Complex Dynamics
Advanced System Dynamics in Complex Variables
(Sprache: Englisch)
Readers learn to predict and control low- and high-dimensional as well as continuous- and discrete-time nonlinear systems dynamics in complex variables. In the final chapter, all the previously developed methods are used to present the "Holy Grail" of...
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Produktinformationen zu „Complex Dynamics “
Readers learn to predict and control low- and high-dimensional as well as continuous- and discrete-time nonlinear systems dynamics in complex variables. In the final chapter, all the previously developed methods are used to present the "Holy Grail" of modern physical and cosmological science, the search for the "theory of everything" and the "true" cosmological dynamics.
Klappentext zu „Complex Dynamics “
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 toolthat 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
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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. Readers learn to predict and control low- and high-dimensional as well as continuous- and discrete-time nonlinear systems dynamics in complex variables. In the final chapter, all the previously developed methods are used to present the "Holy Grail" of modern physical and cosmological science, the search for the "theory of everything" and the "true" cosmological dynamics.
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Inhaltsverzeichnis zu „Complex Dynamics “
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 Space 2 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 Henon Maps 2.2.4 Smale Horseshoes 3 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 Principle 4 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 Kahler 4.2.2 Dolbeault Cohomology and Hodge Numbers 4.3 Basics of Kahler Geometry 4.3.1 The Kahler Ricci Flow 4.3.2 Kahler Orbifolds 4.3.3 Kahler Ricci Flow on Kahler--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
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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 Kahler Ricci Solitons 5.4.1 Introduction 5.4.2 Associated Holomorphic Quantities 5.4.3 Potentials and Local Generality5.5 Monge--Ampere Equations 5.5.1 Monge--Ampere 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 Kahler Manifolds as Classical Phase Spaces 5.6.4 Complex--Structure Deformations 5.6.5 Kahler Deformations 5.6.6 Dynamics on Kahler 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 Neurodynamics 6 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 More General Actions 6.3.5 Transition Amplitude for a Single Point Particle 6.3.6 Witten's Open String Field Theory6.3.7 Topological Strings 6.3.8 Geometrical Transitions 6.3.9 Topological Strings and Black Hole Attractors 6.4 Other Applications of Path Integrals 6.4.1 Stochastic Optimal Control6.4.2 Nonlinear Dynamics of Option Pricing 6.4.3 Nonlinear Dynamics of Complex Nets 6.4.4 Dissipative Quantum Brain Model 6.4.5 Cerebellum as a Neural Path--Integral 6.4.6 Topological Phase Transitions and Hamiltonian Chaos6.4.7 Force--Field Psychodynamics 7 Quantum Gravity and Cosmological Dynamics 7.1 Search for Quantum Gravity 7.1.1 What Is Quantum Gravity? 7.1.2 Main Approaches to Quantum Gravity7.1.3 Traditional Approaches to Quantum Gravity 7.2 Loop Quantum Gravity 7.2.1 Introduction to Loop Quantum Gravity 7.2.2 Formalism of Loop Quantum Gravity7.2.3 Loop Algebra 7.2.4 Loop Quantum Gravity 7.2.5 Loop States and Spin Network States 7.2.6 Diagrammatic Representation of the States 7.2.7 Quantum Operators 7.2.8 Loop v.s. Connection Representation7.3 Cosmological Dynamics 7.3.1 Hawking's Cosmology in 'Plain English' 7.3.2 Theories of Everything, Anthropic Principle and Wave Function of the Universe 7.3.3 Quantum Gravity and Black Holes 7.3.4 Generalized Quantum Mechanics 7.3.5 Anthropic String Landscape 7.3.6 Top--Down Cosmology 7.3.7 Cosmology in the String Landscape 7.3.8 Brane Cosmology7.3.9 Hawking's Brane New World References Index
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Bibliographische Angaben
- Autoren: Vladimir G. Ivancevic , Tijana T. Ivancevic
- 2007, XVIII, 810 Seiten, Maße: 16,5 x 24,2 cm, Gebunden, Englisch
- Verlag: Springer Netherlands
- ISBN-10: 140206411X
- ISBN-13: 9781402064111
Sprache:
Englisch
Rezension zu „Complex Dynamics “
From the reviews:"The book under review presents, in a concise and simple manner, complex dynamical systems, both low- (such as flows in the complex-plane and Riemann sphere) and high-dimensional (flows in complex manifolds), within the continuous- and discrete-time pictures. ... a broad reflection of the current state of the science of complex dynamics. It is a valuable book to read for serious readers who are interested in the whole spectrum of problems related to complex-valued nonlinear dynamics and in how to directly apply them in the research." (Eugene Kryachko, Zentralblatt MATH, Vol. 1134 (12), 2008)
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