Unit(s) of Assessment	Weighting Towards Module Mark (%)
Unseen examination	70
Course work	30
Qualifying Condition(s)    A weighted aggregate mark of 40% is required to pass the module

An introduction to system dynamics.  We start with linear systems and progress to non-linear mappings and then simple non-linear flows.  We take a brief look at numerical methods for ODEs and note their similarity to non-linear mappings. Next we review ways of representing system dynamics including bifurcation diagrams, Poincare sections and attractor reconstructions.  These methods are then tried out on some stochastic systems to further emphasis that chaotic does not mean random.  We then proceed to an overview of fractals and their application to chemical engineering. Cellular automata are introduced in the context of diffusion and reaction systems and we finish with a brief introduction to Monte Carlo as a way of dealing with stochastic systems.  We finish the module with an extended worked example on MC methods.  Excel and Maple are used throughout.

·         To introduce the students to non-linear systems in a qualitative approach that avoids any major mathematical details. ·         To provide students with an overview of the dynamics of linear and non-linear systems. ·         To introduce the concepts of deterministic and stochastic behaviour and their relevance to chemical engineering. ·         To provide access to basic tools such as numerical integration of ODEs and Monte Carlo techniques. ·         Use of both symbolic and numerical computational tools via Maple and Excel.

Upon successful completion of the module you will be able to: ·         Identify flows and maps, linear and non-linear ·         Describe the response of linear systems to simple periodic inputs ·         Give examples of chaotic systems and the non-linearity driving them ·         Describe and use simple techniques to characterise complicated dynamics, such as bifurcation diagrams, Poincare sections and attractor reconstruction techniques. ·         Distinguish chaotic from purely random systems ·         Describe fractals and give appropriate examples ·         Describe and use simple methods to determine fractal dimension ·         Describe and give examples of the mathematical concept of complexity ·         Calculate the mean, variance, skewness and kurtosis of a list of numbers ·         Define and give examples of common probability density functions ·         Calculate a list of numbers that follow a given pdf ·         Set up a Monte Carlo simulation for a simple system ·         Use Maple to manipulate simple equations, solve them numerically and plot the results. ·         Use Excel for a variety of simple numerical and statistical tasks.

Introduction to linear systems ·         Solutions in the time domain An example of linear feedback Introduction to Maple/MathCad Numerical solutions Graphing resulting dynamics Mappings: the logistic map Excel vs Maple/MathCad example Chaos and period doubling routes thereto The bifurcation diagram The Poincare section Reconstructing the attractor Chaos from a simple ODE system What do we get when we reconstruct the attractor? Fractals and fractal dimensions What are fractals? How do we characterise them? Box counting measures of fractal dimension Structure of attractors in chaotic dynamics Complexity: on the edge of chaos The Game of Life Multi-cellular organisms and self-assembly Cellular automata A model of diffusion Statistics and Monte Carlo Simulation Introduction to probability density functions Generating numbers to fit a required distribution (Box Muller and Acceptance/Rejection Tests) Monte Carlo simulation Example of MC: a simple waste water bioreactor and the inspectors from the EA

Essential Reading: None Required Reading: ·         Hilborn RC, ‘Chaos and Nonlinear Dynamics’, OUP, Oxford, 1994. ·         Peitgen HO, Jurgens H and Saupe D, Chaos and Fractals, Springer-Verlag, New York, 1992.  ·         Scott SK, Chemical Chaos, Clarendon Press, Oxford, 1994 Recommended Reading: None

30th September 2010