Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems

610 Pages·2014·8.996 MB·English

by  Vladimir G Ivancevic, Darryn J Reid

Complexity and Control
Towards a Rigorous Behavioral Theory
of Complex Dynamical Systems
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SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Founder and Editor: Ardéshir Guran
Co-Editors: M. Cloud & W. B. Zimmerman
About the Series
Rapid developments in system dynamics and control, areas related to many other topics
in applied mathematics, call for comprehensive presentations of current topics. This series
contains graduate level textbooks, monographs, and a collection of thematically organized
research or pedagogical articles addressing key topics in applied dynamics.
The material is ideal for a general scientific and engineering readership, and is also
mathematically precise enough to be a useful reference for research specialists in mechanics
and control, nonlinear dynamics, and in applied mathematics and physics.
Reporting on academic/industrial research from institutions around the world,
the SVCS series reflects technological advances in mechanics and control. Particular
emphasis is laid on emerging areas such as modeling of complex systems, bioengineering,
mechatronics, structronics, fluidics, optoelectronic sensors, micromachining techniques, and
intelligent system design.
Selected Volumes in Series A
Vol. 14 Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities
Authors: V. A. Yakubovich, G. A. Leonov and A. Kh. Gelig
Vol. 15 Process Modelling and Simulation with Finite Element Methods
Author: W. B. J. Zimmerman
Vol. 16 Design of Nonlinear Control Systems with the Highest Derivative in Feedback
Author: V. D. Yurkevich
Vol. 17 The Quantum World of Nuclear Physics
Author: Yu. A. Berezhnoy
Vol. 18 Multiphysics Modeling with Finite Element Methods
Author: W. B. J. Zimmerman
Vol. 19 New Trends in Control Theory
Authors: V. G. Ivancevic and T. T. Ivancevic
Vol. 20 Complexity and Control: Towards a Rigorous Behavioral Theory of Complex
Dynamical Systems
Authors: Vladimir G. Ivancevic and Darryn J. Reid
Selected Volume in Series B
Vol. 14 Impact and Friction of Solids, Structures and Intelligent Machines
Editor: A. Guran
Vol. 15 Advances in Mechanics of Solids: In Memory of Prof. E. M. Haseganu
Editors: A. Guran, A. L. Smirnov, D. J. Steigmann and R. Vaillancourt

