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2022/23 Taught Postgraduate Module Catalogue
MATH5395M Advanced Dynamical Systems
20 creditsClass Size: 30
Module manager: Dr Jitse Niessen
Email: J.Niesen@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2023/24
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
| 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
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 |
| Open Book exam | 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
Reading list
The reading list is available from the Library websiteLast updated: 28/04/2023 14:54:43
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