This course provides a mathematical introduction to fractal geometry and nonlinear dynamics with focus on biological modelling and the geometry of real world images.

What do models for the structure of ferns and complicated behaviour of the weather have in common?

Both involve the iterative application of functions that map from a space to itself. Both can be treated from the classical geometrical point of view of Felix Klein. Invariants, such as fractal dimension, of important groups of transformations acting on two-dimensional spaces, pictures, and measures are explored.

Deep mathematical ideas are explained in an intuitive and practical manner.  Laboratory work includes projects related to digital imaging and biological modelling. A high point in the course is an introduction to fractal homeomorphisms: what they are and how to work with them in the laboratory.

Topics to be covered include:

• Affine, projective and Möbius geometries
• Iterated function systems
• Metric spaces
• Elementary topology
• Contraction mapping theorem
• Collage theorem
• Orbits of points, sets and pictures
• Local behaviour of transformations
• Code space and the shift transformation
• Julia sets and the Mandelbrot set
• Superfractals
• Escape-time algorithms for constructing fractal sets
• Regular and chaotic behaviour in nonlinear systems
• Characterization and measures of chaos
• Stability and bifurcations
• Routes to chaos
• Feigenbaum's "universal" constant
• Poincare sections
• The relation of fractal structures to simple nonlinear dynamical systems

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the basic concepts and have a practical familiarity with fractal geometry and chaotic dynamics.
2. Be able to formulate and analyze fractal geometric models in biology and computer graphics.
3. Have a deep understanding of affine IFS theory. 

Assessment will be based on:

• Assignments (25%; LO 1-3)
• Notebooks (25%; LO 1-3)
• Exams (50%; LO 1-3)