Lecture 2

In this lecture we introduce the concept of a phase portrait. Phase portraits can be used to visualise the behaviour of nonlinear systems with two states. They will provide the intuition for many of the more advanced concepts that we will encounter throughout the rest of the course.

By the end of the lecture, you should be able to:

1. Sketch the phase portrait of simple systems.
2. Classify singularities, and explain their characteristics.

In addition to the video material, you can make use of:

• Lecture slides that support the course material: lec02.pdf Download lec02.pdf
• The notes that we took together in class published under lecture notes 2022 FRTN05_Lecture02_annotated.pdf Download FRTN05_Lecture02_annotated.pdf.
• Code for drawing phase portraits and simulating trajectories: matlab-lec-2.zip Download matlab-lec-2.zip

We also encourage you to try out some other tools for simulating nonlinear systems, in particular simulink. An introductory video and manual can be found here, as well as in these lecture slides: simulink_lec.pdf Download simulink_lec.pdf

Introduction

In this video we introduce the concept behind a phase portrait.

Equilibrium points and linearisation

In this video we discuss equilibrium points, and the idea behind linearisation.

Nodes and saddles

In systems with two states, the types of equilibria can be classified based on the linearisation. We introduce the first two types of equilibrium points: nodes and saddles.

Foci

The next type of equilibrium point is the focus.

Special cases

When the A matrix in the linearisation has repeated eigenvalues, a more detailed analysis is required. We discuss the steps here.

Eigenvalues on the imaginary axis

When the linearisation has eigenvalues on the imaginary axis, linearisation fails. Here we investigate this case.

Sketching phase portraits

We put it all together, and give an example of sketching a phase portrait.