Feigenbaum Scaling

University essay from Linnéuniversitetet/Institutionen för matematik (MA)

Author: Janek Sendrowski; [2020]

Keywords: Feigenbaum; Feigenbaum constant; Feigenbaum point; superstable; self-similarity; fractals; final-state diagram; logistic map; period-doubling operator; linearized period-doubling operator; Cantor set; Sharkovsky sequence; Chebyshev interpolation; stable manifold; unstable manifold; orbits; fixed points; scaling factor; pitchfork bifurcation; cobweb plot; renormalization; iterates; one-hump map; unimodal map; bifurcation cascade; Feigenbaum universality; chaos theory; spectrum; generalized Feigenvalues; bifurcation points; superstable points; fixed function; universal function; eigenvalues; eigenvectors; eigenfunctions; universality conditions; Schwarzian derivative; Newton’s method; power method; Arnoldi’s method; attractive fixed point; repellent fixed point; Banach fixed-point theorem; collocation method; functional analysis; topology;

Abstract: In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.

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