Chapter 1: Basic definitions
- Ordinary differential equations.
- Systems of ordinary differential equations.
- Ordinary differential equations of higher order.
Chapter 2: Explicit techniques
- Autonomous ordinary differential equations.
- Separation of variables.
- Change of variables
- Homogeneous differential equations.
Chapter 3: Existence and uniqueness
- Fixed point theorem of Banach
- Theorem of Picard-Lindelöf and the Theorem of Peano.
- Continuous/smooth dependence on initial conditions.
Chapter 4: Linear systems
- Homogeneous and inhomogeneous systems.
- Systems with constant coefficients.
- Classification in 2 dimensions.
Chapter 5: Boundary value problems
- Boundary value problem of Sturm-Liouville.
- Construction of solutions.
Hoofdstuk 6: Basic properties of dynamical systems
- Discrete and continuous dynamical systems.
- Vector fields and flows.
- Fixed points, periodic and non-periodic orbits.
- Commuting flows.
Chapter 7: Local and global behaviour of dynamical systems
- Stability and invariant sets.
- Local behaviour close to hyperbolic fixed points: the theorem of Hartman-Grobman.
- Limit sets.
- Dynamics in 2 dimensions: the theorem of Poincaré-Bendixon.
Chapter 8: Chaos: Strange attractors
- What is chaos?
- The Lorenz system.
- Strange attractors.