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SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Series A Volume 20
Founder & Editor: Ardéshir Guran
Co-Editor: M. Cloud & W. B. Zimmerman
Complexity and Control
Towards a Rigorous Behavioral Theory
of Complex Dynamical Systems
Vladimir G Ivancevic
Darryn J Reid
Joint & Operations Analysis Division
Defence Science & Technology Organisation, Australia
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Series on Stability, Vibration and Control of Systems, Series A — Vol. 20
COMPLEXITY AND CONTROL
Towards a Rigorous Behavioral Theory of Complex Dynamical Systems
Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 978-981-4635-86-8
In-house Editor: Rhaimie Wahap
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Printed in Singapore
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Preface
Complexity and Control is a graduate-level monographic textbook, intended
to be a novel and rigorous contribution to modern Complexity Theory.
Thebookcontains11Chapters.MostChaptershavetheirownAppendices
(of graduate-leveltextbook nature).Chapter 1.is anIntroduction,setting up
the stage for rigorous and controllable complexity theory. Chapter 2. intro-
ducesageneralgeometricalmachineryforcomplexityandcontrol,intheform
ofdynamicsonKa¨hlermanifolds;itincludesLagrangiandynamicsonRieman-
nianmanifolds,Hamiltoniandynamicsonsymplecticmanifolds,andquantum
dynamics in complex Hilbert spaces.Chapter 3. introduces globalcategorical
frameworkforcomplexityandcontrol,frommathematical,computer-scientific
and(bio)physicalperspectives.Chapter4.introducesthebasiccrowddynam-
icsandcontrolinthecomplexplaneandfromitdevelopsauniqueframework
forsimulation,optimization,control,learningandlogicofgenericcomplexsys-
tems. Chapter 5. is devoted to hierarchicalself-similarity in group and crowd
behaviors. Chapter 6. presents hybrid topological adaptation in evolving en-
ergy landscapes. Chapter 7. explores complexity and control in solitary pulse
conductionofvariousnature.Chapter8.developsquantum-computationalar-
chitecture for perceptual control theory. Chapter 9. explores complexity and
control in entropic and stochastic self-organization. Chapter 10. presents the
latestresearchinhumanbiodynamicsandcrashsimulation.Chapter11.gives
the additionalcodesamples(inseveralprogramminglanguages)usedincom-
plexity research.
The book Complexity and Control is designed as a one-semester course
forengineers,appliedandpuremathematicians,theoreticalandexperimental
October10,2014 11:10 ComplexityandControl 9inx6in b1966-fm pageVI
VI Preface
physicists, computer and economic scientists, theoretical chemists and biol-
ogists, as well as all mathematically educated scientists and students, both
in industry and academia, interested in predicting and controlling complex
dynamical systems of arbitrary nature.
Adelaide, Dr. Vladimir Ivancevic and Dr. Darryn Reid
May 2014 Command and Control Branch
Joint and Operations Analysis Division
Defence Science & Technology Organisation
Australia
[email protected]
[email protected]
October10,2014 11:10 ComplexityandControl 9inx6in b1966-fm pageVII
Acknowledgments
The authors are very grateful to Joint & Operations Analysis Division,
Defence Science & Technology Organisation, Australia, and in particular to
Dr. Jason Scholz, Research Leader Command and Control, for their valuable
support.
We also express our gratitude to World Scientific Publishing Company,
and especially to Professor Ardeshir Guran, Editor of the Series on Stability,
Vibration and Control of Systems.
Our deepest thanks goes to our wives and families for their love and en-
durance.
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Contents
1 Introduction............................................... 1
2 Local Geometrical Machinery for Complexity and Control . 5
2.1 Introduction: Why Ka¨hler Manifolds? ...................... 5
2.1.1 Dynamical Prologue:Complex Hamiltonian Dynamics . 6
2.1.2 Unified BehavioralPicture: Ka¨hler Geometrodynamics . 8
2.2 Complex Manifolds and Their Vector Bundles............... 10
2.2.1 Behavioral Dynamics on Complex Plane.............. 10
2.2.2 3D Rotations: Spinors.............................. 14
2.2.3 Attractor Dynamics on Riemann Surfaces ............ 16
2.2.4 Complex Manifolds and Vector Bundles .............. 17
2.2.5 (Co)Tangent Spaces to the Complex Manifold......... 20
2.2.6 Complex Vector Bundles ........................... 21
2.3 From Kalman Systems to Riemann Manifolds ............... 22
2.4 Basic Ka¨hler Geometry .................................. 23
2.4.1 Essential Ka¨hler Tensors and Cohomology Groups..... 23
2.4.2 Global Ka¨hler Geometry ........................... 27
2.4.3 Local Ka¨hler Geometry ............................ 30
2.4.4 Invariant Hamiltonian Dynamics on Ka¨hler Manifolds.. 32
2.5 Dynamics on Quaternion-Ka¨hler Manifolds ................. 33
2.5.1 Quaternion-Ka¨hler Manifolds ....................... 33
2.5.2 Hamiltonian Dynamics on Quaternion-Ka¨hlerManifolds 36
2.5.3 LagrangianDynamics on Quaternion-Ka¨hler Manifolds. 37
2.6 Ka¨hler-Ricci-Flow Framework............................. 38
2.6.1 Motivation for the Ka¨hler-Ricci Flow ................ 38
2.6.2 Ricci Flow on a Ka¨hler Manifold .................... 39
2.6.3 Definition of the Ka¨hler-Ricci Flow .................. 40
2.6.4 Evolution of the Ka¨hler-PerelmanEntropy............ 41
2.7 Summary of Ka¨hler-Ricci Geometrical Dynamics ............ 42
2.8 Appendix .............................................. 49