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...) |

I |

II |

III |

IV |

V |

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 |

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.