PX391 Nonlinearity, chaos and complexity

Weighting: 7.5 CATS

The module introduces non-linear phenomena in science. Examples from physics, chemistry and biology are discussed (little previous knowledge of these subjects will be assumed).

A discussion of phase transitions and the elements of bifurcation theory is followed by the theory of first and second order non-linear differential equations. Such phenomena as simple attractors (limit cycles) are discussed. It is shown how non-linear systems can ‘self-organize’ to produce structures which have interesting time and space dependences. The main ideas from the theory of chaos will are introduced using one-dimensional difference equations as working examples.

Aims:
To introduce non-linearity and its treatment in scientific modelling.

Objectives:
At the end of this module you should:

• Be able to obtain basic qualitative features of the solutions of first and second order non-linear ordinary differential equations
• Be aware that simple, but non-linear, equations can describe complicated (chaotic) behaviour and know how to analyse this behaviour
• Be familiar with the concepts for emergent behaviour in complex systems. (computer algorithms).

Syllabus:

1. General introduction to Non-Linear Phenomena and universality.
2. Landau theory of phase transitions, order parameters. Bifurcation diagrams first and second order phase transitions.
3. First order non-linear differential equations. Fixed points and linear stability analysis. Global stability (1D phase plane).
4. Second order non-linear differential equations. Phase plane analysis and classification of fixed points. Limit cycles (Attractor).
5. Difference equations and maps. The tent map and global chaos, Lyapunov exponents. The logistic map, fixed points and bifurcation sequence to chaos. Feigenbaum universality.
6. Self organisation, and emergent behaviour many degree of freedom systems. Examples by computer: avalanche and forest fire models, preferential attachment, flocking, segregation. Concept of few order parameters, critical behaviour, phase transitions and scaling.

Commitment: 15 Lectures

Assessment: 1.5 hour examination