Dynamical Sys Stability Symbolic Dynamics & Chaos
1st Edition
Published on 22. December 1994
Book
Hardback
480 pages
978-0-8493-8493-6 (ISBN)
Description
This new text/reference treats dynamical systems from a mathematical perspective, centering on multidimensional systems of real variables. Background material is carefully reviewed as it is used throughout the book, and ideas are introduced through examples. Numerous exercises help the reader understand presented theorems and master the techniques of the proofs and topic under consideration.
The book treats the dynamics of both iteration of functions and solutions of ordinary differential equations. Many concepts are first introduced for iteration of functions where the geometry is simpler, but results are interpreted for differential equations. A proof of the existence and continuity of solutions with respect to initial conditions is included. Explicit formulas for the various bifurcations are included, and a treatment of the Henon map and the Melnikov method is provided.
The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. Even the more local theory which is treated deals with characterizing types of solutions under various hypothesis. Later chapters deal more directly with more global aspects, with one chapter discussing various examples and later chapters giving the global theory.
The book treats the dynamics of both iteration of functions and solutions of ordinary differential equations. Many concepts are first introduced for iteration of functions where the geometry is simpler, but results are interpreted for differential equations. A proof of the existence and continuity of solutions with respect to initial conditions is included. Explicit formulas for the various bifurcations are included, and a treatment of the Henon map and the Melnikov method is provided.
The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. Even the more local theory which is treated deals with characterizing types of solutions under various hypothesis. Later chapters deal more directly with more global aspects, with one chapter discussing various examples and later chapters giving the global theory.
More details
Language
English
Place of publication
Bosa Roca
United States
Publishing group
Taylor & Francis Inc
Target group
College/higher education
Professional and scholarly
Graduate students and professionals interested in various aspects of dynamical systems
Dimensions
Height: 254 mm
Width: 178 mm
Weight
1030 gr
ISBN-13
978-0-8493-8493-6 (9780849384936)
Schweitzer Classification
Other editions
New editions
Book
11/1998
2nd Edition
CRC Press
€271.20
Shipment within 15-20 days
Content
Introduction
Population Growth Models, One Population
Iteration of Real Valued Functions as Dynamical systems
Higher Dimensional Systems
Outline of the Topics of the Chapters
One Dimensional Dynamics by Iteration
Calculus Prerequisites
Periodic Points
Limit Sets and recurrence for Maps
Invariant Cantor Sets for the Quadratic Family
Symbolic Dynamics for the Quadratic Map
Conjugacy and Structural Stability
Conjugacy and Structural Stability of the Quadratic Map
Homeomorphisms of the Circle
Exercises
Chaos and Its Measurement
Sharkovskii's Theorem
Subshifts of Finite Type
Zeta Function
Period Doubling Cascade
Chaos
Liapunov Exponents
Exercises
Linear Systems
Review: Linear Maps and the Real Jordan Canonical Form
Linear Differential Equations
Solutions for Constant Coefficients
Phase Portraits
Contracting Linear Differential Equations
Hyperbolic Linear Differential Equations
Topologically Conjugate Linear Differential Equations
Nonhomogeneous Equations
Linear Maps
Exercises
Analysis Near Fixed Points and Periodic Orbits
Review: Differentiation in Higher Dimensions
Review: The Implicit Function Theorem
Existence of Solutions for Differential Equations
Limit Sets and Recurrence for Flows
Fixed Points for Nonlinear Differential Equations
Stability of Periodic Points for Nonlinear Maps
Proof of the Hartman-Grobman Theorem
Periodic Orbits for Flows
Poincare-Bendixson Theorem
Stable Manifold Theorem for a Fixed Point of a Map
The Inclination Lemma
Exercises
Bifurcation of Periodic Points
Saddle-Node
Saddle-Node Bifurcation in Higher Dimensions
Period Doubling: Bifurcation
