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Unstable Singularities and Randomness - 1st Edition - ISBN: 9780444516138, 9780080474694

Unstable Singularities and Randomness

1st Edition

Their Importance in the Complexity of Physical, Biological and Social Sciences

Author: Joseph Zbilut
Hardcover ISBN: 9780444516138
eBook ISBN: 9780080474694
Imprint: Elsevier Science
Published Date: 18th June 2004
Page Count: 252
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Traditionally, randomness and determinism have been viewed as being diametrically opposed, based on the idea that causality and determinism is complicated by “noise.” Although recent research has suggested that noise can have a productive role, it still views noise as a separate entity. This work suggests that this not need to be so. In an informal presentation, instead, the problem is traced to traditional assumptions regarding dynamical equations and their need for unique solutions. If this requirement is relaxed, the equations admit for instability and stochasticity evolving from the dynamics itself. This allows for a decoupling from the “burden” of the past and provides insights into concepts such as predictability, irreversibility, adaptability, creativity and multi-choice behaviour. This reformulation is especially relevant for biological and social sciences whose need for flexibility a propos of environmental demands is important to understand: this suggests that many system models are based on randomness and nondeterminism complicated with a little bit of determinism to ultimately achieve concurrent flexibility and stability. As a result, the statistical perception of reality is seen as being a more productive tool than classical determinism. The book addresses scientists of all disciplines, with special emphasis at making the ideas more accessible to scientists and students not traditionally involved in the formal mathematics of the physical sciences. The implications may be of interest also to specialists in the philosophy of science.

Key Features

· Presents the ideas in an informal language. · Provides tools for exploring data for singularities.


Physical scientists, biological scientists, social scientists, psychologists, psychiatrists, cognitive scients, neuroscientists and humanists (artists/historians/writiers/critics).

Table of Contents


  1. Probability and Dynamics
    1. A Dichotomy
    1. Historical Perspective
    1. Probabilities
    1. Randomness
    1. Singularities
    1. Models and Reality
  2. Singularities and Instability
    1. Dynamics
  3. 1.1. Attractors
  4. 1.2. Liapunov Exponents
    1. Limitations of the Classical Approach
    1. Dynamical Instability
    1. Lipschitz Conditions
    1. Basic Concepts
  5. 5.1. Dissipation
  6. 5.2. Terminal Dynamics Limit Sets
  7. 5.3. Interpretation of Terminal Attractors
  8. 5.4. Unpredictability in Terminal Dynamics
  9. 5.5. Irreversibility of Terminal Dynamics
  10. 5.6. Probabilistic Structure
  11. 5.7. Self-Organization in Terminal Dynamics
  12. Noise and Determinism
    1. Experimental Determinations
    1. The Larger Metaphor
    1. Non-Equilibrium Singularities
  13. 3.1. Simple Harmonic Oscillator
  14. 3.2. A Physically Motivated Example
  15. 3.3. Uncertainty in Piecewise Deterministic Dynamics
  16. 3.4. Nondeterminism and Predictability
  17. 3.5. Controlling Nondeterministic Chaos
  18. 3.6. Implications
    1. Classification of Nondeterministic Systems
  19. Singularities in Biological Sciences
    1. An Alternative Approach
    1. Nonstationary Features of the Cardio-Pulmonary System
  20. 2.1. Tracheal Pressures
  21. 2.2. Lung Sounds
  22. 2.3. Heart Beat
    1. Neural (Brain) Processes
  23. 3.1. Electroencephalograms and Seizures
  24. 3.2. Terminal Neurodynamics
  25. 3.3. Creativity and Neurodynamics
  26. 3.4. Collective Brain
  27. 3.5. Stochastic Attractor as a Tool for Generalization
  28. 3.6. Collective Brain Paradigm
  29. 3.7. Model of Collective Brain
  30. 3.8. Terminal Comments
    1. Arm Motion
    1. Protein Folding
  31. 5.1. Two General Contemporary Schemata
  32. 5.2. A Different View
  33. 5.3. Proteins from a Signal Analysis Perspective
  34. 5.4. Singularities of Protein Hydrophobicity
    1. Compartment Models
    1. Biological Complexity
  35. Singularities in Social Science/Arts
    1. Economic Time Series
  36. 1.1. Stock Market Indexes
  37. 1.2. Exchange Rates
    1. Art (The Science of Art?)
  38. 2.1. Examples
    1. Psychology
  39. 3.1. Examples
    1. Sociology
  40. 4.1. Examples
  41. Conclusions
  42. Glossary
  43. Mathematical Appendix
    1. Nondeterministic System with Singularities
    1. Recurrence Quantification Analysis (RQA)
    1. Recurrence Plots
    1. Recurrence Quantification
  44. 4.1. Determining Parameters for Nonstationary Series
  45. 4.2. Choice of Embedding
  46. 4.3. Choice of Lag
  47. 4.4. Choice of Radius
    1. Detecting Singularities
  48. 5.1. Maxline (Liapunov exponent)
  49. 5.2. Orthogonal Vectors
  50. 5.3. Some Observations
    Bibliography Books Book Chapters, Proceedings Journals Index


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© Elsevier Science 2004
18th June 2004
Elsevier Science
Hardcover ISBN:
eBook ISBN:

About the Author

Joseph Zbilut


"This book is a clear text for understanding unstable singularities and randomness and their importance in the complexity of different application fields." Prof. Nicoletta Sala, University of Lugano in: Chaos and Complexity Letters, No. 4, 2004

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