Table Of ContentHow Complexity
Shapes the World
How Complexity
Shapes the World
By
Georg Franz Weber
How Complexity Shapes the World
By Georg Franz Weber
This book first published 2021
Cambridge Scholars Publishing
Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Copyright © 2021 by Georg Franz Weber
All rights for this book reserved. No part of this book may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.
ISBN (10): 1-5275-7187-4
ISBN (13): 978-1-5275-7187-7
CONTENTS (CONCISE)
Introduction ....................................................................................... 1
I. Background and History ............................................................. 9
1. A Science Discovered Late ........................................................ 11
1.1. Linear Perception and Linearization ................................... 11
1.2. The Roots of Complexity Research .................................... 15
1.3. Contemporary Dynamic Knowledge ................................... 21
II. The Nature of Complex Phenomena ....................................... 25
2. Characteristics of Complexity ..................................................... 28
2.1. Hallmarks ............................................................................ 28
2.2. Bifurcations ......................................................................... 45
2.3. Strange Attractors ............................................................... 50
2.4. The Edge of Chaos............................................................. 60
2.5. Unique Responses to External Force................................. 63
3. Sources of Complexity ................................................................ 71
3.1. Disturbance of Periodicity ................................................... 71
3.2. Destabilizing Feedback ...................................................... 75
3.3. Energy Transfer from Micro- to Macro-Scale ..................... 78
3.4. Fractal Geometry ................................................................ 88
III. Common Features of Complex Occurrences ..................... 117
4. Complexity and the Progression of Time ................................. 120
4.1. Parameter Time ................................................................ 122
4.2. Coordinate Time ............................................................... 122
4.3. Operator Time .................................................................. 126
vi Contents (Concise)
5. Depiction of Complexity in Conceptual Space ......................... 131
5.1. Abstract Spaces ............................................................... 131
5.2. State Space ...................................................................... 136
5.3. Phase Space .................................................................... 137
5.4. Event Space ..................................................................... 140
5.5. The Multiplicative Ergodic Theorem in Various Spaces .... 141
6. Depiction of Complexity on Adaptive Landscapes ................... 146
6.1. Basic Rules for Movements across the Landscape ......... 146
6.2. Mathematical Models ....................................................... 148
6.3. Plasticity of the Landscape ............................................... 153
7. Depiction of Complexity in Harmonics ...................................... 156
7.1. Wave Relations ................................................................ 158
7.2. Fourier Series ................................................................... 163
7.3. Wavelet Analysis .............................................................. 171
7.4 Autocorrelation .................................................................. 172
8. Common Measures of Complexity ........................................... 175
8.1. Dimensionality in Conceptual Space ................................ 179
8.2. Information Evolution ........................................................ 184
8.3. Entropy ............................................................................. 190
IV. Analytics of Complexity ....................................................... 199
9. Algorithms ................................................................................. 203
9.1. Differential Equations or Difference Equations ................ 203
9.2. Integration Methods .......................................................... 209
9.3. Matrix Calculations ........................................................... 211
9.4. Probabilistic Approaches .................................................. 214
9.5. Geometry in Abstract Space ............................................ 231
9.6. Comparison to Surrogate Data ......................................... 237
10. Visualization ............................................................................ 242
10.1. Time Series .................................................................... 242
10.2. Bifurcation Diagrams ...................................................... 243
10.3. Maps ............................................................................... 245
10.4. Reduction of Dimensions ............................................... 251
How Complexity Shapes the World vii
V. Applications ............................................................................ 259
11. Complex Phenomena in the World ......................................... 264
11.1. Motion and Progression ................................................. 264
11.2. Nature ............................................................................. 274
11.3. Networks ......................................................................... 292
11.4. Economy ......................................................................... 312
12. Complexity in the Human Mind ............................................... 322
12.1. Self-Reflexive Brain ........................................................ 323
12.2. Epistemology .................................................................. 328
12.3. Ethics .............................................................................. 331
12.4. Music .............................................................................. 337
13. A Unified Open-Ended Cosmos ............................................. 344
13.1. Loss of Categorical Divisions ......................................... 344
13.2. Undecidable Dichotomies ............................................... 347
13.3. Conclusions .................................................................... 352
Index ............................................................................................. 361
CONTENTS (DETAILED)
Introduction
I. Background and History
1. A Science Discovered Late
1.1. Linear Perception and Linearization
1.2. The Roots of Complexity Research (philosophy,
mathematics, physics, chemistry)
1.3. Contemporary Dynamic Knowledge
II. The Nature of Complex Phenomena
2. Characteristics of Complexity
2.1. Hallmarks
non-periodic flow
sensitive dependence on starting conditions
emergence of properties from non-equilibrium
(physics, chemistry, evolution, artificial life)
generation of new information to replace all old
information
2.2. Bifurcations (local bifurcations, global bifurcations)
2.3. Strange Attractors
stability (Lyapunov’s second method, robustness of
the phase portrait)
point attractors, periodic attractors and limit cycles
complex attractors
2.4. The Edge of Chaos
2.5. Unique Responses to External Force (equilibrium
versus non-equilibrium)
3. Sources of Complexity
3.1. Disturbance of Periodicity
period doubling (Feigenbaum (cid:71) constant,
Feigenbaum (cid:68) constant, U-sequence)
intermittency
quasi-periodicity
3.2. Destabilizing Feedback
How Complexity Shapes the World ix
3.3. Energy Transfer from Micro- to Macro-Scale
dissipation
turbulence (forces, multifractals, information rates,
vector fields)
Taylor-Couette system
Rayleigh-Bénard convection (density gradient,
surface tension)
3.4. Fractal Geometry
dimension (topological dimension, similarity
dimension, Hausdorff-Besicovich dimension, fractal
dimension, curdling dimension)
fractal generation (initiators and generators,
iterated function systems, inversions, Mandelbrot
and Julia sets)
natural fractals (approximated lengths, mass
dimension, percolation, diffusion-limited
aggregation)
power laws (homogeneous power laws,
heterogeneous power laws)
scaling
III. Common Features of Complex Occurrences
4. Complexity and the Progression of Time
4.1. Parameter Time
4.2. Coordinate Time
4.3. Operator Time
5. Depiction of Complexity in Conceptual Space
5.1. Abstract Spaces
vectors
vector spaces
system behavior in vector space
5.2. State Space
5.3. Phase Space
5.4. Event Space
5.5. The Multiplicative Ergodic Theorem in Various Spaces
(in phase space, in event space, time horizon, non-
uniformity factor)
6. Depiction of Complexity on Adaptive Landscapes
6.1. Basic Rules for Movements across the Landscape
6.2. Mathematical Models (adaptive walks, pair correlation
functions, number of meta-stable states)
