# PX391 Nonlinearity, chaos and complexity

##### Lecturer: Sandra Chapman

##### 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:**

- General introduction to Non-Linear Phenomena and universality.
- Landau theory of phase transitions, order parameters. Bifurcation diagrams first and second order phase transitions.
- First order non-linear differential equations. Fixed points and linear stability analysis. Global stability (1D phase plane).
- Second order non-linear differential equations. Phase plane analysis and classification of fixed points. Limit cycles (Attractor).
- Difference equations and maps. The tent map and global chaos, Lyapunov exponents. The logistic map, fixed points and bifurcation sequence to chaos. Feigenbaum universality.
- 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

This module has a home page.

**Recommended Text: **G Rowlands, *Non-Linear Phenomena in Science and Engineering*, Ellis Horwood