3
1
Please contact admin.teaching.msi@anu.edu.au for consent to enrol in this course.
true
MATH6116
<p>This course provides a mathematical introduction to fractal geometry and nonlinear dynamics with focus on biological modelling and the geometry of real world images. </p><p>What do models for the structure of ferns and complicated behaviour of the weather have in common? </p><p>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. </p><p>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.</p><p>Topics to be covered include:</p><ul><li>Affine, projective and Möbius geometries</li><li>Iterated function systems</li><li>Metric spaces</li><li>Elementary topology</li><li>Contraction mapping theorem</li><li>Collage theorem</li><li>Orbits of points, sets and pictures</li><li>Local behaviour of transformations</li><li>Code space and the shift transformation</li><li>Julia sets and the Mandelbrot set</li><li>Superfractals</li><li>Escape-time algorithms for constructing fractal sets</li><li>Regular and chaotic behaviour in nonlinear systems</li><li>Characterization and measures of chaos</li><li>Stability and bifurcations</li><li>Routes to chaos</li><li>Feigenbaum's "universal" constant</li><li>Poincare sections</li><li>The relation of fractal structures to simple nonlinear dynamical systems</li></ul><span><p>Note: Graduate students attend joint classes with undergraduates but will be assessed separately.</p></span>
Bachelor degree; with first year Mathematics.
1
false
12645
<p>Assessment will be based on:</p><ul><li>Assignments (25%; LO 1-3)</li><li>Notebooks (25%; LO 1-3)</li><li>Exams (50%; LO 1-3)</li></ul>
1
1
<p>On satisfying the requirements of this course, students will have the knowledge and skills to:</p><p>1. Explain the basic concepts and have a practical familiarity with fractal geometry and chaotic dynamics.<br />2. Be able to formulate and analyze fractal geometric models in biology and computer graphics.<br />3. Have a deep understanding of affine IFS theory. </p>
2
Fractal Geometry and Chaotic Dynamics
6
6
6
First year Mathematics is required.
Fractals and Chaos
Band 2 NP
Mathematics
u8606170
2
2010