Complex Dynamics: Advanced System Dynamics in Complex Variables

by Ivancevic, Vladimir G.; Ivancevic, Tijana T.

• ISBN13: 9781402064111
• ISBN10: 140206411X
• Format: Hardcover
• Copyright: 2007-09-03
• Publisher: Springer Verlag

Summary

Complex Dynamics: Advanced System Dynamics in Complex Variables is a graduate-level monographic textbook. It is designed as a comprehensive introduction into methods and techniques of modern complex-valued nonlinear dynamics with its various physical and non-physical applications. This book is a complex-valued continuation of our previous two monographs, Geometrical Dynamics of Complex Systems and High-Dimensional Chaotic and Attractor Systems, Volumes 31 and 32 in the Springer book series Intelligent Systems, Control and Automation: Science and Engineering, where we had developed the most powerful mathematical machinery to deal with high-dimensional nonlinear, attractor and chaotic real-valued dynamics. The present monograph is devoted to understanding, prediction and control of both low- and high-dimensional, as well as both continuous- and discrete-time, nonlinear systems dynamics in complex variables. Its objective is to provide a serious reader with a serious scientific tool that will enable him/her to actually perform a competitive research in modern complex-valued nonlinear dynamics. This book has seven Chapters. The first, introductory Chapter explains 'in plain English' the objective of the book and provides the preliminaries in complex numbers and variables; it also gives a soft introduction to quantum dynamics. The second Chapter develops low-dimensional dynamics in the complex plane, theoretical and computational, continuous- and discrete-time. The third Chapter presents a modern introduction to quantum dynamics, mainly following Dirac's notation. The fourth Chapter develops geometrical machinery of complex manifolds, essential for the further text. The fifth Chapter develops high-dimensional complex continuous dynamics, which takes place on complex manifolds. The sixth Chapter develops the formalism of complex path integrals, which extends the continuous dynamics to the general high-dimensional dynamics, which can be both discrete and stochastic. In the last, seventh Chapter, all previously developed methods are employed to present the 'Holy Grail' of modern physical and cosmological science, the search for the 'theory of everything' and the 'true' cosmological dynamics.

