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2022/23 Taught Postgraduate Module Catalogue

MATH5395M Advanced Dynamical Systems

20 creditsClass Size: 30

Module manager: Dr Jitse Niessen

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2022/23

Pre-requisite qualifications

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

This module is mutually exclusive with

MATH3396Dynamical 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.


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.


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 typeNumberLength hoursStudent hours
Private study hours156.00
Total Contact hours44.00
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

Assessment typeNotes% of formal assessment
ProjectReport and Presentation40.00
Total percentage (Assessment Coursework)40.00

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

Exam typeExam duration% of formal assessment
Open Book exam2 hr 00 mins60.00
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 website

Last updated: 13/09/2022


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