Ordinary differential equations and dynamical systems 1st Edition by Gerald Teschl ISBN 147047641X 9781470476410 by Teschl G. instant download after payment.

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ISBN 10: 147047641X 
ISBN 13: 9781470476410
Author: Gerald Teschl

Ordinary differential equations and dynamical systems 1st Table of contents:

Part 1: First-Order Ordinary Differential Equations

1. Introduction to Differential Equations
   • What is a Differential Equation?
   • Classification of ODEs (Order, Linearity, Homogeneity)
   • Solutions, Initial Value Problems, and General Solutions
   • Direction Fields and Integral Curves
2. Techniques for Solving First-Order ODEs
   • Separation of Variables
   • Exact Equations and Integrating Factors
   • Linear First-Order Equations
   • Homogeneous Equations
   • Bernoulli Equations
   • Applications (Growth and Decay, Mixing Problems, Newton's Law of Cooling, Population Dynamics)
3. Qualitative Analysis of First-Order Equations
   • Equilibrium Solutions (Fixed Points) and Stability
   • Phase Line Analysis
   • Bifurcations in One-Dimensional Systems

Part 2: Higher-Order Linear Ordinary Differential Equations

4. Second-Order Linear Homogeneous Equations
   • Homogeneous Equations with Constant Coefficients (Characteristic Equation)
   • Real Distinct Roots, Complex Conjugate Roots, Repeated Roots
   • Method of Reduction of Order
   • Applications (Mass-Spring Systems, RLC Circuits)
5. Second-Order Linear Nonhomogeneous Equations
   • Method of Undetermined Coefficients
   • Method of Variation of Parameters
6. Higher-Order Linear Equations
   • Homogeneous Equations with Constant Coefficients
   • Nonhomogeneous Equations

Part 3: Systems of Ordinary Differential Equations

7. Introduction to Systems of ODEs
   • First-Order Linear Systems
   • Phase Plane Analysis for Linear Systems
   • Homogeneous Linear Systems with Constant Coefficients (Eigenvalue Method)
   • Real Distinct Eigenvalues, Complex Eigenvalues, Repeated Eigenvalues
8. Nonhomogeneous Linear Systems
   • Method of Variation of Parameters for Systems
9. Nonlinear Systems and Qualitative Methods
   • Linearization Around Equilibrium Points
   • Stability of Nonlinear Systems
   • Phase Plane Analysis for Nonlinear Systems
   • Limit Cycles and Poincaré-Bendixson Theorem (for 2D systems)
   • Bifurcations in Two-Dimensional Systems

Part 4: Introduction to Dynamical Systems

10. Autonomous Systems and Flow
    • Definition of Dynamical Systems
    • Flows and Maps
    • Phase Space
    • Poincaré Maps (for higher-dimensional systems)
11. Stability Theory
    • Lyapunov Stability (Asymptotic Stability, Global Stability)
    • LaSalle's Invariance Principle
    • Stability of Limit Cycles
12. Bifurcation Theory (More Advanced)
    • Local Bifurcations (Saddle-Node, Transcritical, Pitchfork, Hopf)
    • Global Bifurcations (Homoclinic, Heteroclinic)
13. Chaos and Complex Dynamics
    • Sensitive Dependence on Initial Conditions
    • Strange Attractors
    • Fractal Dimensions (brief introduction)
    • Examples of Chaotic Systems (Lorenz System, Duffing Oscillator)
14. Discrete Dynamical Systems (Maps)
    • Iteration of Functions
    • Fixed Points and Cycles for Maps
    • Stability of Fixed Points for Maps
    • Bifurcations in Maps (e.g., Period-Doubling Bifurcation and Chaos)
    • Logistic Map as a Case Study

Part 5: Advanced Topics and Applications (Optional, depending on the book's depth)

15. Boundary Value Problems
    • Sturm-Liouville Theory
    • Eigenvalues and Eigenfunctions
16. Series Solutions of Linear Equations
    • Power Series Solutions (around ordinary and singular points)
    • Frobenius Method
17. Laplace Transforms
    • Definition and Properties
    • Solving Initial Value Problems with Laplace Transforms
18. Perturbation Methods
    • Regular Perturbation
    • Singular Perturbation
19. Numerical Methods for ODEs
    • Euler's Method, Runge-Kutta Methods
    • Stability of Numerical Methods
20. Applications in Science and Engineering
    • Biology (Predator-Prey Models, Epidemics)
    • Physics (Pendulum, Planetary Motion, Circuits)
    • Engineering (Control Systems, Mechanical Vibrations)
    • Economics

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