## Module and Programme Catalogue

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# 2018/19 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 30

Module manager: Dr Jon Ward
Email: j.a.ward@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2018/19

### Pre-requisite qualifications

(MATH2016 or MATH2017) and (MATH1012 or MATH1060 or MATH1331). MATH2391 is helpful but not required.

### This module is mutually exclusive with

 MATH3396 Dynamical Systems

This module is not approved as an Elective

### Module summary

This course continues the study of nonlinear dynamics begun in MATH 2391, but for maps rather than differential equations. Maps are the natural setting for understanding the nature of chaotic dynamics, which arise in a variety of contexts in biology, chemistry, physics, economics and engineering.

### Objectives

On completing this module, students should be able to:
a) find fixed points, periodic orbits and other invariant sets in maps and compute their stability;
b) understand the structure of chaos in maps;
c) use a computer to investigate the behaviour of families of one-dimensional maps;
d) transform between the dynamics of a one-dimensional maps (the Lorenz map, the tent map and the logistic map) and symbolic dynamics;
e) identify codimension-one bifurcations in maps and sketch bifurcation diagrams;
f) use renormalisation techniques to understand the cascades of bifurcations involved in the transition to chaos.
g) be familiar with an advanced topic in the theory of discrete dynamical systems, like two-dimensional maps, ergodic theory or complex dynamics.

### Syllabus

One-dimensional maps: fixed points, periodic points, asymptotic and Lyapunov stability, Lyapunov exponent, omega-limit sets, conjugate maps, topological entropy, topological chaos and horse-shoes, Period-three implies chaos, sensitive dependence on initial conditions, Schwartzian derivative, renormalisation, the period-doubling cascade and Feigenbaum's constant. Maple programs will be used throughout to demonstrate important principles.

Two-dimensional maps: fixed points, stability, examples of bifurcations and chaotic dynamics. Transverse homoclinic orbits. Ergodic theory: measure-preserving map, mixing and ergodicity of maps, recurrence and ergodic theorems.

Complex dynamics: quadratic maps in the complex plane. Julia sets and the Mandelbrot set.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

95 hours of private study, 8 hours preparation for workshops and 55 hours preperation and writing of the report.

### Opportunities for Formative Feedback

There are five example sheets containing a mixture of pedagogical questions (feedback is given but the questions are not included in the assessment). The progress in the projects is discussed within the workshops.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment Project Report and Presentation 40.00 Total percentage (Assessment Coursework) 40.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 60.00 Total percentage (Assessment Exams) 60.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated