LECTURES

2

1

Introduction

1.0

Some famous ordinary differential equations, some basic methods, and some mathematical background

1.1

First order systems

1.2

Existence and Uniqueness Theorems, including Grönwall's Inequality

Dependence on Parameters, and Continuation, Theorems

4

2

Linear Equations

2.1

Existence and Uniqueness

2.2

Homogeneous Linear Systems

2.3

Inhomogeneous Linear Systems

2.4

Second order linear equations

2.5

Linear equations with constant coefficients

4

3

Linear Equations with Periodic Coefficients

3.1

Floquet Theory

3.2

Parametric resonance

3.3

Perturbation methods for the Mathieu equation

3.4

The Mathieu equation with damping

4

4

Stability

4.1

Preliminary definitions

4.2

Stability for linear systems

4.3

Principle of linearized stability

4.4

Stability for autonomous systems

4.5

Liapunov functions

3

5

Planar Autonomous Systems

5.1

Critical points

5.2

Linear autonomous systems

5.3

Autonomous, nonlinear perturbations of linear autonomous

2

6

Periodic Solutions of Plane Autonomous Systems

6.1

Preliminary results

6.2

The index of a critical point

6.3

The Van der Pol equation

6.4

Conservative systems

???

7

Perturbation Methods for Periodic Solutions

7.1

Poincaré-Lindstedt method

7.2

Poincaré-Lindstedt method, continued

???

8

Perturbation Methods for Forced Oscillations

8.1

Non-resonant case

8.2

Non-resonant case, continued

???

The method of Liapunov-Schmidt