### Synopses & Reviews

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.

#### Review

From the reviews: "As it is mentioned in the preface this is 'a graduate-level monographical textbook. It represents a comprehensive introduction into rigorous geometrical dynamics of complex systems of various natures'. ... This book has to be recommended for graduates in applied mathematics who are interested in basics of modern mathematical methods mostly based on geometry." (Iskander A. Taimanov, Zentralblatt MATH, Vol. 1092 (18), 2006)

#### Review

From the reviews:

"As it is mentioned in the preface this is 'a graduate-level monographical textbook. It represents a comprehensive introduction into rigorous geometrical dynamics of complex systems of various natures'. ... This book has to be recommended for graduates in applied mathematics who are interested in basics of modern mathematical methods mostly based on geometry." (Iskander A. Taimanov, Zentralblatt MATH, Vol. 1092 (18), 2006)

### Table of Contents

Modern Geometrical Machinery; 1 .1 Introduction; 1 .2 Smooth Manifolds; 1.2.1 Intuition Behind a Smooth Manifold; 1.2.2 Definition of a Smooth Manifold; 1.2.3 Smooth Maps Between Manifolds; 1.2.4 (Co)Tangent Bundles of a Smooth Manifold; 1.2.5 Tensor Fields and Bundles of a Smooth Manifold; 1.2.6 Lie Derivative on a Smooth Manifold; 1.2.7 Lie Groups and Associated Lie Algebras; 1.2.8 Lie Symmetries and Prolongations on Manifolds ;1.2.9 Riemannian Manifolds; 1.2.10 Finsler Manifolds; 1.2.11 Symplectic Manifolds; 1.2.12 Complex and Kähler Manifolds; 1.2.13 Conformal Killing--Riemannian Geometry; 1.3 Fibre Bundles; 1.3.1 Intuition Behind a Fibre Bundle; 1.3.2 Definition of a Fibre Bundle ;1.3.3 Vector and Affine Bundles; 1.3.4 Principal Bundles; 1.3.5 Multivector--Fields and Tangent--Valued Forms; 1.4 Jet Spaces; 1.4.1 Intuition Behind a Jet Space ; 1.4.2 Definition of a 1--Jet Space; 1.4.3 Connections as Jet Fields; 1.4.4 Definition of a 2--Jet Space; 1.4.5 Higher--Order Jet Spaces; 1.4.6 Jets in Mechanics;1.4.7 Jets and Action Principle; 1.5 Path Integrals: Extending Smooth Geometrical Machinery; 1.5.1 Intuition Behind a Path Integral; 1.5.2 Path Integral History; 1.5.3 Standard Path--Integral Quantization; 1.5.4 Sum over Geometries/Topologies; 1.5.5 TQFT and Stringy Path Integrals; 2 Dynamics of High--Dimensional Nonlinear Systems; 2.1 Mechanical Systems; 2.1.1 Autonomous Lagrangian/Hamiltonian Mechanics; 2.1.2 Non--Autonomous Lagrangian/Hamiltonian Mechanics; 2.1.3 Semi--Riemannian Geometrical Dynamics; 2.1.4 Relativistic and Multi--Time Rheonomic Dynamics; 2.1.5 Geometrical Quantization; 2.2 Physical Field Systems; 2.2.1 n--Categorical Framework; 2.2.2 Lagrangian Field Theory on Fibre Bundles; 2.2.3 Finsler--Lagrangian Field Theory; 2.2.4 Hamiltonian Field Systems: Path--Integral Quantization; 2.2.5 Gauge Fields on Principal Connections; 2.2.6 Modern Geometrodynamics; 2.2.7 Topological Phase Transitions and Hamiltonian Chaos; 2.2.8 Topological Superstring Theory; 2.2.9 Turbulence and Chaos Field Theory; 2.3 Nonlinear Control Systems; 2.3.1 The Basis of Modern Geometrical Control;2.3.2 Geometrical Control of Mechanical Systems ;2.3.3 Hamiltonian Optimal Control and Maximum Principle; 2.3.4 Path--Integral Optimal Control of Stochastic Systems; 2.4 Human--Like Biomechanics; 2.4.1 Lie Groups and Symmetries in Biomechanics; 2.4.2 Muscle--Driven Hamiltonian Biomechanics; 2.4.3 Biomechanical Functors; 2.4.4 Biomechanical Topology; 2.5 Neurodynamics; 2.5.1 Microscopic Neurodynamics and Quantum Brain; 2.5.2 Macroscopic Neurodynamics; 2.5.3 Oscillatory Phase Neurodynamics ;2.5.4 Neural Path--Integral Model for the Cerebellum; 2.5.5 Intelligent Robot Control; 2.5.6 Brain--Like Control Functor in Biomechanics; 2.5.7 Concurrent and Weak Functorial Machines; 2.5.8 Brain--Mind Functorial Machines; 26 Psycho--Socio--Economic Dynamics; 2.6.1 Force--Field Psychodynamics; 2.6.2 Geometrical Dynamics of Human Crowd; 2.6.3 Dynamical Games on Lie Groups ; 2.6.4 Nonlinear Dynamics of Option Pricing; 2.6.5 Command/Control in Human--Robot Interactions; 2.6.6 Nonlinear Dynamics of Complex Nets; 2.6.7 Complex Adaptive Systems: Common Characteristics; 2.6.8 FAM Functors and Real--Life Games; 2.6.9 Riemann--Finsler Approach to Information Geometry; 3 Appendix: Tensors and Functors; 3.1 Elements of Classical Tensor Analysis; 3.1.1 Transformation of Coordinates and Elementary Tensors; 3.1.2 Euclidean Tensors; 3. 1 .3 Tensor Derivatives on Riemannian Manifolds; 3.1.4 Tensor Mechanics in Brief ; 3.1.5 The Covariant Force Law in Robotics and Biomechanics; 3.2 Categories and Functors; 3.2.1 Maps; 3.2.2 Categories; 3.2.3 Functors; 3.2.4 Natural Transformations; 3.2.5 Limits and Colimits; 3.2.6 The Adjunction; 3.2.7 ri--Categories; 3.2.8 Abelian Functorial Algebra; References; Index.