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273027 Introduction to Dynamical Systems (5 sp)

The course treats fundamental notions in the theory of discrete-time dynamical systems, such as periodicity, attraction/repulsion, bifurcations, chaos and sensitive dependence on initial conditions ("the butterfly effect"). In the multidimensional case we give a survey of classical stability theory for linear systems.

The course is given, in English, by professor Göran Högnäs on Mondays and Fridays 10-12 in Tillståndsrummet (Axelia I, 3rd floor) and Thursdays 14-16 in Aud. Salin (Axelia II).

The course starts on Thursday, September 3 at 2 p.m. and ends by the end of October. There are weekly exercise sessions, but in addition some larger individual project work will also be assigned. The examination is a 24-hour open-book examination.

Prerequisites: Calculus and some course in linear algebra (matrices) and differential equation.

Course material: R. L. Devaney: An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 2nd ed. 1989 (or later editions). Additional handouts will be distributed.

The handouts can now be downloaded as pdf-files below:
Handouts, pages 1-12, 13-18, 19-27, 28-35, 36-45, 46-53, 54, 55-60,
61-68, 68-75, 76-86, 81, 87-92, 93-99, A1-10, Ex. 4 (25.9)

Terminology and exercises. Some exercises from the book.

More course material:
A picture of an Allee model
Bifurcation diagrams for the logistic model: the interval (3, 3.2), the interval (3, 3.6) and the interval (3.8, 4)
Bifurcation diagram for the Ricker model
The behaviour of the deterministic Ricker model, fixed point 57 and fixed point 1900
Approximation formula for iterations of sin(x)
Pictures of periods
On Intervals, Transitivity = Chaos by Vellekoop and Berglund

273028 Special course in Dynamical Systems (5 sp)

During period 2, a follow-up course will be taught by Professor Högnäs and Docent Gunnar Söderbacka. In this course we will briefly look at dynamical systems in continuous time and apply it to some simple malaria models. The lecture hours are Mondays 10-12 in Tillståndsrummet, Thursdays 13-15 in Ringbom and Fridays 10-12 in Tillståndsrummet. The course starts on Monday, October 26.

Course material:
Lecture material for Monday 26.10: Course outline, Lecture notes, Book excerpts
Lecture material for Thursday 29.10 - Friday 6.11: Simple dynamical systems and epidemical modelling, Error sheet
Lecture material: Book excerpts, section 4.5, Book excerpts, chapter 5
Lecture material for Wednesday 11.11: Lecture notes
Lecture material for Thursday 12.11: Lecture notes
Lecture material: Book excerpts, section 5.11
Lecture material: Book excerpts, section 5.7-5.8 and appendix
Lecture material for Friday 11.12: Final lecture notes

Further course material:
Notes on one-dimensional discrete systems
Phase portraits for one-dimensional systems
Phase portraits for linear systems
Population models, two examples
Exercises with answers

Constructing phase portraits
Phase portraits for equations of a certain type
Phase portraits: an example
The Rozov exercises
Some other exercises
Types of differential equations

Stability and Lyapunov functions
Stability of fixed points

Senast uppdaterad: 11.12.2009
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