微分方程动态系统和混沌导论

CHAPTER 1 First-Order Equations
1.1 The Simplest Example
　1.2 The Logistic Population Model
　1.3 Constant Harvesting and Bifurcations
　1.4 Periodic Harvesting and Periodic Solutions
　1.5 Computing the Poincard Map
　1.6 Exploration：A Two-Parameter Family
CHAPTER 2 Planar Linear Systems
　2.1 Second-Order Differential Equations
　2.2 Planar Systems
　2.3 Preliminaries from Algebra
　2.4 Planar Linear Systems
　2.5 Eigenvalues and Eigenvectors
　2.6 Solving Linear Systems
　2.7 The Linearity Principle
CHAPTER 3 Phase Portraits for Planar Systems
　3.1 Real Distinct Eigenvalues
　3.2 Complex Eigenvalues
　3.3 Repeated Eigenvalues
　3.4 Changing Coordinates
CHAPTER 4 Classification of Planar Systems
　4.1 The Trace-Determinant Plane
　4.2 Dynamical Classification
　4.3 Exploration：A 3D Parameter Space
CHAPTER 5 Higher Dimensional Linear Algebra
　5.1 Preliminaries from Linear Algebra
　5.2 Eigenvalues and Eigenvectors
　5.3 Complex Eigenvalues
　5.4 Bases and Subspaces
　5.5 Repeated Eigenvalues
　5.6 Genericity
CHAPTER 6 Higher Dimensional Linear Systems
　6.1 Distinct Eigenvalues
　6.2 Harmonic Oscillators
　6.3 Repeated Eigenvalues
　6.4 The Exponential of a Matrix
　6.5 Nonautonomous Linear Systems
CHAPTER 7 Nonlinear Systems
　7.1 Dynamical Systems
　7.2 The Existence and Uniqueness Theorem
　7.3 Continuous Dependence of Solutions
　7.4 The Variational Equation
　7.5 Exploration：Numerical Methods
CHAPTER 8 Equilibria in Nonlinear Systems
　8.1 Some Nustrative Examples
　8.2 Nonlinear Sinks and Sources
　8.3 Saddles
　8.4 Stability
　8.5 Bifurcations
　8.6 Exploration：Complex Vector Fields
CHAPTER 9 Global Nonlinear Techniques
　9.1 Nullclines
　9.2 Stability of Equilibria
　9.3 Gradient Systems
　9.4 Hamiltonian Systems
　9.5 Exploration：The Pendulum with Constant Forcing
CHAPTER 10 Closed Orbits and Limit Sets
　10.1 Limit Sets
　10.2 Local Sections and Flow Boxes
　10.3 The Poincare Map
　10.4 Monotone Sequences in Planar Dynamical Systems
　10.5 The Poincare-Bendixson Theorem
　10.6 Applications of Poincare-Bendixson
　10.7 Expl0ration：Chemical Reactions That Oscillate
CHAPTER 11 Applications in Biology
　11.1 Infectious Diseases
　11.2 Predator／Prey Systems
　11.3 Competitive Species
　11.4 Exploration：Competition and Harvesting
CHAPTER 12 Applications in Circuit Theory
　12.1 An RLC Circuit
　12.2 The Lienard Equation
　12.3 The van der Pol Equation
　12.4 A Hopf Bifurcation
　12.5 Exploration：Neurodynamics
CHAPTER 13 Applications in Mechanics
　13.1 Newton’S Second Law
　13.2 Conservative Systems
　13.3 Central Force Fields
　13.4 The Newtonian Central Force System
　13.5 Kepler’s First Law
　13.6 The Two-Body Problem
　13.7 Blowing Up the Singularity
　13.8 Exploration：Other Central Force Problems
　13.9 Exploration：Classical Limits of Quantum Mechanical Systems
CHAPTER 14 The Lorenz System
　14.1 Introduction to the Lorenz System
　14.2 Elementary Properties of the Lorenz System
　14.3 The Lorenz Attractor
　14.4 A Model for the Lorenz Attractor
　14.5 The Chaotic Attractor
　14.6 Exploration：The Rossler Attractor
CHAPTER 15 Discrete Dynamical Systems
　15.1 Introduction to Discrete Dynamical Systems
　15.2 Bifurcations
　15.3 The Discrete Logistic Model
　15.4 Chaos
　15.5 Symbolic Dynamics
　15.6 The Shift Map
　15.7 The Cantor Middle-Thirds Set
　15.8 Exploration：Cubic Chaos
　15.9 Exploration：The Orbit Diagram
CHAPTER 16 Homoclinic Phenomena
　16.1 The Shil’nikov System
　16.2 The Horseshoe Map
　16.3 The Double Scroll Attractor
　16.4 Homoclinic Bifurcations
　16.5 Exploration：The Chua Circuit
CHAPTER 17 Existence and Uniqueness Revisited
　17.1 The Existence and Uniqueness Theorem
　17.2 Proof of Existence and Uniqueness
　17.3 Continuous Dependence on Initial Conditions
　17.4 Extending Solutions
　17.5 Nonautonomous Systems
　17.6 Differentiability of the Flow
Bibliography
Index