Module description
This module introduces a number of concepts which are fundamental in all branches of Physics. The representation of a data series in terms of complementary parameters (such as time and frequency, or distance and angle) is used to introduce Fourier theory and the relationship between a function and its Fourier transform, autocorrelation function and power spectrum.

The difference between linear and non-linear equations is demonstrated, including the generation of sum and difference frequencies in non-linear processing. The use of dimensionless numbers in solving non-linear equations is illustrated and the relationship between non-linear equations and chaos theory is explored through solutions of the difference equation.

Finally, modelling techniques are introduced and examples of applications to specific problems are presented.

Learning outcomes
After taking this module students should be able to:

• Understand the concept of Fourier analysis of signals
• Understand the concept of Convolution and Correlation
• Appreciate the role of Fourier analysis in a number of physical systems
• Develop simple models to approximate physical situations
• Understand the difference between the evolution of linear and non-linear systems
• Appreciate the significance of higher order terms and perturbations on the evolution of physical systems and models

Additional learning activities
Workshops to provide practice in the development of modelling and approximation techniques.

Outline syllabus
Fourier series
Fourier transforms
Convolution and correlations

Introduction to the concepts and philosophy of modelling
Development of models as a problem solving tool in physics.

Simple pendulum as a linear oscillator
The concept of phase space
Trajectories in phase space: periodic and non-periodic behaviour; deterministic chaos
Modelling the atmosphere and the Lorentz attractor
Non-linear electrical circuit
Logistic difference equation
Bifurcation and period doubling, the Feigenbaum number
Relationship between Chaos and Fractals; Concept of a non-integer dimension
Implications on non-linear behaviour in deterministic systems