaos in artificial neural networks
View the documentV. Discussion
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View the document6. The impact of chaos on physics
close this folder7. Chaos and physics
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View the documentDeterminism versus probabilism
View the documentA class of ubiquitous phenomena
View the documentThe impact of physics on chaos
View the documentThe problem of quantum chaos
View the documentIs there new physics in chaos?
View the documentDoes chaos bring a new fundamental principle into physics?
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close this folder8. Irreversibility and quantum chaos
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View the documentI. Introduction
View the documentII. Quantum suppression of classical chaos
View the documentIII. Recovery of chaos
View the documentIV. Stationary dissipation
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close this folder9. Impact of high-dimensional chaos: A further step towards dynamical complexity
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View the documentI. From chaos to high-dimensional chaos
View the documentII. From spatio-temporal chaos to turbulence
View the documentIII. High-dimensional chaos as the basis of statistical mechanics
View the documentIV. Network of chaotic elements
View the documentV. Neural information processing with high-dimensional chaos
View the documentVI. Homeochaos in biological networks
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View the documentNotes
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close this folder10. The impact of chaos on biology: Promising directions for research
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View the documentAbstract
View the documentIntroduction
View the documentPersistence and extinction in animal populations
View the documentPeriodicity in chaos
View the documentConclusion
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View the documentAppendix. The difficulties of finding chaos in biological data
close this folder11. Dynamical disease - The impact of nonlinear dynamics and chaos on cardiology and medicine
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View the documentAbstract
View the documentI. Introduction - Chaos and dynamical disease
View the documentII. Chaos in physiological experiments and medicine
View the documentIII. Nonlinear dynamics in cardiology
View the documentIV. Summary and conclusions
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close this folder12. The impact of chaos on meteorology
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View the documentI. Introduction
View the documentII. Local and global properties
View the documentIII. The middle-latitude jet as a dynamical system
View the documentIV. Conclusions
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close this folder13. The concept of chaos in the problem of earthquake prediction
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View the documentNonlinear dynamics and earthquake-prone faults
View the documentModelling
View the documentPrediction
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close this folder14. The impact of chaos on engineering
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View the documentIntroduction
View the documentThe role of geometrical theory in applied mechanics
View the documentTransient failure
View the documentThe influence of chaotic transients
View the documentConclusions
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close this folder15. The impact of chaos on economic theory
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View the documentI. Introduction
View the documentII. Impediments to chaos in economics
View the documentIII. Empirical investigations
View the documentIV. Theoretical investigations
View the documentV. Conclusions
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close this folder16. Chaos in society: Reflections on the impact of chaos theory on sociology
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close this folder17. Strange attractors and the origin of chaos
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View the documentI. Prologue
View the documentII. The oldest chaos in a non-autonomous system - A shattered egg
View the documentIII. Encounter with the Japanese attractor
View the documentIV. The Hayashi Laboratory at the time of the ''McGraw-Hill Book''
View the documentV. From the harmonic balance method to the mapping method
View the documentVI. The true value of an advisor: A scion of chaos
View the documentVII. The end of the Chihiro Hayashi Laboratory
View the documentVIII. The original data that were preserved
View the documentIX. Epilogue
View the documentAcknowledgements
View the documentReferences
View the documentPanel discussion: The impact of chaos on science and society
View the documentOpening address
View the documentContributors
View the documentOther titles of interest

(introduction...)

Renate Mayntz

This paper draws heavily on three previous publications (Mayntz 1988, 1990, 1991), some passages from these publications having been adopted virtually unchanged.

Over the past decades, attention in the natural sciences has increasingly turned to phenomena that defy analysis in terms of the traditional physical world view with its assumptions of linearity and reversibility, i.e., to the behaviour of systems remote from equilibrium and to discontinuous processes resulting from nonlinearity. After the recognition of the stochastic nature of many real processes, the attention paid to nonlinear processes means a further step away from the traditional mechanistic world view. Nonlinear systems display a number of behaviours that can be widely observed in the natural world. Their state variables can change discontinuously, producing phase jumps, i.e., sudden changes of state, as in the phenomenon of ferro-magnetism or in superconductivity. In such discontinuous processes, threshold and critical mass phenomena often play a decisive role. A threshold phenomenon exists where a dependent variable initially does not react at all, or only very little, to continuous changes of an independent variable, but beyond a given point it reacts suddenly and strongly. The threshold may be defined by a critical mass, e.g. the number of particles of a specific kind that must be present before a reaction sets in, but other kinds of threshold also exist. There are phase transitions from order to disorder and in the reverse direction. The behaviour of nonlinear systems can become completely irregular - or "chaotic" - if the values of given parameters move into a particular range. On the other hand, nonlinear systems can also move spontaneously from disorder to order, a stationary state far from equilibrium; this is called a dissipative structure, or self-organization. Furthermore, systems characterized by nonlinear dynamics can display a specific kind of irreversibility, i.e. hysteresis (path dependency of phase jumps), and a specific kind of indeterminateness expressed in the term bifurcation: a point where a trajectory can proceed in different directions. The analysis of nonlinear dynamics has been enhanced by the development of new mathematical methods, as such René Thom's catastrophe theory, and by the computational power of modern EDP,1 which for instance made it possible to discover and formalize the phenomenon of (deterministic) chaos.

The label "chaos theory" is presently being used both in a narrower and a more comprehensive sense. The narrowest interpretation of the term equates it with the mathematical theory of deterministic chaos (and its applications), and hence with the preconditions of a phase transition from order to disorder in nonlinear systems. In a wider interpretation, chaos theory may refer to discontinuous processes moving either from order to chaos or from chaos (or disorder) to order; the term would thus also cover phenomena of self-organization. Sometimes, the term chaos theory seems to be used in an even wider sense, referring to the whole field of research into nonlinear dynamics of non-equilibrium systems. In this paper, the second understanding of the term "chaos theory" is used. I shall be considering discontinuous processes, or phase transitions, going in both directions - from order to chaos and from chaos to order. However, I shall rather use the terms self-organization or nonlinear dynamics except where we deal with phase transitions from order to chaos, the narrowest meaning of "chaos theory."

The following argument can be summarized in the form of a few theses:

1. Natural science theories of nonlinear dynamics were not necessary to turn the attention of social scientists to phenomena of sudden disruptions of order and system breakdown.

2. Natural science theories and models of nonlinear dynamics have nevertheless had an impact both on formal modelling attempts and substantive theorizing in sociology.

3. The potential relevance of natural science theories of nonlinear dynamics lies in the promise to gain a better understanding of discontinuous changes at the macro-level as a consequence of micro-level processes.

4. The nature of social reality nevertheless seriously limits the potential applicability of natural science models of nonlinear dynamics.