Lecture 1
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Welcome to the course! In this lecture we explain what makes a nonlinear system nonlinear: nonlinear behaviours! By the end of this lecture you should have a basic understanding of common nonlinear behaviours including:
- Limit cycles
- Regions of attraction
- Multiple equilibria
We will also introduce the basic framework of nonlinear state-space models that we will use for studying nonlinear systems throughout the course.
Lecture slides that can be used to support and complement the video material can be downloaded here: [lec01.pdf].
The notes that we took together in class published under lecture notes 2022 [FRNT05_Lecture01 - annotated.pdf].
We also strongly encourage you to make a start on the exercise problems, and to start simulating nonlinear systems yourself!
Introduction
In this video we provide a quick overview of a class of interesting nonlinear behaviours.
A recap on linear systems
In this video we revisit the linear world, and concepts of scaling and superposition of solutions.
A modelling framework for nonlinear systems
In this video we introduce the basic modelling framework for nonlinear systems used throughout the course.
Limit cycles
In this video we discuss limit cycles in a bit more depth through the Van der Pol oscillator. We explain what it means for a system to have a limit cycle, and why linear systems cannot have them.
Regions of attraction
In this video we discuss what it means for an equilibrium point to only be locally stable from the perspective of a simple model for disease spread.
Multiple equilibria
In this video we introduce a simple mechanical model that has multiple stable equilibria, and explain why no linear system can display this behaviour.
In case you needed any more motivation...
I thoroughly recommend the following video for an introduction to other interesting nonlinear behaviours. We won't be able to cover them in the course unfortunately, but well worth a watch!
Nicky Case has also made some nice interactive tools for playing around with nonlinear behaviours. Feel free to check out this link for more!