Outline

1.

The classical Peano existence theorem.

2.

The classical existence & uniqueness theorem.

3.

Phase plane analysis.

4.

Limit cycles.

5.

The Poincare Bendixson theorem.

6.

Linearization.

7.

Classification of equilibrium solutions of linear and non-linear systems.

8.

Energy methods and estimates including Gronwall's inequality.

9.

Lyapunov stability.

10.

Definition and properties of stiff systems.

11.

Basic theory of approximation schemes for ODEs:

(a)

Local truncation error.

(b)

Global error estimates.

(c)

Stability.

(d)

Stiff systems.

12.

Practical implementation issues:

(a)

Function evaluation count.

(b)

Local error estimation and adaptive stepsize control.

13.

Order and stability of the following schemes:

(a)

Runge-Kutta schemes including the classical embedded scheme.

(b)

Multistep schemes including the classical Adams-Bashforth-Moulton predictor corrector and BDF schemes.

(c)

Extrapolation schemes including Richardson extrapolation and the Bulirsch-Stoer method.

(d)

Stiff schemes including Rosenbrock methods.