Lecture 15: Fractals and fractal geometry, coastline paradox, spectral characteristics of dynamical systems, 1-D complex valued maps, Mandelbrot set and nonlinear dynamical systems, introduction to application of fractal geometry and chaos Dmitri Kartofelev, PhD Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of Solid Mechanics Week 15 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 1 / 34
Lecture outline (discussed during the lecture) Fractal similarity dimension and the coastline paradox Definition of fractal Spectral characteristics of periodic, quasi-periodic and chaotic systems Second look at 1-D and 2-D maps, complex valued maps Mandelbrot set and Julia sets, connection to nonlinear dynamical systems Generation of Mandelbrot set and corresponding Julia sets Buddhabrot Multibrot sets Examples of fractal geometry in nature and applications Introduction to application of fractals and chaos Synchronisation D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 2 / 34
Coastline paradox Example of fractal geometry in practical applications: The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded by Benoit Mandelbrot. Read: B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, New Series, 156(3775), 1967, pp. 636 638. See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 3 / 34
Coastline paradox x = b x = a Slope of the log-log plot d = ln(l( x)) ln( x) = ln(l( x)) ln(1/ x), (1) where L is resulting measurement and x is measurement resolution (accuracy). D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 4 / 34
Coastline paradox D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 5 / 34
Coastline paradox, Estonia Resulting length = 809 km See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 6 / 34
Coastline paradox, Estonia Resulting length = 1473 km See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 7 / 34
Coastline paradox, Kock snowflake Measured length 3.16879 Measured length 4.91986 See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 8 / 34
Coastline paradox Great Britain d = 1.25; Norway d = 1.52; Estonia d = 1.2; South African coast d = 1.0 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 9 / 34
Definition of fractal Fractal Endless and complex geometric shape with fine structure at arbitrarily small scales. In other words magnification of tiny features of fractal are reminiscent of the whole. Similarity can be exact (invariant), more often it is approximate or statistical. Examples: Cantor set, von Kock curve, Hilbert curve, L-systems, etc. See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 10 / 34
Spectral characteristics of dynamical systems Spectral Power Level 5 0-5 -10-15 Power spectrum of x(t), periodic -20 0 50 100 150 200 Frequency ω 0 Spectral Power Level Power spectrum of sin(ω(t)), quasi-periodic 0-5 -10-15 -20-25 Power spectrum of x(t), chaotic -30 0 5 10 15 20 25 30 35 Frequency ω Spectral Power Level -5-10 -15 0 10 20 30 40 Frequency ω D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 11 / 34
Dynamics analysis methods Discrete time analysis Orbit diagram long term behaviour. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 12 / 34
1-D complex maps, non-trivial dynamics Fixing the polynomial System dynamics D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 13 / 34
Mandelbrot set and dynamical systems Mandelbrot set M is defined as follows z n+1 = zn 2 + c, {z, c} C, n Z + z 0 = 0 c M lim sup z n+1 2 n (2) 1.0 0.5 Im(c) 0.0-0.5-1.0-2.0-1.5-1.0-0.5 0.0 0.5 Re(c) See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 14 / 34
Mandelbrot set, self-similar properties (video) No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 15 / 34
Julia sets and dynamical systems Julia set J is defined as follows z n+1 = zn 2 + c, {z, c} C, n Z + c = const = c 2 z 0 J lim sup z n+1 2 n (3) 0.5 Im(z) 0.0-0.5-1.5-1.0-0.5 0.0 0.5 1.0 1.5 Re(z) Figure: Filled Julia set where c = 1.1 0.1i. See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 16 / 34
Mandelbrot set and filled Julia sets D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 17 / 34
Mandelbrot set and Julia sets D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 18 / 34
Fractal dimension of the edge D = 2.0 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 19 / 34
Mandelbrot set and period-p cycles D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 20 / 34
Mandelbrot set, Buddhabrot D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 21 / 34
Multibrot sets z n+1 = z p n + c, z 0 = 0 (4) -2-1 0 1 2 2 2-2 -1 0 1 2 2 2-2 -1 0 1 2 2 2 1 1 1 1 1 1 Im(c) 0 0 Im(c) 0 0 Im(c) 0 0-1 -1-1 -1-1 -1-2 -2-2 -1 0 1 2 Re(c) -2-2 -2-1 0 1 2 Re(c) -2-2 -2-1 0 1 2 Re(c) p = 3 p = 4 p = 5 See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 22 / 34
Multibrot sets z n+1 = z p n + c, z 0 = 0-2 -1 0 1 2 2 2-2 -1 0 1 2 2 2-2 -1 0 1 2 2 2 1 1 1 1 1 1 Im(c) 0 0 Im(c) 0 0 Im(c) 0 0-1 -1-1 -1-1 -1-2 -2-2 -1 0 1 2 Re(c) -2-2 -2-1 0 1 2 Re(c) -2-2 -2-1 0 1 2 Re(c) p = 5.5 p = 10 p = 45 See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 23 / 34
Fractal geometry of nature D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 24 / 34
Fractal geometry of nature D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 25 / 34
Fractal geometry of nature D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 26 / 34
Fractal geometry of nature (computer graphics) D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 27 / 34
Fractal geometry of nature (computer graphics) Brownian noise with d = 2.0 topography. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 28 / 34
Fractal geometry and technology D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 29 / 34
Synchronisation: metronomes No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 30 / 34
Synchronisation: fireflies No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 31 / 34
Synchronisation: Millennium bridge (2000) No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 32 / 34
Conclusions Fractal similarity dimension and the coastline paradox Definition of fractal Spectral characteristics of periodic, quasi-periodic and chaotic systems Second look at 1-D and 2-D maps, complex valued maps Mandelbrot set and Julia sets, connection to nonlinear dynamical systems Generation of Mandelbrot set and corresponding Julia sets Buddhabrot Multibrot sets Examples of fractal geometry in nature and applications Introduction to application of fractals and chaos Synchronisation D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 33 / 34
Revision questions Explain the coastline paradox. Can a coastline be described with Euclidean geometry? What determines spectral characteristics of dynamical systems? What is 1-D complex valued map? What are Mandelbrot set and Julia sets? Physical meaning of the Mandelbrot set? Physical meaning of Julia sets? What is the Multibrot set? Name an example of self-similar phenomena in nature. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 34 / 34