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Programme & Unit Catalogues

## MA30060: Dynamics and chaos

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Honours (FHEQ level 6)
Period: Semester 1
Assessment Summary: EX 100%
Assessment Detail:
• Exam (EX 100%)
Supplementary Assessment: MA30060 - Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA20218 and take MA20221
Description: Aims:
This course provides an elementary and accessible introduction to dynamical systems and chaos. The course treats discrete-time systems in one dimension. This permits results to be presented rigorously while relying only on core material from the second year.

Learning Outcomes:
Students should be conversant with the generation of complicated dynamics in simple nonlinear systems. They should appreciate in particular the dynamics of iterated maps of the line and of the circle.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).

Content:
Continuous maps of the interval, the sawtooth, tent and logistic maps. Fixed points and periodic orbits. Sensitive dependence on initial conditions, topological transitivity. Symbolic dynamics. Horseshoes. Period three implies chaos.
Continuous invertible maps of the circle. Rotation number; dynamics in the rational and irrational cases. Arnold tongues.
Additional topics chosen from: Period M orbits imply existence of period N orbits. Sharkovskii's theorem. Unimodal maps and Feigenbaum's constant. Topological entropy.
Non-invertible circle maps. Rotation intervals. Strange 'non-chaotic' attractors.
Programme availability:

#### MA30060 is Optional on the following programmes:

Department of Mathematical Sciences
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
• USMA-AFB05 : BSc(Hons) Statistics (Year 3)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
• USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)

Notes:
* This unit catalogue is applicable for the 2014/15 academic year only. Students continuing their studies into 2015/16 and beyond should not assume that this unit will be available in future years in the format displayed here for 2014/15.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.