MAT-61506 Dynamical systems and chaos (spring 2014)


Introduction to nonlinear dynamical systems and chaotic dynamics. Several examples are given in biology, meteorology, discrete systems, etc. The course includes a project work on computational aspects. This year's application theme is biology.
Note: the course on differential equations (or equiv. basic knowledge on systems of differential equations) is an essential prerequisite for the course.
This course is for graduate and advanced undergraduate studies.

Lectures

Lectures are given in English by professor Mikko Kaasalainen and Dr Ilya Potapov. Period 4: Wednesday 10-12 (TD308) and Friday 12-14 (TD308).

The lecturer is available for questions on Fridays at 11-12 in office TD321.

No lecture on Fri 21 March.

Part II: Biology Applications.

NOTE: there is NO lecture on Fri Apr 11.

NOTE: Friday lectures time changed to 10.00-12.00, starting from Week 16.

NOTE: Lecture 4 is on Fri, Apr 25 due to the Easter holiday break.

The lectures of the Part II: Biology Applications are available online. The links to the video lectures as well as slides and board notes are provided below.

Lecture 1. Slides. Notes.

Lecture 2. Slides. Notes. (Logistic Map M-file, Function plots).

Lecture 3. Slides. Notes.

Lecture 4. Slides. Notes. (Lotka M-file).

Lecture 5. Slides. Notes. (Genetic and Volterra bistable switch M-file).

Lecture 6. Slides. Notes. (Harmonic oscillator M-file, Brusselator M-file).

Lecture 7. Slides. Notes. (Integrate-and-fire neuron, Hodgkin-Huxley model).

Lecture 8. Slides. Notes.

Course material

The course follows mainly the book M. Hirsch, S. Smale & R. Devaney: Differential Equations, Dynamical Systems and an Introduction to Chaos, ISBN 0123497035, Academic Press 2002. (Chapters 8-).
Available as an e-book. Notes for chaos in Hamiltonian systems can be found here.
Other useful books are, e.g., Regular and Stochastic Dynamics by Lichtenberg and Lieberman (Springer), or Perspectives of nonlinear dynamics by Jackson (Cambridge).
"Biology" books: JD Murray, Mathematical Biology I: An Introduction; CP Fall, Computational Cell Biology; E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting.

Project work

You can choose one of the Explorations of chapters 11.4, 14.6 (items 1-4, or explore the Lorenz system similarly) or 15.9 of HSD (you can investigate the systems in your own way; the numbered items in HSD are suggestions). You can also explore orbits in a Hamiltonian system (instructions here ).

Within the biological application framework you can choose between: 1) two species interaction (see instructions), 2) discrete population models (for this go, as much as you can, through the Exercises section of Chapter 2 of the Murray's book), or 3) Morris-Lecar model: excitability phenomena (see instructions).

You can return the project work any time this year. This can be in the form of a standard report (intro, problem setup, results, discussion; pdf or doc file), or you can arrange a time with either of the lecturers and show your findings (particularly useful if you want to run a program).

The project work is optional but highly recommended. If you return a decent report (addressing the essential questions/issues in the instructions), your grade by exam is raised by 1; if you discuss the topic more extensively, by 2.

Exam material

Exam chapters: HSD 8.5, 9.2-4, 10.1, 11.1, 14.1-3, 15.1-4, 15.7, Hamiltonian chaos notes 1, 2, 3.2, 4, 5, and the material on biological systems: J.D. Murray, Mathematical Biology I: An Introduction, Chapters 2 and 3; E. Izhikevich, Dynamical Systems in Neuroscience, first 2 chapters.


Page updated 17 May 2014.