Unstable Singularities and Randomness

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


  • Joseph Zbilut

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


Book information

  • Published: June 2004
  • Imprint: ELSEVIER
  • ISBN: 978-0-444-51613-8


"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

Table of Contents

1. Probability and Dynamics1.1. A Dichotomy1.2. Historical Perspective1.3. Probabilities1.4. Randomness1.5. Singularities1.6. Models and Reality
2. Singularities and Instability2.1. Dynamics2.1.1. Attractors2.1.2. Liapunov Exponents2.2. Limitations of the Classical Approach2.3. Dynamical Instability2.4. Lipschitz Conditions2.5. Basic Concepts2.5.1. Dissipation2.5.2. Terminal Dynamics Limit Sets2.5.3. Interpretation of Terminal Attractors2.5.4. Unpredictability in Terminal Dynamics2.5.5. Irreversibility of Terminal Dynamics2.5.6. Probabilistic Structure2.5.7. Self-Organization in Terminal Dynamics
3. Noise and Determinism3.1. Experimental Determinations3.2. The Larger Metaphor3.3. Non-Equilibrium Singularities3.3.1. Simple Harmonic Oscillator3.3.2. A Physically Motivated Example3.3.3. Uncertainty in Piecewise Deterministic Dynamics3.3.4. Nondeterminism and Predictability3.3.5. Controlling Nondeterministic Chaos3.3.6. Implications3.4. Classification of Nondeterministic Systems
4. Singularities in Biological Sciences4.1. An Alternative Approach4.2. Nonstationary Features of the Cardio-Pulmonary System4.2.1. Tracheal Pressures4.2.2. Lung Sounds4.2.3. Heart Beat4.3. Neural (Brain) Processes4.3.1. Electroencephalograms and Seizures4.3.2. Terminal Neurodynamics4.3.3. Creativity and Neurodynamics4.3.4. Collective Brain4.3.5. Stochastic Attractor as a Tool for Generalization4.3.6. Collective Brain Paradigm4.3.7. Model of Collective Brain4.3.8. Terminal Comments4.4. Arm Motion4.5. Protein Folding4.5.1. Two General Contemporary Schemata4.5.2. A Different View4.5.3. Proteins from a Signal Analysis Perspective4.5.4. Singularities of Protein Hydrophobicity4.6. Compartment Models4.7. Biological Complexity
5. Singularities in Social Science/Arts5.1. Economic Time Series5.1.1. Stock Market Indexes5.1.2. Exchange Rates5.2. Art (The Science of Art?)5.2.1. Examples5.3. Psychology5.3.1. Examples5.4. Sociology5.4.1. Examples
6. Conclusions
7. Glossary
8. Mathematical Appendix8.1. Nondeterministic System with Singularities8.2. Recurrence Quantification Analysis (RQA)8.3. Recurrence Plots8.4. Recurrence Quantification8.4.1. Determining Parameters for Nonstationary Series8.4.2. Choice of Embedding8.4.3. Choice of Lag8.4.4. Choice of Radius8.5. Detecting Singularities8.5.1. Maxline (Liapunov exponent)8.5.2. Orthogonal Vectors8.5.3. Some Observations
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