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Not available in 2020/21

MA303 Half Unit

Chaos in Dynamical Systems

**This information is for the 2020/21 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 and BSc in Mathematics, Statistics and Business. 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.

**Important information in response to COVID-19**

Please note that during 2020/21 academic year some variation to teaching and learning activities may be required to respond to changes in public health advice and/or to account for the situation of students in attendance on campus and those studying online during the early part of the academic year. For assessment, this may involve changes to mode of delivery and/or the format or weighting of assessments. Changes will only be made if required and students will be notified about any changes to teaching or assessment plans at the earliest opportunity.

** Key facts **

Department: Mathematics

Total students 2019/20: 11

Average class size 2019/20: 11

Capped 2019/20: No

Value: Half Unit

**Personal development skills**

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