The Impact of Chaos on Science and Society (UNU, 1997, 415 pages): 16. Chaos in society: Reflections on the impact of chaos theory on sociology: (introduction...)

The Impact of Chaos on Science and Society (UNU, 1997, 415 pages)

(introduction...)

Preface

1. Chaotic dynamics

(introduction...)

Abstract

I. Introduction

II. Measuring Chaos

III. Routes to chaos

IV. Chaos in physical systems

V. Noise and computer round-off errors

VI. Hamiltonian systems

VII. Symbolic dynamics

VIII. Concluding remarks

References

2. Chaos and politics: applications of nonlinear dynamics to socio-political issues

(introduction...)

Abstract

I. Introduction

II. The evolution of simple models for population dynamics

III. Predicting the weather: An intuitive example of chaotic dynamics

IV. Chaotic dynamics and arms-race models

V. Future outlook

VI. Discussion and conclusions: The lessons of nonlinearity

Notes

References

3. Is the EEG a strange attractor? Brain stem neuronal discharge patterns and electroencephalographic rhythms

(introduction...)

I. Introduction

II. The EEG as a global nonlinear oscillator: Quasiperiodic, (

III. The neocortical source of the EEG signal

IV. Hierarchical noise driving of the hierarchical modes of the EEG by brain stem neurons

V. Deterministic and random models of hierarchical neuronal discharge patterns

VI. Stochastic resonance and quasiperiodicity in single neuron-neocortical dynamics

VII. Single neuron dynamics and the EEG: Two clinical examples

VIII. Summary

References

4. The impact of chaos on mathematics

(introduction...)

Abstract

I. Introduction: A historical view

II. The Lorenz attractor

III. The Feigenbaum bifurcation

IV. Hydrodynamic turbulence

V. Ergodic theory of differentiable dynamical systems: Axiom A systems

VI. Ergodic theory of differentiable dynamical systems: General systems

VII. Quadratic maps of the interval and the Hénon attractor

VIII. Zeta functions

IX. Conclusion

References

5. Chaos in neural networks

(introduction...)

Abstract

I. Introduction

II. Chaotic dynamics in nerve membranes

III. Chaos in biological neural networks

IV. Chaos in artificial neural networks

V. Discussion

References

6. The impact of chaos on physics

7. Chaos and physics

(introduction...)

Determinism versus probabilism

A class of ubiquitous phenomena

The impact of physics on chaos

The problem of quantum chaos

Is there new physics in chaos?

Does chaos bring a new fundamental principle into physics?

Acknowledgements

Notes

References

8. Irreversibility and quantum chaos

(introduction...)

Abstract

I. Introduction

II. Quantum suppression of classical chaos

III. Recovery of chaos

IV. Stationary dissipation

References

9. Impact of high-dimensional chaos: A further step towards dynamical complexity

(introduction...)

I. From chaos to high-dimensional chaos

II. From spatio-temporal chaos to turbulence

III. High-dimensional chaos as the basis of statistical mechanics

IV. Network of chaotic elements

V. Neural information processing with high-dimensional chaos

VI. Homeochaos in biological networks

Acknowledgements

Notes

References

10. The impact of chaos on biology: Promising directions for research

(introduction...)

Abstract

Introduction

Persistence and extinction in animal populations

Periodicity in chaos

Conclusion

Acknowledgements

Notes

References

Appendix. The difficulties of finding chaos in biological data

11. Dynamical disease - The impact of nonlinear dynamics and chaos on cardiology and medicine

(introduction...)

Abstract

I. Introduction - Chaos and dynamical disease

II. Chaos in physiological experiments and medicine

III. Nonlinear dynamics in cardiology

IV. Summary and conclusions

Acknowledgements

References

12. The impact of chaos on meteorology

(introduction...)

I. Introduction

II. Local and global properties

III. The middle-latitude jet as a dynamical system

IV. Conclusions

References

13. The concept of chaos in the problem of earthquake prediction

(introduction...)

Abstract

Nonlinear dynamics and earthquake-prone faults

Modelling

Prediction

Conclusion

References

14. The impact of chaos on engineering

(introduction...)

Introduction

The role of geometrical theory in applied mechanics

Transient failure

The influence of chaotic transients

Conclusions

Acknowledgements

References

15. The impact of chaos on economic theory

(introduction...)

I. Introduction

II. Impediments to chaos in economics

III. Empirical investigations

IV. Theoretical investigations

V. Conclusions

References

16. Chaos in society: Reflections on the impact of chaos theory on sociology

(introduction...)

III

Notes

References

17. Strange attractors and the origin of chaos

(introduction...)

I. Prologue

II. The oldest chaos in a non-autonomous system - A shattered egg

III. Encounter with the Japanese attractor

IV. The Hayashi Laboratory at the time of the ''McGraw-Hill Book''

V. From the harmonic balance method to the mapping method

VI. The true value of an advisor: A scion of chaos

VII. The end of the Chihiro Hayashi Laboratory

VIII. The original data that were preserved

IX. Epilogue

Acknowledgements

References

Panel discussion: The impact of chaos on science and society

Opening address

Contributors

Other 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.