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Fractals and Fractal Geometry, Coastline Paradox, Spectral Lecture №14: Fractals and fractal geometry, coastline paradox, spectral characteristics of dynamical systems, 1-D complex valued maps, Mandelbrot set and nonlinear dynamical systems, introduction to applications of fractal geometry and chaos Dmitri Kartofelev, PhD Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of Solid Mechanics D. Kartofelev YFX1520 1 / 41 Lecture outline 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 the Mandelbrot set and the corresponding Julia sets Buddhabrot Multibrot sets Examples of fractal geometry in nature and applications Introduction to applications of fractals and chaos Fractal similarity dimension and the coastline paradox Synchronisation D. Kartofelev YFX1520 2 / 41 Definition of fractal1 Fractal Endless and complex geometric shape with fine structure at arbitrarily small scales. In other words magnification of tiny features of a 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. 1See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 3 / 41 Spectral characteristics of dynamical systems Power spectrum of x(t), periodic Power spectrum of sin(ω(t)), quasi-periodic 0 5 -5 0 -10 -5 -15 -10 -20 -15 Spectral Power Level Spectral Power Level -25 -20 -30 0 50 100 150 200 0 5 10 15 20 25 30 35 Frequency ω Frequency ω Power spectrum of x(t), chaotic 0 -5 -10 Spectral Power Level -15 0 10 20 30 40 Frequency ω Figure: (Top left) Sine wave shown with the red and a periodic solution of the Lorenz attractor shown with the blue. (Top right) Quasi-periodic solution. (Bottom) Chaotic solution. D. Kartofelev YFX1520 4 / 41 Dynamics analysis methods ( x0 = f (x0 ; y0 ) n+1 1 n n ) r = f (r ) (1) 0 0 0 n+1 3 n yn+1 = f2(xn; yn) ~ 0 0 0 0 0 T Construction of the Poincar´emap P (~x ) = (f1(x ; y ); f2(x ; y )) (1). Mapping of the Poincar´esection points where r is the radial distance from the origin (“flat” attractor). D. Kartofelev YFX1520 5 / 41 Dynamics analysis methods ) Orbit diagram { long-term discrete-time behaviour analysis. D. Kartofelev YFX1520 6 / 41 Mandelbrot set and dynamical systems The Mandelbrot set2 M is defined as follows 8 2 + >zn+1 = zn + c; fz; cg 2 C; n 2 Z <> z0 = 0 (2) > :>c 2 M () lim sup jzn+1j ≤ 2 n!1 1.0 0.5 ) c ( 0.0 Im -0.5 -1.0 -2.0-1.5-1.0-0.5 0.0 0.5 Re(c) 2See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 7 / 41 Mandelbrot set The complex square map given in the form 2 zn+1 = zn + c; (3) where z = x + iy, c = r + is, and z; c 2 C, can be represented as a 2-D real valued map. The component form of (3) is 2 xn+1 + iyn+1 = (xn + iyn) + r + is; (4) 2 2 xn+1 + iyn+1 = xn + 2ixnyn − yn + r + is; (5) 2 2 xn+1 + iyn+1 = xn − yn + r + i(2xnyn + s): (6) Separation of the real and imaginary parts, and elimination of the imaginary unit i yields ( ( x = x2 − y2 + r x = x2 − y2 + r; n+1 n n ) n+1 n n (7) ¡iyn+1 = ¡i(2xnyn + s) yn+1 = 2xnyn + s; where x, y, r, s 2 R. D. Kartofelev YFX1520 8 / 41 1-D complex maps, non-trivial dynamics Fixing the polynomial System dynamics D. Kartofelev YFX1520 9 / 41 Mandelbrot set, self-similar properties (video) No embedded video files in this pdf D. Kartofelev YFX1520 10 / 41 Julia sets and dynamical systems The Julia set3 J is defined as follows 8 2 + >zn+1 = zn + c; fz; cg 2 C; n 2 Z <> c = const. = jcj ≤ 2 (8) > :>z0 2 J () lim sup jzn+1j ≤ 2 n!1 0.5 ) z ( 0.0 Im -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. 