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Geometrical Dynamics of Complex Systems A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho-Socio-Economical Dynamics

ISBN-10: 1402045441

ISBN-13: 9781402045448

Edition: 2006

Authors: Vladimir G. Ivancevic, Tijana T. Ivancevic

List price: $339.00
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Description:

This volume presents a comprehensive introduction into rigorous geometrical dynamics of complex systems of various natures. By "complex systems", in this book are meant high-dimensional nonlinear systems, which can be (but not necessarily are) adaptive. This monograph proposes a unified geometrical approach to dynamics of complex systems of various kinds: engineering, physical, biophysical, psychophysical, sociophysical, econophysical, etc. As their names suggest, all these multi-input multi-output (MIMO) systems have something in common: the underlying physics. Using sophisticated machinery composed of differential geometry, topology and path integrals, this book proposes a unified approach to complex dynamics of predictive power much greater than the currently popular "soft-science" approach to complex systems. The main objective of this book is to show that high-dimensional nonlinear systems in "real life" can be modeled and analyzed using rigorous mathematics, which enables their complete predictability and controllability, as if they were linear systems. The book has two chapters and an appendix. The first chapter develops the geometrical machinery in both an intuitive and rigorous manner. The second chapter applies this geometrical machinery to a number of examples of complex systems, including mechanical, physical, control, biomechanical, robotic, neurodynamical and psycho-social-economical systems. The appendix gives all the necessary background for comprehensive reading of this book.
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Book details

List price: $339.00
Copyright year: 2006
Publisher: Springer
Publication date: 1/18/2006
Binding: Hardcover
Pages: 824
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 3.564
Language: English

Preface
Acknowledgments
Glossary of Frequently Used Symbols
Modern Geometrical Machinery
Introduction
Smooth Manifolds
Intuition Behind a Smooth Manifold
Definition of a Smooth Manifold
Smooth Maps Between Manifolds
(Co)Tangent Bundles of a Smooth Manifold
Tensor Fields and Bundles of a Smooth Manifold
Lie Derivative on a Smooth Manifold
Lie Groups and Associated Lie Algebras
Lie Symmetries and Prolongations on Manifolds
Riemannian Manifolds
Finsler Manifolds
Symplectic Manifolds
Complex and K�hler Manifolds
Conformal Killing-Riemannian Geometry
Fibre Bundles
Intuition Behind a Fibre Bundle
Definition of a Fibre Bundle
Vector and Affine Bundles
Principal Bundles
Multivector-Fields and Tangent-Valued Forms
Jet Spaces
Intuition Behind a Jet Space
Definition of a 1-Jet Space
Connections as Jet Fields
Definition of a 2-Jet Space
Higher-Order Jet Spaces
Jets in Mechanics
Jets and Action Principle
Path Integrals: Extending Smooth Geometrical Machinery
Intuition Behind a Path Integral
Path Integral History
Standard Path-Integral Quantization
Sum over Geometries and Topologies
TQFT and Stringy Path Integrals
Dynamics of Complex Systems
Mechanical Systems
Autonomous Lagrangian/Hamiltonian Mechanics
Non-Autonomous Lagrangian/Hamiltonian Mechanics
Semi-Riemannian Geometrical Dynamics
Relativistic and Multi-Time Rheonomic Dynamics
Geometrical Quantization
Physical Field Systems
n-Categorical Framework
Lagrangian Field Theory on Fibre Bundles
Finsler-Lagrangian Field Theory
Hamiltonian Field Systems: Path-Integral Quantization
Gauge Fields on Principal Connections
Modern Geometrodynamics
Topological Phase Transitions and Hamiltonian Chaos
Topological String Theory
Turbulence and Chaos Field Theory
Nonlinear Control Systems
The Basis of Modern Geometrical Control
Geometrical Control of Mechanical Systems
Hamiltonian Optimal Control and Maximum Principle
Path-Integral Optimal Control of Stochastic Systems
Life: Complex Dynamics of Gene Regulatory Networks
Human-Like Biomechanics
Lie Groups and Symmetries in Biomechanics
Muscle-Driven Hamiltonian Biomechanics
Biomechanical Functors
Biomechanical Topology
Neurodynamics
Microscopic Neurodynamics and Quantum Brain
Macroscopic Neurodynamics
Oscillatory Phase Neurodynamics
Neural Path-Integral Model for the Cerebellum
Intelligent Robot Control
Brain-Like Control Functor in Biomechanics
Concurrent and Weak Functorial Machines
Brain-Mind Functorial Machines
Psycho-Socio-Economic Dynamics
Force-Field Psychodynamics
Geometrical Dynamics of Human Crowd
Dynamical Games on Lie Groups
Nonlinear Dynamics of Option Pricing
Command/Control in Human-Robot Interactions
Nonlinear Dynamics of Complex Nets
Complex Adaptive Systems: Common Characteristics
FAM Functors and Real-Life Games
Riemann-Finsler Approach to Information Geometry
Appendix: Tensors and Functors
Elements of Classical Tensor Analysis
Transformation of Coordinates and Elementary Tensors
Euclidean Tensors
Tensor Derivatives on Riemannian Manifolds
Tensor Mechanics in Brief
The Covariant Force Law in Robotics and Biomechanics
Categories and Functors
Maps
Categories
Functors
Natural Transformations
Limits and Colimits
The Adjunction
n-Categories
Abelian Functorial Algebra
References
Index