Andronov-Hopf Bifurcation for Diffeomorphisms
Andronov-Hopf Bifurcation for Differential Equations
Exercises
Examples of Hyperbolic Sets and Attractors
Definition of a Manifold
Transitivity Theorems
Geometric Horseshoe
Toral Anosov Diffeomorphisms
Attractors
The Solenoid Attractor
The DA Attractor
Plykin Attractors in the Plane
Attractor for the Henon Map
Lorenz Attractor
Morse-Smale Systems
Exercises
Measurement of Chaos in Higher Dimensions
Topological Entropy
Liapunov Exponents
Sinai-Bowen-Ruelle Measure for an Attractor
Fractal Dimension
Exercises
Global Theory of Hyperbolic Systems
Fundamental Theorem of Dynamical Systems
Stable Manifold Theorem for a Hyperbolic Invariant Set
Shadowing and Expansiveness
Anosov Closing Lemma
Decomposition of Recurrent Points
Markov Partition fro a Hyperbolic Invariant Set
Local Stability and Stability of Anosov Diffeomorphisms
Stability of Anosov Flows
Global Stability Theorems
Exercises
Generic Properties
Kupka-Smale Theorem
Transversality
Proof of the Kupka-Smale Theorem
Necessary Conditions for Structural Stability
Nondensity of Structural Stability
Exercises
Smoothness of Stable Manifolds and Applications
Fiber Contractions
Differentiability of Invariant Splitting
Differentiability of the Center Manifold
Persistence of Normally Contracting Manifolds
Exercises
References
Index
Population Growth Models, One Population
Iteration of Real Valued Functions as Dynamical systems
Higher Dimensional Systems
Outline of the Topics of the Chapters
One Dimensional Dynamics by Iteration
Calculus Prerequisites
Periodic Points
Limit Sets and recurrence for Maps
Invariant Cantor Sets for the Quadratic Family
Symbolic Dynamics for the Quadratic Map
Conjugacy and Structural Stability
Conjugacy and Structural Stability of the Quadratic Map
Homeomorphisms of the Circle
Exercises
Chaos and Its Measurement
Sharkovskii's Theorem
Subshifts of Finite Type
Zeta Function
Period Doubling Cascade
Chaos
Liapunov Exponents
Exercises
Linear Systems
Review: Linear Maps and the Real Jordan Canonical Form
Linear Differential Equations
Solutions for Constant Coefficients
Phase Portraits
Contracting Linear Differential Equations
Hyperbolic Linear Differential Equations
Topologically Conjugate Linear Differential Equations
Nonhomogeneous Equations
Linear Maps
Exercises
Analysis Near Fixed Points and Periodic Orbits
Review: Differentiation in Higher Dimensions
Review: The Implicit Function Theorem
Existence of Solutions for Differential Equations
Limit Sets and Recurrence for Flows
Fixed Points for Nonlinear Differential Equations
Stability of Periodic Points for Nonlinear Maps
Proof of the Hartman-Grobman Theorem
Periodic Orbits for Flows
Poincare-Bendixson Theorem
Stable Manifold Theorem for a Fixed Point of a Map
The Inclination Lemma
Exercises
Bifurcation of Periodic Points
Saddle-Node
Saddle-Node Bifurcation in Higher Dimensions
Period Doubling: Bifurcation
Andronov-Hopf Bifurcation for Diffeomorphisms
Andronov-Hopf Bifurcation for Differential Equations
Exercises
Examples of Hyperbolic Sets and Attractors
Definition of a Manifold
Transitivity Theorems
Geometric Horseshoe
Toral Anosov Diffeomorphisms
Attractors
The Solenoid Attractor
The DA Attractor
Plykin Attractors in the Plane
Attractor for the Henon Map
Lorenz Attractor
Morse-Smale Systems
Exercises
Measurement of Chaos in Higher Dimensions
Topological Entropy
Liapunov Exponents
Sinai-Bowen-Ruelle Measure for an Attractor
Fractal Dimension
Exercises
Global Theory of Hyperbolic Systems
Fundamental Theorem of Dynamical Systems
Stable Manifold Theorem for a Hyperbolic Invariant Set
Shadowing and Expansiveness
Anosov Closing Lemma
Decomposition of Recurrent Points
Markov Partition fro a Hyperbolic Invariant Set
Local Stability and Stability of Anosov Diffeomorphisms
Stability of Anosov Flows
Global Stability Theorems
Exercises
Generic Properties
Kupka-Smale Theorem
Transversality
Proof of the Kupka-Smale Theorem
Necessary Conditions for Structural Stability
Nondensity of Structural Stability
Exercises
Smoothness of Stable Manifolds and Applications
Fiber Contractions
Differentiability of Invariant Splitting
Differentiability of the Center Manifold
Persistence of Normally Contracting Manifolds
Exercises
References
Index