3See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 11 / 41 Mandelbrot set and filled Julia sets D. Kartofelev YFX1520 12 / 41 Mandelbrot set and Julia sets D. Kartofelev YFX1520 13 / 41 Fractal dimension of the edge d = 2:0 Credit: CC BY-SA 3.0 Adam Majewski, Wolf Jung, J.C. Sprott D. Kartofelev YFX1520 14 / 41 Mandelbrot set and period-p orbits Credit: CC BY-SA 3.0 Hoehue commonswiki D. Kartofelev YFX1520 15 / 41 Cardioid Certain caustics can take the shape of cardioid. Figure: Caustic in a coffee cup. Credit: CC BY-SA 3.0 G´erard Janot D. Kartofelev YFX1520 16 / 41 Mandelbrot set, Buddhabrot The image is rotated in a clockwise direction by 90◦. Credit: CC BY-SA 3.0 Purpy Pupple, Evercat, Michael Pohoreski D. Kartofelev YFX1520 17 / 41 Generalised Mandelbrot set, Multibrot set4 p zn+1 = zn + c; z0 = 0 (9) -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 2 2 2 2 2 2 1 1 1 1 1 1 ) ) ) c c c ( 0 0 ( 0 0 ( 0 0 Im Im Im -1 -1 -1 -1 -1 -1 -2 -2 -2 -2 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Re(c) Re(c) Re(c) p = 3, p = 4 p = 5 main bulb is nephroid 4See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 18 / 41 Generalised Mandelbrot set, Multibrot set p zn+1 = zn + c; z0 = 0 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 2 2 2 2 2 2 1 1 1 1 1 1 ) ) ) c c c ( 0 0 ( 0 0 ( 0 0 Im Im Im -1 -1 -1 -1 -1 -1 -2 -2 -2 -2 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Re(c) Re(c) Re(c) p = 5:5 p = 10 p = 45 D. Kartofelev YFX1520 19 / 41 Fractal geometry and nature D. Kartofelev YFX1520 20 / 41 Fractal geometry and nature Yarlung Tsangpo River, China. Credit: NASA/GSFC/LaRC/JPL, MISR Team. D. Kartofelev YFX1520 21 / 41 Fractal geometry and nature D. Kartofelev YFX1520 22 / 41 Fractal geometry and nature No embedded video files in this pdf D. Kartofelev YFX1520 23 / 41 Fractal geometry and nature D. Kartofelev YFX1520 24 / 41 Fractal geometry and nature (computer graphics) D. Kartofelev YFX1520 25 / 41 Fractal geometry and nature (computer graphics) Brownian noise with d = 2:0 ! topography. D. Kartofelev YFX1520 26 / 41 Fractal geometry and technology D. Kartofelev YFX1520 27 / 41 Coastline paradox The coastline paradox5 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. 5See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 28 / 41 Coastline paradox ∆x = b ∆x = a Slope of the resulting graph ln(L(∆x)) ln(L(∆x)) d = − = ; (10) ln(∆x) ln(1=∆x) where L is the resulting measurement and ∆x is the measurement resolution, i.e., length of the measuring stick. D. Kartofelev YFX1520 29 / 41 Coastline paradox D. Kartofelev YFX1520 30 / 41 Coastline paradox, Estonia6 Resulting length= 809 km 6See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 31 / 41 Coastline paradox, Estonia Resulting length= 1473 km D. Kartofelev YFX1520 32 / 41 Coastline paradox, von Kock snowflake7 Measured length 3.16879 Measured length 4.91986 7See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 33 / 41 Coastline paradox Great Britain d = 1:25; Norway d = 1:52; Estonia∗ d = 1:2; South African coast d = 1:0 D. Kartofelev YFX1520 34 / 41 Synchronisation: metronomes No embedded video files in this pdf D. Kartofelev YFX1520 35 / 41 Synchronisation: fireflies No embedded video files in this pdf D. Kartofelev YFX1520 36 / 41 Synchronisation: Millennium bridge (2000) No embedded video files in this pdf D. Kartofelev YFX1520 37 / 41 Synchronisation Aperiodicity of chaos ! bifurcation/s ! periodic solution Conceptual model φ_ = µ − sin φ, (11) where µ ≥ 0 is the system parameter and φ is the phase difference/s. ϕ′ ϕ′ ϕ′ 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 ϕ ϕ ϕ -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 µ > 1 0 < µ < 1 µ = 0 D. Kartofelev YFX1520 38 / 41 Conclusions 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 the Mandelbrot set and the corresponding Julia sets Buddhabrot Multibrot sets Examples of fractal geometry in nature and applications Introduction to applications of fractals and chaos Fractal similarity dimension and the coastline paradox Synchronisation D. Kartofelev YFX1520 39 / 41 Revision questions Define fractal (technical definition).
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