A mathematical model is simply a set of equations which represent the behaviour of some ‘real-life’ process.  By analysing these equations mathematically we can then seek to understand and predict the behaviour of the process.  An example of this is Fibonacci's famous (if somewhat crude) model for the growth of a rabbit population. However, many other phenomena can be modelled mathematically, and modern applications cover a wide range of topics from the physical, biological and social sciences.

This module provides an introduction to the modelling process through the study of linear and non-linear dynamical systems in discrete time. This leads on to the study of the chaotic behaviour in dynamical systems and of the ways how deterministic chaos can arise even in simple models. The list of examples of the dynamical systems under study includes linear recurrent equations, rotations of the circle, tent map, Bernoulli shift and the logistic map. The questions like bifurcations diagrams, Lyapunov exponents and invariant measures are discussed in some detail. Fractals and their characterizations through fractal dimensions is another topic of the module.

In addition to providing a foundation for later studies in modelling and applied mathematics, this module is intended to consolidate topics covered in core mathematics modules by using the material in contexts which lend themselves to visualisation.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• analyse and solve first and second order recurrence equations
• see the difference between regular and chaotic behaviours of nonlinear dynamical systems
• demonstrate an understanding of the qualitative features of the chaotic behaviour in dynamical systems
• describe fractals and some of their features

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:

Solution and qualitative analysis of recurrence equations

Qualitative and quantitative analysis of simple nonlinear dynamical systems

Qualitative and quantitative characterizations of fractals

Ability to generate chaos and fractals using standard software 

Transferable Skills:

Modelling skills.  Appreciation of chaotic behaviour and fractals.

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.  

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Recurrent equations.
• Practical problems leading to these equations.
• Generating functions and its properties.
• Solving recurrences using the technique of generating functions.
• Examples of non-linear discrete dynamical systems and practical problems leading to these systems.
• Rotations of the circle. Introduction of the notion of invariant measure.
• Piece-wise linear transformations of the unit interval. Introduction of the concepts of the Lyapunov exponent and entropy.
• The logistic map. Bifurcation diagram. Routes to chaos.
• Fractals and fractal dimensions.

Concrete Mathematics, Graham, R. L., Knuth, D. E., Patashnik, O., Addison-Wesley, 1994.

An Introduction to Chaotic Dynamical Systems, Devaney, R .L., Westview Press, 2003.