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
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
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.
[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)
([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.
No comments:
Post a Comment