Unstable Singularities and Randomness

1st Edition

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

Authors: 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 1.1. 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


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© Elsevier Science 2004
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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