Introductory description

Many phenomena in nature can be modelled using non-linear mathematics, for example population dynamics, evolution, heartbeats, cell growth, animal locomotion, snowflakes as well as other phenomena which we cannot model with any degree of certainty, but the mathematics can help us to try and understand why certain things can happen, for example weather or apparent random properties of epidemics. The idea of this course is to introduce the techniques which mathematicians use to study, and to try to explain, the diversity and apparent unpredictability of nature. The idea is not to go through specific biological examples in any great detail, nor to rigorously prove all the techniques we use but rather to explain how the tools can be used, with some relevant examples thrown in for good measure. By the end of the course you should have enough background to be able to at least have a good guess at the techniques that should be tried when given a natural phenomena with apparent non-linear effects to model.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Evolutionary Game Theory: Hawk-Dove and Hawk-Dove-Bully-Retaliator games used to introduce simple Game Theory techniques. Evolutionary Stable Strategies, corresponding dynamical systems, fixed points and dynamics on symplices. Essentially how population dynamics can be explained through how individuals interact with each other.

Nonlinear Oscillations: Overview of linear oscillations (by mechanical and electrical examples), self excited oscillations (including Van Der Pol oscillator), Duffings Equation including forcing and non-forcing cases, vector fields, extended phase space, Poincare Maps, stability, Homoclinic Tangles (leading to chaos).

Epidemics: SEIR equations, discussion of contact rates and attractors for measles and chicken-pox, are such epidemics periodic or chaotic? Kermack-McKendrick model as a more analytical example.

Dynamical Systems: Discrete and continuous dynamical systems, fixed points and periodic points/orbits and stabilities. Examples include Hawk-Dove and Henon map. Invariant sets, attractors, Strange Attractors, structural stability.

Sensitive Dependence: Lyapunov Characteristic Exponents, Lyapunov Spectrum, chaotic attractors, practical method for computing LCE. Discussion on implications of chaos, and ways to ’control’ chaotic systems.

Bifurcations and Catastrophes: How small changes to parameters can lead to large (’catastrophic’) changes to a physical system. Euler Strut, cusp catastrophes, decision making (intelligent vs. non-intelligent), the Canonical Cusp Catastrophe, catastrophe set, singularities, the Spruce-Budworm and brief mention of the seven elementary catastrophes. Hopf bifurcation (introduced by wavy rolls in convection), linearisation and eigenvalues at a Hopf bifurcation, Hopf theorems (a Hopf bifurcation is a process which forms periodic orbits).

Coupled Oscillators: Fireflies as an example of coupled integrate and fire models, linearly coupled relaxation oscillators, introduction to patterns observed in symmetrically coupled oscillators through the symmetries of animal gaits. This year there will also be a discussion of how symmetry affects ’steady-state’solutions.

Fractals: A rather basic introduction to Fractals through the Middle Third Cantor Set, Koch Curve, Julia Sets, spaces of Fractals, chaotic repellors, iterated function systems, dimension (counting and Hausdorff dimensions).

Learning outcomes

By the end of the module, students should be able to:

• Be able to demonstrate how large changes to solutions can be caused by small changes to initial conditions or parameters.
• Show an understanding of how nonlinearity can explain many natural phenomena through mathematical modelling.
• Be able to analyse various nonlinear systems both analytically and numerically.

Indicative reading list

J.D. Murray, Mathematical Biology Vol I, Springer Verlag..
Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag.
P. Drazin, Nonlinear Systems, Cambridge.
M. Golubitsky, I. Stewart, S. Schaeffer, Singularities and Groups in Bifurcation Theory Vol II, Springer
Arrowsmith and Place, An Introduction to Dynamical Systems, Cambridge.

Research element

20% of the assessment is the beginning of the building of a model for a problem from the natural world to demonstrate an understanding of the principal aims of the module.

Interdisciplinary

This module relies on mathematical modelling of phenomena from biology, engineering and economics.

Subject specific skills

Students will acquire a basic understanding of many mathematical concepts which can be explored further in later modules. Including, but not limited to, dynamical systems, chaos, epidemiology, game theory, symmetry, catastrophes, fractal geometry. Students will learn techniques to model phenomena in the natural world and understand how the underlying mathematics can lead to unexpected results.

Transferable skills

Mastering modelling skills of a wide range of phenomena and how to apply mathematics to areas of study outside of mathematics. Problem solving skills and appreciation of how large scale phenomena can be affected by small changes to parameters and/or initial conditions.