Unstable Singularities and Randomness

Unstable Singularities and Randomness

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

1st Edition - June 18, 2004

Write a review

  • Author: Joseph Zbilut
  • eBook ISBN: 9780080474694

Purchase options

Purchase options
DRM-free (PDF)
Sales tax will be calculated at check-out

Institutional Subscription

Free Global Shipping
No minimum order


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

  • Preface

    1. Probability and Dynamics
    1.1. A Dichotomy
    1.2. Historical Perspective
    1.3. Probabilities
    1.4. Randomness
    1.5. Singularities
    1.6. Models and Reality

    2. Singularities and Instability
    2.1. Dynamics
    2.1.1. Attractors
    2.1.2. Liapunov Exponents
    2.2. Limitations of the Classical Approach
    2.3. Dynamical Instability
    2.4. Lipschitz Conditions
    2.5. Basic Concepts
    2.5.1. Dissipation
    2.5.2. Terminal Dynamics Limit Sets
    2.5.3. Interpretation of Terminal Attractors
    2.5.4. Unpredictability in Terminal Dynamics
    2.5.5. Irreversibility of Terminal Dynamics
    2.5.6. Probabilistic Structure
    2.5.7. Self-Organization in Terminal Dynamics

    3. Noise and Determinism
    3.1. Experimental Determinations
    3.2. The Larger Metaphor
    3.3. Non-Equilibrium Singularities
    3.3.1. Simple Harmonic Oscillator
    3.3.2. A Physically Motivated Example
    3.3.3. Uncertainty in Piecewise Deterministic Dynamics
    3.3.4. Nondeterminism and Predictability
    3.3.5. Controlling Nondeterministic Chaos
    3.3.6. Implications
    3.4. Classification of Nondeterministic Systems

    4. Singularities in Biological Sciences
    4.1. An Alternative Approach
    4.2. Nonstationary Features of the Cardio-Pulmonary System
    4.2.1. Tracheal Pressures
    4.2.2. Lung Sounds
    4.2.3. Heart Beat
    4.3. Neural (Brain) Processes
    4.3.1. Electroencephalograms and Seizures
    4.3.2. Terminal Neurodynamics
    4.3.3. Creativity and Neurodynamics
    4.3.4. Collective Brain
    4.3.5. Stochastic Attractor as a Tool for Generalization
    4.3.6. Collective Brain Paradigm
    4.3.7. Model of Collective Brain
    4.3.8. Terminal Comments
    4.4. Arm Motion
    4.5. Protein Folding
    4.5.1. Two General Contemporary Schemata
    4.5.2. A Different View
    4.5.3. Proteins from a Signal Analysis Perspective
    4.5.4. Singularities of Protein Hydrophobicity
    4.6. Compartment Models
    4.7. Biological Complexity

    5. Singularities in Social Science/Arts
    5.1. Economic Time Series
    5.1.1. Stock Market Indexes
    5.1.2. Exchange Rates
    5.2. Art (The Science of Art?)
    5.2.1. Examples
    5.3. Psychology
    5.3.1. Examples
    5.4. Sociology
    5.4.1. Examples

    6. Conclusions

    7. Glossary

    8. Mathematical Appendix
    8.1. Nondeterministic System with Singularities
    8.2. Recurrence Quantification Analysis (RQA)
    8.3. Recurrence Plots
    8.4. Recurrence Quantification
    8.4.1. Determining Parameters for Nonstationary Series
    8.4.2. Choice of Embedding
    8.4.3. Choice of Lag
    8.4.4. Choice of Radius
    8.5. Detecting Singularities
    8.5.1. Maxline (Liapunov exponent)
    8.5.2. Orthogonal Vectors
    8.5.3. Some Observations

    Book Chapters, Proceedings

Product details

  • No. of pages: 252
  • Language: English
  • Copyright: © Elsevier Science 2004
  • Published: June 18, 2004
  • Imprint: Elsevier Science
  • eBook ISBN: 9780080474694

About the Author

Joseph Zbilut

Ratings and Reviews

Write a review

There are currently no reviews for "Unstable Singularities and Randomness"