## MA30060: Dynamics and chaos

[Page last updated: 15 October 2020]

Owning Department/School: Department of Mathematical Sciences
Credits: 6      [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Honours (FHEQ level 6)
Period:
Semester 1
Assessment Summary: EX 100%
Assessment Detail:
• Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module 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, including thesawtooth, tent and logistic maps.
* Fixed points and periodic orbits. Sensitivedependence on initial conditions, topological transitivity.
* Bifurcations and their classification
* Symbolic dynamics. Horseshoes. Period threeimplies chaos.
* Circle maps: continuous invertible maps of the circle. Rotation number; dynamics in the rational and irrational cases.Arnold tongues.
* Period M orbits imply existence of period Norbits. Sharkovskii's theorem. Unimodal maps and Feigenbaum's constant. Topological entropy.
* 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-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)
• 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)

 Notes: This unit catalogue is applicable for the 2020/21 academic year only. Students continuing their studies into 2021/22 and beyond should not assume that this unit will be available in future years in the format displayed here for 2020/21. Programmes and units are subject to change 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.Find out more about these and other important University terms and conditions here.