Fractals and Chaos: An Illustrated Course

摘要: INTRODUCTION Introduction A matter of fractals Deterministic chaos Chapter summary and further reading REGULAR FRACTALS AND SELF-SIMILARITY The Cantor set Non-fractal dimensions: the Euclidean topological dimension similarity Koch curve quadratic island Curves in plane with exceeding 2 Sierpinski gasket carpet Menger Sponge Revision questions tasks RANDOM Randomizing Fractal boundaries box counting Hausdorff structured walk technique divider perimeter-area relationship FRACTIONAL BROWNIAN MOTION Regular Brownian motion Fractional motion: time traces surfaces spatial trajectories Diffusion limited aggregation color power noise ITERATIVE FEEDBACK PROCESSES CHAOS Population growth Verhulst model logistic map effect variation control parameter General form iterated solutions Graphical iteration Bifurcation, stability Feigenbaum number two dimensional map: Henon Iterations complex plane: Julia sets Mandelbrot CHAOTIC OSCILLATIONS simple nonlinear mechanical oscillator: Duffing oscillator Chaos weather: Lorenz Rossler systems Phase space, attractor Spatially extended systems: coupled oscillators fluids Mathematical routes to turbulence CHARACTERIZING Preliminary characterization: visual inspection frequency spectra Characterizing chaos: Lyapunov exponents estimates Attractor reconstruction embedding for Regions behavior on characterization limitations task APPENDIX 1: Computer Program Equations 2: Illustrative Papers 3: Experimental SOLUTIONS REFERENCES