Enhanced coursework deadline 22 January 2010

For all MSc and M4 students, please note that the deadline for the Enhanced Coursework (see left-hand side margin) is 22 January 2010;  hand-in by 4pm to the student office (Huxley 649). 

Final progress test week 8

The third and last test will be on wed 2/12; 10:00-10:30. Material: [HK] sections 7.1, 7.2, 7.3, 7.4 + lecture notes of Oleg Makarenkov (except section 3 of his notes). The emphasis  will be on the examples that were not included in the material of the last test: nonlinear expanding circle maps, 2d examples (like horseshoe and "cat map"), and interval maps (like logistic map).

Arnold "cat map" applet

http://amanuensis.co.uk/arnold.html

MSc/M4 Enhanced coursework

The enhanced coursework concerns Sharkovsky's Theorem, see problem sheet and relevant notes from [BS] section 7.3 in the left-hand side margin.

Logistic map applets

During the lectures I made use of the following applets to illustrate the behaviour of the logistic map:
http://math.la.asu.edu/~chaos/logistic.html
http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/BifArea/

Problem sheets

Please note the links to problem sheets in the lhs margin. Most problems are refs to problems in [HK] and [BS].

Week 5 test

Thu 12 Nov, 9-9.30am (please be early)
Material: expanding circle maps, topological mixing and chaos.
notes [HK] 7.1.1, 7.1.3, 7.2.1, 7.2.2, 7.2.3, 7.2.5, 7.3.1, 7.3.2, 7.3.4, 7.3.5, 7.3.6, and 7.3.7. (previously I indicated also 7.4.1 and 7.4.2 but as I did not manage to treat them this week, the material there will not be part of this test).

22/10 lectures

The 22/10 lecture will be given by Cristina Sargent and Dmitry Turaev:
9-10am: Mode-locking in the Arnold circle map (Sargent), relevant notes [AP] Sec 5.2
10-11am: Denjoy's theorem (Turaev), relevant notes [BS] Sec 7.2

Week 3 test

There will be 3 or 2 short tests during term with a combined weight of 10% of the total exam mark. It should help you see how you are doing and provide some motivation to keep up to speed with the course material. It will also help me find out how you are doing.


The first test will be thu 29 Oct 9.00-9.30am. I will not be in a position to make arrangements for late arrivals as I will start the lecture at 9.30am, so make sure you come EARLY.


The test will concern (basic) material on invertible circle maps. Preparation should consist of a thorough studying of the relevant notes ([BS] section 7.1 and [HK] sections 4.3.1, 4.3.2, 4.3.4, 4.3.5, 4.3.6) with the aim to understand rather then to remember. It will of course also be useful to attempt doing the associated exercises.

Note: I did not finish all of the above material in the lecture concerning proof of the nature of the w-limit set in the irrational case and the construction of the (semi-)conjugacy so these parts will not feature in the test (and I will discuss them in the lecture on wednesday).

Prerequisites

Contraction mapping theorem

It would be good if all students attending this course have a look through Chapter 3 of the course notes for the 2nd year course M2AA1 (see also link in left-hand-side margin). The main result discussed there is the contraction mapping theorem and some immediate consequences like the implicit and inverse function theorems. It would be advisable to be familiar with the contraction mapping theorem (in detail inclusive of proof) and with the implicit and inverse function theorems (without proofs).

Other general background

Please see the appendix of [HK] in the left-hand-side margin for more useful background material.

Week 1

The relevant notes for week 1 (on invertble circle maps) can be found in [BS] sections 7.1 and 7.2 and [HK] section 4.3 (see left-hand-side margin for links to pdfs).

In the first lecture, I showed the following Videos of pendula and a simulator for the vertically driven pendulum . Some more more simulators of well known dynamical systems can be found here.

Welcome to M3/4A23 version 2009

Lectures: room 341 (Huxley), wednesday 9-11am, thurday 9-11am

The aim of this course is to provide an introduction to basic concepts and ideas underlying the modern qualitative theory of ordinary differential equations (dynamical systems), also popularly known as Chaos Theory.

This course is strongly recommended for those students intending to take Ergodic Theory (M4A36), Bifurcation Theory (M3A24/M4A23) and Advanced Dynamical Systems (M4A38).

Suggested literature:

Main texts:

[BS] Michael Brin and Garrett Stuck. Introduction to Dynamical Systems. 2002. (recommended buy)

[HK] Boris Hasselblatt and Anatole Katok. A first course in Dynamics. 2003.

Other:

John Guckenheimer and Philip Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. 1983. (somewhat dated but inspiring in scope and context)

Anatole Katok and Boris Hasselblatt. Introduction to the Modern Theory of Dynamical Systems.1995. (reference text)

Clark Robinson. Dynamical Systems. Stability, Symbolic Dynamics and Chaos. 1995. (advanced textbook)