MAM2046W - Second year nonlinear dynamics

Section 0.0: 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.

This is the second year course which follows the first year course, MAM1043H. Notes for MAM1043H can be found here. The next couple of paragraphs are the same as those in the previous course.

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:

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: Two dimensional nonlinear systems (Strogatz chapter 6)

1.1 Phase portraits in two dimensions 1.2 Existence and uniqueness in two dimensional systems 1.3 Linearisation, fixed points and classification 1.4 Conservative systems 1.5 Reversible systems 1.6 Index theory 1.7 Exercises for part 1

Section 2: Limit cycles (Strogatz chapter 7)

2.1 Introduction to limit cycles 2.2 Ruling out closed orbits 2.3 The Poincaré-Bendixson theorem 2.4 Liénard systems 2.5 Relaxation oscillations

2.6 Weakly nonlinear oscillators

2.6 Exercises for part 2

Section 3: Bifurcations (Strogatz chapter 8)

3.1 Saddle-nodes, transcritical and pitchfork bifurcations in two dimension 3.2 Hopf bifurcations 3.3 Oscillating chemical reactions 3.4 Global bifurcations of cycles 3.5 Coupled oscillators and quasiperiodicity 3.6 Poincaré maps 3.7 Exercises for part 3

Section 4: Chaos (Strogatz chapter 9)

4.1 The Lorenz equations and chaos on a Strange Attractor 4.2 The Lorenz map 4.3 Exercises for part 4

Section 5: One dimensional maps (Strogatz chapter 10)

5.1 The logistic map 5.2 Lyapunov exponents 5.3 Hénon maps (Chapter 12) 5.4 Exercises for part 5

Section 6: Fractals (Strogatz chapter 11)

6.1 TBD

Chapter 0: Reminder

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

We started off last year talking about differential equations where we had a single function, of the form:

and we found that there was some interesting phenomena to discover even in this very simple setting. However, there were a number of things which we proved could not happen, for instance you can’t have oscillations in a one dimensional system (remembering that if we add in time dependence this is equivalent to looking at a two dimensional system).

Where

We are actually going to be able to use lots of the machinery that we developed in the first year, so remind yourselves of this here. There is also a very nice set of lecture notes here which gives a detailed overview of such systems. I’d definitely recommend using this as a resource.

This plot alone will help us a lot when we move on in the next section to the non-linear case.

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