An Introduction To Non-Linear Dynamics

Section 1.1: Course Introduction

Department of Maths and Applied Maths - University of Cape Town

If you find any mistakes, or have comments on the notes, please message me at jon.shock@gmail.com.

The textbook that I will be following is: Nonlinear Dynamics and Chaos, by Steven Strogatz.

You do not need to get it. I will aim for this all to be self-contained. However, it is a beautifully written book, and almost certainly

the best introduction to the subject.

Steven Strogatz actually has some of the material that we will be covering on Youtube (Note that we will only really cover the first two his lectures):

here.

He has in the past tweeted to this link which has some Python code for some of the examples in the book.

Here are some more links to online courses which you may want to browse.

∘ Complexity Explorer course on intro to dynamical systems and chaos
∘

ComplexityExplorercourseonintroduction-to-differential-equations

Please note, that I will often put in links to Wikipedia articles. One has to be careful about how accurate Wikipedia pages are, and in general if you are writing a scientific research paper, Wikipedia articles are not good to reference. However, for an educational set of notes like this, where I think that it will add useful content, I will be doing so. Wikipedia articles should be read with a critical hat on, and if you want to dig further, look at the references in the articles themselves.

What is this course about?1) Understanding what a dynamical system is through examples and definitions2) Knowing how to write down the mathematics of a dynamical system through examples and problems2) Knowing how to understand the interactions of dynamical systems through quantitative and qualitative means through many different techniques

I am writing these notes using the Wolfram Mathematica programming language. It means that I can very easily include nice plots and animations which should add to the content. You do not need to know how to use this.

The Course Content:

Section 1: Introduction (Strogatz chapter 1)

1.1 Overview of the course (what you're reading now) 1.2 a) The history of dynamics - The beginnings of dynamics 1.2 b) The history of dynamics - Poincare and chaos

Aside: Difference equations 1.3 Systems of differential equations 1.4 Phase space 1.5 The spectrum of systems of differential equations

1.6 Exercises for part 1

Section 2: Flows on the line (Strogatz chapter 2)

2.1 Fixed points and trajectories 2.2 Fixed Points and their stability 2.3 Linear stability analysis 2.4 Existence and uniqueness of solutions 2.5 Absence of oscillations 2.6 Potentials 2.7 a) Solving equations on the computer

2.7 b) Back to basics: A summary of phase spaces and phase portraits

2.8 Exercises for part 2

Section 3: Bifurcations (Strogatz chapter 3)

3.1 and 3.2 Introduction and saddle-node bifurcations 3.3 Transcritical bifurcations 3.4 Pitchfork bifurcations 3.5 Combinations of bifurcations 3.6 Chaos and the logistic map 3.7 a)

Imperfectbifurcationsandcatastrophes

3.7 b)

Imperfectbifurcationsandcatastrophespart2

3.8 Exercises for part 3

Section 4: Linear systems in two dimensions (Strogatz chapter 5)

4.1

Introductiontolinearsystemsintwodimensions

4.2

Classificationoflinearsystemsandtheiroriginpart1

4.3 Classification of linear systems and their origin part 2 4.4 Application: Love affairs

4.5 Exercises for part 4

Chapter 1: Introduction

All models are wrong, but some are useful
George Box (sort of)

1) Exponential growth:

2) The logistic equation:

3) The pendulum:

All of these differential equations are in terms of time, but a differential equation doesn't have to be. However, in this course, we will be talking about time...and we'll see why. The key is in the word dynamics.

If you’re given a differential equation describing some complicated system, like populations of rabbits and wolves, or planets moving around each other, or neurons firing in the brain, we very often can figure out some behavioural traits without solving the equation. This 2nd half of MAM1043H is about solving these so-called dynamical systems. Systems of interacting elements (things) which are described by differential equations. We will see lots along the way, but the types of things that we might be interested in are:

∘ A planet moving around a star
∘ A population of rhinos interacting with a population of poachers
∘ The propagation of light from a lightbulb through the air
∘ The price of oil
∘ The flow of water through a river
∘ The weather
∘ The evolution of life on Earth
∘ The expansion of the universe
∘ The swinging of a pendulum
∘ The firing of a single neuron in your brain
∘ The firing of all of the neurons in your brain
∘ The growth of the internet
∘ The flow of heat through a piece of metal
∘ The spread of fire through a forest
∘ The running of a computer program
∘ The increase in the number of bacteria in a sample
∘ and so many more!

As you can see, these already come from so many disciplines: physics, biology, economics, computer science, meteorology, sociology, neuroscience...

What do all of these things have in common?

◼

They are all about how something changes in time.

◼

They are all about the interaction, either with itself, or in some environment. Let’s look at that list again:



Now try and complete the list below on your own, as much as you can. There are many ways to answer this, and most of them are very complicated systems. It's more a matter of getting you to think about it than getting a fixed answer.

∘ The weather
∘ The evolution of life on Earth
∘ The expansion of the universe
∘ The swinging of a pendulum
∘ The firing of a single neuron in your brain
∘ The firing of all of the neurons in your brain
∘ The growth of the internet
∘ The flow of heat through a piece of metal
∘ The spread of fire through a forest
∘ The running of a computer program
∘ The increase in the number of bacteria in a sample
∘ and so many more!



Now try and write down some dynamical systems of your own... ie. interacting things which change in time.

You may ask the question “What is a system which is not dynamical”. Well, the definition of system, is one from Wikipedia (though many dictionary definitions are similar):

“A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole”

So, a set of equations which are connected, like:

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