Table of Contents

1 Introduction 1.1 Why Complex Dynamics ? 1.2 Preliminaries: Basics of Complex Numbers and Variables 1.2.1 Complex Numbers and Vectors 1.2.2 Complex Functions1.2.3 Unit Circle and Riemann Sphere 1.3 Soft Introduction to Quantum Dynamics 1.3.1 Complex Hilbert Space2 Nonlinear Dynamics in the Complex Plane 2.1 Complex Continuous Dynamics 2.1.1 Complex Nonlinear ODEs 2.1.2 Numerical Integration of Complex ODEs 2.1.3 Complex Hamiltonian Dynamics 2.1.4 Dissipative Dynamics with Complex Hamiltonians 2.1.5 Classical Trajectories for Complex Hamiltonians 2.2 Complex Chaotic Dynamics: Discrete and Symbolic 2.2.1 Basic Fractals and Biomorphs2.2.2 Mandelbrot Set 2.2.3 Hènon Maps 2.2.4 Smale Horseshoes3 Complex Quantum Dynamics 3.1 Non-Relativistic Quantum Mechanics 3.1.1 Dirac''s Canonical Quantization 3.1.2 Quantum States and Operators 3.1.3 Quantum Pictures 3.1.4 Spectrum of a Quantum Operator 3.1.5 General Representation Model 3.1.6 Direct Product Space 3.1.7 State-Space for n Quantum Particles 3.2 Relativistic Quantum Mechanics and Electrodynamics 3.2.1 Difficulties of the Relativistic Quantum Mechanics 3.2.2 Particles of Half-Odd Integral Spin3.2.3 Particles of Integral Spin 3.2.4 Dirac''s Electrodynamics Action Principle4 Complex Manifolds 4.1 Smooth Manifolds 4.1.1 Intuition and Definition of a Smooth Manifold 4.1.2 (Co)Tangent Bundles of a Smooth Manifold 4.1.3 Lie Derivatives, Lie Groups and Lie Algebras 4.1.4 Riemannian, Finsler and Symplectic Manifolds 4.1.5 Hamilton-Poisson Geometry and Human Biodynamics 4.2 Complex Manifolds 4.2.1 Complex Metrics: Hermitian and Ka''hler 4.2.2 Dolbeault Cohomology and Hodge Numbers 4.3 Basics of Ka''hler Geometry 4.3.1 The Ka''hler Ricci Flow 4.3.2 Ka''hler Orbifolds 4.3.3 Ka''hler Ricci Flow on Ka''hler-Einstein Orbifolds 4.3.4 Induced Evolution Equations 4.4 Conformal Killing-Riemannian Geometry4.4.1 Conformal Killing Vector-Fields and Forms on M 4.4.2 Conformal Killing Tensors and Laplacian Symmetry on M 4.5 Stringy Manifolds 4.5.1 Calabi-Yau Manifolds 4.5.2 Orbifolds 4.5.3 Mirror Symmetry 4.5.4 String Theory in ''Plain English''5 Nonlinear Dynamics on Complex Manifolds 5.1 Gauge Theories 5.1.1 Classical Gauge Theory 5.2 Monopoles5.2.1 Monopoles in R35.2.2 Spectral Curve 5.2.3 Twistor Theory of Monopoles 5.2.4 Nahm Transform and Nahm Equations 5.3 Hermitian Geometry and Complex Relativity 5.3.1 About Space-Time Complexification 5.3.2 Hermitian Geometry 5.3.3 Invariant Action5.4 Gradient Ka''hler Ricci Solitons 5.4.1 Introduction 5.4.2 Associated Holomorphic Quantities 5.4.3 Potentials and Local Generality5.5 Monge-Ampére Equations 5.5.1 Monge-Ampére Equations and Hitchin Pairs 5.5.2 The @-Operator 5.6 Quantum Mechanics Viewed as a Complex Structure on a Classical Phase Space 5.6.1 Introduction 5.6.2 Varying the Vacuum 5.6.3 Ka''hler Manifolds as Classical Phase Spaces 5.6.4 Complex-Structure Deformations 5.6.5 Ka''hler Deformations 5.6.6 Dynamics on Ka''hler Spaces 5.6.7 Interpretations 5.7 Geometric Quantization 5.7.1 Quantization of Ordinary Hamiltonian Mechanics 5.7.2 Quantization of Relativistic Hamiltonian Mechanics 5.8 K-Theory and Complex Dynamics 5.8.1 Topological K-Theory 5.8.2 Algebraic K-Theory 5.8.3 Chern Classes and Chern Character5.8.4 Atiyah''s View on K-Theory 5.8.5 Atiyah-Singer Index Theorem 5.8.6 The Infinite-Order Case5.8.7 Twisted K-Theory and the Verlinde Algebra 5.8.8 Stringy and Brane Dynamics via K-Theory 5.9 Self-Similar Liouville Neurodynamics6 Path Integrals and Complex Dynamics6.1 Path Integrals: Sums Over Histories 6.1.1 Intuition Behind a Path Integral 6.1.2 Path Integral History6.1.3 Standard Path-Integral Quantization6.1.4 Sum over Geometries and Topologies 6.2 Complex Dynamics of Quantum Fields6.2.1 Topological Quantum Field Theory 6.2.2 Seiberg-Witten Theory and TQFT 6.2.3 TQFTs Associated with SW-Monopoles 6.3 Complex Stringy Dynamics 6.3.1 Stringy Actions and Amplitudes 6.3.2 Transition Amplitudes for Strings 6.3.3 Weyl Invariance and Vertex Operator Formulation 6.3.4