Chaotic dynamical systems

Dr Matt Finn, University of Adelaide
matthew {dot} finn AT adelaide [.] edu (dot) au
Feel free to e-mail me if you have any questions about the course.

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Synopsis

Many complex physical systems can be modelled accurately by a small number of deterministic coupled differential or difference equations. Examples include neuron dynamics, population modelling, chemical reactions, stirring and mixing, particle interactions, forced pendulums, weather modelling, and even dripping taps, to name just a few. Though these models are often simple to write down, their solutions often exhibit very complicated behaviours (which are observed in reality), including chaos, meaning extreme sensitivity to initial conditions.

This course will provide an introduction to the theory of continuous and discrete dynamical systems, with a particular emphasis on bifurcations and routes to chaotic behaviour. The first few lectures will set the scene, showing how low-dimensional dynamical systems arise from various approximations to complex physical systems. We will observe that these systems can exhibit exotic changes in behaviour as governing parameters are varied smoothly, including chaotic dynamics. To understand this behaviour we will study in detail the stability and bifurcations of periodic structures in these systems, bringing together results from linear algebra, multivariable calculus, differential equations, topology and group theory.

Topics covered will include: an introduction to flows and maps; phase plane analysis of autonomous flows; periodic orbits and limit cycles; the Poincare–Bendixson theorem; saddle–node, transcritical, pitchfork and Hopf bifurcations; conservative versus dissipative systems; hysteresis; dimensional reduction and Poincare sections; analysis of interval maps; period doubling bifurcations and cascades; chaos; Sharkovskii's theorem; area preserving maps and homoclinic tangles; the Thurston–Nielsen theorem; mapping classes and braid groups (time permitting).

Contact hours

There will be eight lectures and one tutorial class in each of the first three weeks. In the fourth week there will be no new material and students will be encouraged to attend the ANZIAM applied mathematics conference from Monday to Thursday. On the last Friday there will be a tutorial as usual.

The tutorial classes will explore all aspects of the course material. Some questions will require an element of Matlab programming, so the tutorials will take place in a computer laboratory. In the first fifteen minutes of each tutorial there will be a multiple choice test of the week's material. The tests (which do not count towards assessment) are designed to provide you and me with some extra feedback on understanding of the material.

I will be available in my office for consulting during business hours during the first three weeks. Either stop by unannounced and I will be happy to help if I am not busy, or email ahead to arrange a time. In week four I will be at the ANZIAM Conference, so grab me there if you want to talk.

Prerequisites

Familiarity with basic linear algebra (especially eigenvalues and eigenvectors), multivariable calculus (especially differentiability of vector fields and Taylor series) and differential equations (especially solution techniques for systems of linear ordinary differential equations) is essential. Some prior knowledge of numerical methods for solving nonlinear ordinary differential equations and experience programming with Matlab would be helpful, but is not essential. No prior experience of topology or group theory is required.

I will make a prerequisite "revision" test available on the web page near the start of the course. If you want to revise your core undergraduate mathematics you can find all the background material in Advanced Engineering Mathematics (by Kreyszig) or any similar textbook.

Assessment

The course will be assessed by four written assignments (10% each) and a take-home examination (60%).

Assignment questions will be available on the Monday of each week and solutions will be due by 5pm on the following Monday. Each coursework will involve a combination of pen-and-paper analysis and Matlab computing. You can seek help with the assignments at any time, including during the tutorial classes on the Friday of each week. I will mark all assignments (whether you take the course for credit or not) and return them in the next tutorial class. You may submit your solutions by email, in lectures, or to my office. I am very strict with assignment deadlines, and under normal circumstances a late assignment will be awarded a mark of zero.

Plagiarism and copying are serious offences in any academic environment. Students are strongly encouraged to discuss the coursework among themselves, but the solutions presented must be their own work written in their own words. Substantially similar works from different students will not be acceptable. Plagiarism may incur penalties as severe as failure in the subject.

Students taking the course for credit will receive a take-home examination in the final week of the course. Students must not collaborate or seek assistance from anyone regarding the examination questions. Solutions may be submitted in person, by email, or in the post, and must arrive to me before 5pm on Friday 18 February. Late submissions will receive a mark of zero unless there are extenuating circumstances. (If there are, it is very important that you notify me as soon as possible, and certainly before the due date.)

Alternative assessment arrangements for disabled students may be made through the Summer School Director.

Resources

A comprehensive web page will be available containing all of the material for the course, including tutorial questions and solutions and a full set of lecture notes. There are many good textbooks on dynamical systems. I have listed some useful references below. You are certainly not required to purchase any of these books, as all the necessary material will appear in the lecture notes. You may wish to borrow one or more from your library to have access to further material on certain topics. Please share your books with your coursemates.

  • Nonlinear Systems (Drazin)
  • Stability, Instability and Chaos (Glendinning)
  • Nonlinear Dynamics and Chaos (Strogatz)
  • Dynamical Systems in Neuroscience (Izhikevich)
  • The Topology of Chaos (Gilmore and Lefranc)
  • Chaos: A Mathematical Introduction (Banks, Dragan and Jones)
  • Chaotic Behaviour of Deterministic Dissipative Systems (Marek and Schreiber)
  • Differential Equations, Dynamical Systems and an Introduction to Chaos (Hirsch, Smale and Devaney)
  • An Introduction to Chaotic Dynamical Systems (Devaney)
  • The Essence of Chaos (Lorenz)

About Matt Finn

I am originally from the United Kingdom. I obtained a Masters degree and PhD in Applied Mathematics from the University of Nottingham. Since receiving my doctorate in 2003 I have been a researcher at the University of Nottingham, Imperial College London, and (currently) the University of Adelaide. I am a fluid dynamicist by training. I am particularly interested in the topology of chaotic fluid mixing, a field that naturally brings together several interesting areas of contemporary applied and pure mathematics.


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