Third (and last) test on Friday 12/12, 10-11am, in room 642

The test covers the following material from [F03]:

• Introduction
• Chapters 2, 3 and 4: sections 2.1, 2.2, 2.3, 3.1, 3.2, and 4.1

and the following exercises

• Exercises #4

The prototypical question in this test will be "determine the Hausdorff and/or Box dimension of the following fractal". This requires giving upper and lower bounds 

Preservation of topological mixing under semiconjugacy

Relevant notes can be found here. This result is part of the course, but not included in the [HK] notes.

Second test on Friday 21/11, 10-11am, in room 642

The test will cover the following material:

[HK] 7.3 and 7.4
Note on "Preservation of topological mixing under semiconjugacy" as in the 14/11 post, above.
Exercises #3

The focus here lies on the properties of symbolic dynamics, and its application to one-dimensional and two-dimensional examples (along the lines discussed in the lectures and [HK] 7.3 and 7.4).

Your will receive back the marked scripts for these tests on Friday 28/11.

Exercises #4

[F03]  exercises 2.1-2.16
[F03]  exercises 3.1, 3.3, 3.4, 3.5, 3.6, 3.8, 3.11
[F03]  exercises 4.1-4.6

First test on Friday 31/10, 10am-11am, in room 642

The test will cover the following material:
[HK] 7.1.1, 7.1.2, 7.1.3.
[HK] 7.2.1 (inclusive of 7.2 initial text), 7.2.2, 7.2.3, 7.2.5.
[HK] 7.3.1, 7.3.2
Exercises #1
Exercises #2

This test will be marked by 14/11.

Exercises #3

[HK] 7.3.3, 7.3.4, 7.3.5, 7.3.6, 7.3.7, 7.3.8, 7.3.9, 7.3.10, 7.3.11, 7.3.12 7.4.2, 7.4.3, 7.4.4, 7.4.5

Logistic map applets

Some applets to experiment with the quadratic logistic map that I discussed today
http://www.geom.uiuc.edu/~math5337/ds/applets/burbanks/Logistic.html
http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/BifArea/
for more links see
http://mat.uab.cat/~alseda/LogisticApplets.html

Exercises #2

• Verify the statements and proofs of [HK] section 7.2.1, lemma 7.1.8, propositions 7.1.3, 7.1.9 and lemma 7.2.6
• Exercises [HK] 7.2.1, 7.2.3 
• Exercises [HK] 7.3.1, 7.3.2 
• Determine the exponential growth rate p(E_m) with m in Z\{0}.
• With reference to Prop 7.1.5 of [HK], p 199, give examples of maps f satisfying the conditions set out in the first sentence of this proposition such that P_n(f)>2^n (for some n, and for all n).

Exercises #1

The first set of  exercises consists of 7.1.1 until 7.1.9 on page 204 of [HK] Chapter 7.

Office hour

The office hour for this course is thursday 12-1pm in my office Huxley 638. Please come at the start of the hour. If no-on has arrived within 15 mins of the start of the office hour, I may not be available the remainder of the hour. If you know you will be late, please advice me by e-mail (jswlamb@imperial.ac.uk).

Welcome to M345PA46

Lecturer: Prof. Jeroen S.W. Lamb 
Lectures: friday 10am-1pm in 642

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 complementary to M345PA23 Dynamical Systems, lectured by Dr Dmitry Turaev, and strongly recommended for those students intending to take Ergodic Theory (M45PA36), Bifurcation Theory (M345A24) or Advanced Dynamical Systems (M45PA38). For more information on these courses and other activities or opportunities in Dynamical Systems, see the DynamIC website.

There will be 3 in-class tests concerning the course material. The results of these tests together weigh as 10% of the total course mark. Details of these tests are announced in class and on this blog. The dates of the tests are 31/10, 21/11 and 12/12.

The course consists roughly of two parts with the following content: 
Part 1: CHAOS
(1) expanding maps
(2) chaos and mixing.
(3) Markov partitions and symbolic dynamics
(4) Smale's horseshoe and Arnold's cat map
Part 2: FRACTALS
(5) dynamically generated cantor sets
(6) Hausdorff measure and dimension
(7) Iterated Function Systems: limit sets and their Hausdorff dimension
(8) Probabilistic Iterated Function Systems and the Collage Theorem for image compression

Suggested literature (including links to essential notes):

Part 1: CHAOS

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

[BS] Michael Brin and Garrett Stuck. Introduction to Dynamical Systems, 2002.
([BS] recommended buy, although for Part 1 I follow [HK] chapter 7)

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)
Part 2: FRACTALS
[F85] Kenneth Falconer, The geometry of fractal sets, 1985
[F03] Kenneth Falconer, Fractal geometry: mathematical foundations and applications, 2003 (or 1990 ed).
[B] Michael Barnsley, Fractals everywhere, 2000.