Chaos in Dynamical Systems

**This information is for the 2019/20 session.**

**Teacher responsible**

Prof Graham Brightwell

**Availability**

This course is available on the BSc in Business Mathematics and Statistics, BSc in Mathematics and Economics, BSc in Mathematics with Economics, BSc in Mathematics, Statistics, and Business and BSc in Statistics with Finance. This course is available as an outside option to students on other programmes where regulations permit. This course is available with permission to General Course students.

**Pre-requisites**

**Mathematical Methods** (MA100) and **Introduction to Abstract Mathematics **(MA103) are pre-requisites. It is highly desirable to have also taken the course **Real Analysis** (MA203).

**Course content**

Given a function f:X->X, what is the (iterative) behaviour of fn (x)=f(f(...(f(x)) )) (f applied n times). What do 'orbits' of f look like - an orbit has the form x, f(x), f(f(x)).,,, Particular emphasis is given to long-term evolution and stability analysis of such systems f:X->X. We explore connections with so-called fractal sets, which are roughly sets that look the same at any degree of magnification, and may have a dimension which is not an integer. We also discuss chaotic maps f, where fn (x) cannot be determined if there is any small uncertainty about x. Topics covered: Iteration of discrete maps. Orbit analysis. Contraction mapping theorem. Quadratic maps. Bifurcations. Definition of chaos. Sarkovskii's theorem. Fractal sets.

**Teaching**

22 hours of lectures and 10 hours of classes in the LT.

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**Formative coursework**

Written answers to set problems will be expected on a weekly basis.

**Indicative reading**

Full lecture notes will be provided. The following may prove useful, R Devaney, A First Course in Chaotic Dynamical Systems; R Devaney, An Introduction to Chaotic Dynamical Systems; E Scheinerman, Invitation to Dynamical Systems. A fun and interesting read might be Glieck, J.,Chaos: Making a New Science.

**Assessment**

Exam (100%, duration: 2 hours) in the summer exam period.

**
Key facts
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Department: Mathematics

Total students 2018/19: 4

Average class size 2018/19: 4

Capped 2018/19: No

Value: Half Unit

**Personal development skills**

- Self-management
- Problem solving
- Application of information skills
- Communication
- Application of numeracy skills
- Specialist skills