This course provides a mathematical introduction to fractal geometry and nonlinear dynamics, with applications to 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, the contraction mapping theorem, the collage theorem, orbits of points, local behaviour of transformations, code space and the shift transformation, Julia sets and the Mandelbrot set, superfractals, deterministic, Markov chain, and 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, crises, Poincare sections, the relation of fractal structures to simple nonlinear systems.

Honours pathway option (HPO)

In the HPO option we will expand on the theoretical aspects of the underlying concepts. Alterative assessment in the assignments and exam will be used to assess these theoretical aspects.

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. (HPO only) Be able to prove the main theorems that underline projective IFS theory.

Assessment may be based on:

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

12 units of A courses in Mathematics, including MATH1003 or MATH1013 or MATH1115; it is assumed that students will have some knowledge of differential equations and several variable calculus.

MATH2062