Ordinary Differential Equations
and
Dynamical Systems

Gerald Teschl

Abstract
This manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear flow.

Then we establish the Frobenius method for linear equations in the complex domain and investigate Sturm-Liouville type boundary value problems including oscillation theory.

Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems.

We prove the Poincare-Bendixson theorem and investigate several examples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed as well.

Finally, there is an introduction to chaos. Beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits.

MSC: 34-01
Keywords: Ordinary differential equations, Dynamical systems, Sturm-Liouville equations.

Download
The text is available as postscript (6.8M), or pdf (3.0M) version. In addition, there is a notebook (1M) with the Mathematica code from the text plus some further extensions. To solve complex equations via the power series method you also need my package DiscreteMath`DiffEqs`. Any comments and bug reports are welcome!
Table of contents
    Part 1: Classical theory
  1. Introduction
    1. Newtons equations
    2. Classification of differential equations
    3. First order equations
    4. Finding explicit solutions
    5. Qualitative analysis of first order equations
    6. Qualitative analysis of first order periodic equations
  2. Initial value problems
    1. Fixed point theorems
    2. The basic existence and uniqueness result
    3. Some extensions
    4. Dependence on the initial condition
    5. Extensibility of solutions
    6. Euler's method and the Peano theorem
  3. Linear equations
    1. The matrix exponential
    2. Linear autonomous first order systems
    3. Linear autonomous equations of order n
    4. General linear first order systems
    5. Periodic linear systems
    6. Appendix: Jordan canonical form
  4. Differential equations in the complex domain
    1. The basic existence and uniqueness result
    2. The Frobenius method for second order equations
    3. Linear systems with singularities
    4. The Frobenius method
  5. Boundary value problems
    1. Introduction
    2. Symmetric compact operators
    3. Regular Sturm-Liouville problems
    4. Oscillation theory
    5. Periodic operators
    Part 2: Dynamical systems
  6. Dynamical systems
    1. Dynamical systems
    2. The flow of an autonomous equation
    3. Orbits and invariant sets
    4. The Poincaré map
    5. Stability of fixed points
    6. Stability via Liapunov's method
    7. Newton's equation in one dimension
  7. Local behavior near fixed points
    1. Stability of linear systems
    2. Stable and unstable manifolds
    3. The Hartman-Grobman theorem
    4. Appendix: Integral equations
  8. Planar dynamical systems
    1. The Poincaré-Bendixson theorem
    2. Examples from ecology
    3. Examples from electrical engineering
  9. Higher dimensional dynamical systems
    1. Attracting sets
    2. The Lorenz equation
    3. Hamiltonian mechanics
    4. Completely integrable Hamiltonian systems
    5. The Kepler problem
    6. The KAM theorem
    Part 3: Chaos
  10. Discrete dynamical systems
    1. The logistic equation
    2. Fixed and periodic points
    3. Linear difference equations
    4. Local behavior near fixed points
  11. Discrete dynamical systems in one dimension
    1. Period doubling
    2. Sarkovskii's theorem
    3. On the definition of chaos
    4. Cantor sets and the tent map
    5. Symbolic dynamics
    6. Strange attractors/repellors and fractal sets
    7. Homoclinic orbits as source for chaos
  12. Periodic solutions
    1. Stability of periodic solutions
    2. The Poincare map
    3. Stable and unstable manifolds
    4. Melnikov's method for autonomous perturbations
    5. Melnikov's method for nonautonomous perturbations
  13. Chaos in higher dimensional systems
    1. The Smale horseshoe
    2. The Smale-Birkhoff homoclinic theorem
    3. Melnikov's method for homoclinic orbits

    Bibliography
    Glossary of notations
    Index