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Chaos and Fractals

Chaos and Fractals

Code	10830
Year	1
Semester	S1
ECTS Credits	12
Workload	TP(45H)
Scientific area	Mathematics
Entry requirements	.
Mode of delivery	Face-to-face.
Work placements	Non applicable.
Learning outcomes	This curricular unit aims to make an introduction to fundamental concepts on dynamical systems and fractal geometry. - To understand some basic concepts of dynamical systems associated with the iterative functions, such as the concept of fixed point, periodic point and set limit. - To take knowledge of some aspects of the dynamics of the quadratic family. - To take knowledge of some aspects of the limit sets of geometric constructions. - To take knowledge of some aspects of dynamical systems in the complex plane. - To take knowledge of some mathematical aspects associated with some of the best-known fractal sets. - To establish links between the contents and the curricula of Mathematics in the 3rd cycle of Basic Education and Secondary Education.
Syllabus	I - Discrete dynamical systems: composition of functions and iteration; orbits, fixed points and periodic points; limit sets. II - Dynamics of quadratic: the quadratic family, period doubling and bifurcations; symbolic dynamics; conjugation; period three and chaos. III - Self-similar sets: self-similarity in nature; Cantor set, Von Koch curve and Sierpinskii sponge; self-similar sets; sensibility to initial conditions; fractal sets; what is dimension?; fractal dimension. IV - Complex dynamic: quadratic transformations in the complex plane; Julia sets and Mandelbrot set; some geometric and dynamic aspects of Julia sets.
Main Bibliography	1) Devaney, R., An introduction to chaotic dynamical systems, Addison-Wesley Publishing Company, 1989. 2) Barnsley, Fractals Everywhere, Morgan Kaufmann Pub. 3) Falconer, Fractal Geometry – Mathematical Foundations And Applications, John-Wiley & Sons, 1990. 4) Mandelbrot, B., Objectos Fractais, Gradiva, 1998. 5) Robinson, R. C., An introduction to dynamical systems, Prentice Hall, 2004.
Language	Portuguese. Tutorial support is available in English.