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Lecture №14: and , , spectral characteristics of dynamical , 1-D complex valued maps, Mandelbrot and nonlinear dynamical systems, introduction to applications of fractal geometry and chaos

Dmitri Kartofelev, PhD

Tallinn University of , School of Science, Department of , Laboratory of Solid

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 and Julia sets, connection to nonlinear dynamical systems Generation of the Mandelbrot set and the corresponding Julia sets Multibrot sets Examples of fractal geometry in nature and applications Introduction to applications of fractals and chaos Fractal similarity and the coastline paradox Synchronisation

D. Kartofelev YFX1520 2 / 41 Definition of fractal1

Fractal Endless and complex geometric 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: , von Kock , , 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) shown with the red and a periodic solution of the Lorenz 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 ⇒ = 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  2 + zn+1 = zn + c, {z, c} ∈ C, n ∈ Z  z0 = 0 (2)  c ∈ M ⇐⇒ lim sup |zn+1| ≤ 2 n→∞

1.0

0.5

0.0 Im(c) -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 ∈ 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 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 ∈ R. D. Kartofelev YFX1520 8 / 41 1-D complex maps, non-trivial dynamics

Fixing the dynamics

D. Kartofelev YFX1520 9 / 41 Mandelbrot set, self-similar (video)

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D. Kartofelev YFX1520 10 / 41 Julia sets and dynamical systems The Julia set3 J is defined as follows  2 + zn+1 = zn + c, {z, c} ∈ C, n ∈ Z  c = const. = |c| ≤ 2 (8)  z0 ∈ J ⇐⇒ lim sup |zn+1| ≤ 2 n→∞

0.5

0.0 Im(z) -0.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Re(z) Figure: Filled 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 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

0 0 0 0 0 0 Im(c) Im(c) Im(c)

-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,

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

0 0 0 0 0 0 Im(c) Im(c) Im(c)

-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 , 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

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D. Kartofelev YFX1520 23 / 41 Fractal geometry and nature

D. Kartofelev YFX1520 24 / 41 Fractal geometry and nature ( graphics)

D. Kartofelev YFX1520 25 / 41 Fractal geometry and nature ()

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 . This results from the fractal-like properties of coastlines. The first recorded observation of this phenomenon was by and it was expanded by .

Read: B. Mandelbrot, “How long is the of Britain? Statistical self-similarity and fractional dimension,” Science, New , 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

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D. Kartofelev YFX1520 35 / 41 Synchronisation: fireflies

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D. Kartofelev YFX1520 36 / 41 Synchronisation: Millennium bridge (2000)

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D. Kartofelev YFX1520 37 / 41 Synchronisation

Aperiodicity of chaos → bifurcation/s → periodic solution 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). Explain the coastline paradox. Can a coastline be described with ? What determines spectral characteristics of dynamical systems? What is a 1-D complex valued map? What are the Mandelbrot set and Julia sets? Assuming z = x + iy, c = r + is, and z, c ∈ C, show that map in the form ( x = x2 − y2 + r, n+1 n n (12) yn+1 = 2xnyn + s, is the real counterpart of the Mandelbrot set.

D. Kartofelev YFX1520 40 / 41 Revision questions

Physical meaning of the Mandelbrot set? Physical meaning of the Julia sets? What is generalised Mandelbrot set also known as the Multibrot set? Name an example of self-similar phenomena in nature.

D. Kartofelev YFX1520 41 / 41