Lecture 14: Fractals and fractal geometry, coastline …dima/YFX1520/Loeng_14no_vid.pdfLecture 14:...

Lecture №14: Fractals and fractal geometry,coastline paradox, spectral characteristics of dynamical

systems, 1-D complex valued maps, Mandelbrot setand 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

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Lecture outline

Definition of fractal

Spectral characteristics of periodic, quasi-periodic and chaoticsystems

Second look at 1-D and 2-D maps, complex valued maps

Mandelbrot set and Julia sets, connection to nonlinear dynamicalsystems

Generation of Mandelbrot set and 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 paradoxSynchronisation

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Definition of fractal1

Fractal Endless and complex geometric shape with finestructure at arbitrarily small scales. In other wordsmagnification of tiny features of a fractal arereminiscent 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 / 39

Spectral characteristics of dynamical systems

0 50 100 150 200-20

-15

-10

-5

0

5

Frequency ω

SpectralPowerLevel

Power spectrum of x(t), periodic

0 5 10 15 20 25 30 35-30

-25

-20

-15

-10

-5

0

Frequency ω

SpectralPowerLevel

Power spectrum of sin(ω(t)), quasi-periodic

0 10 20 30 40

-15

-10

-5

0

Frequency ω

SpectralPowerLevel

Power spectrum of x(t), chaotic

Figure: (Top left) Sine wave shown with the red and a periodic solution ofthe Lorenz attractor shown with the blue. (Top right) Quasi-periodicsolution. (Bottom) Chaotic solution.

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Dynamics analysis methods

Discrete timeanalysis

Orbit diagram –long termbehaviour.

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Mandelbrot set and dynamical systems

Mandelbrot set2 M is defined as followszn+1 = z2n + c, {z, c} ∈ C, n ∈ Z+

z0 = 0

c ∈M ⇐⇒ lim supn→∞

|zn+1| ≤ 2

(1)

-2.0-1.5-1.0-0.5 0.0 0.5

-1.0

-0.5

0.0

0.5

1.0

Re(c)

Im(c)

2See Mathematica .nb file uploaded to course webpage.D. Kartofelev YFX1520 6 / 39

Mandelbrot set

The complex square map given in the form

zn+1 = z2n + c, (2)

where z = x+ iy, c = r + is, and z, c ∈ C, can be represented as a2-D real valued map. The component form of (2) is

xn+1 + iyn+1 = (xn + iyn)2 + r + is, (3)

xn+1 + iyn+1 = x2n + 2ixnyn − y2n + r + is, (4)

xn+1 + iyn+1 = x2n − y2n + r + i(2xnyn + s). (5)

Separation of the real and imaginary parts, and elimination of theimaginary unit i yields{

xn+1 = x2n − y2n + r

�iyn+1 = �i(2xnyn + s)⇒

{xn+1 = x2n − y2n + r,

yn+1 = 2xnyn + s,(6)

where x, y, r, s ∈ R.D. Kartofelev YFX1520 7 / 39

1-D complex maps, non-trivial dynamics

Fixing the polynomial System dynamics

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Mandelbrot set, self-similar properties (video)

No embedded video files in this pdf

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Julia sets and dynamical systems

Julia set3 J is defined as followszn+1 = z2n + c, {z, c} ∈ C, n ∈ Z+

c = const. = |c| ≤ 2

z0 ∈ J ⇐⇒ lim supn→∞

|zn+1| ≤ 2

(7)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-0.5

0.0

0.5

Re(z)

Im(z)

Figure: Filled Julia set where c = −1.1− 0.1i.

3See Mathematica .nb file uploaded to course webpage.D. Kartofelev YFX1520 10 / 39

Mandelbrot set and filled Julia sets

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Mandelbrot set and Julia sets

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Fractal dimension of the edge d = 2.0

Credit: CC BY-SA 3.0 Adam Majewski, Wolf Jung, J.C. Sprott

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Mandelbrot set and period-p orbits

Credit: CC BY-SA 3.0 Hoehue commonswikiD. Kartofelev YFX1520 14 / 39

Cardioid

Certain caustics can take the shape of cardioids.

Figure: Caustic in a coffee cup.

Credit: CC BY-SA 3.0 Gerard JanotD. Kartofelev YFX1520 15 / 39

Mandelbrot set, Buddhabrot

The image is rotatedin a clockwise directionby 90◦.

Credit: CC BY-SA 3.0 PurpyPupple, Evercat, MichaelPohoreski

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Multibrot sets4

zn+1 = zpn + c, z0 = 0 (8)

-2 -1 0 1 2-2

-1

0

1

2-2 -1 0 1 2

-2

-1

0

1

2

Re(c)

Im(c)

-2 -1 0 1 2-2

-1

0

1

2-2 -1 0 1 2

-2

-1

0

1

2

Re(c)

Im(c)

-2 -1 0 1 2-2

-1

0

1

2-2 -1 0 1 2

-2

-1

0

1

2

Re(c)

Im(c)

p = 3 p = 4 p = 5

4See Mathematica .nb file uploaded to course webpage.D. Kartofelev YFX1520 17 / 39

Multibrot sets

zn+1 = zpn + c, z0 = 0

-2 -1 0 1 2-2

-1

0

1

2-2 -1 0 1 2

-2

-1

0

1

2

Re(c)

Im(c)

-2 -1 0 1 2-2

-1

0

1

2-2 -1 0 1 2

-2

-1

0

1

2

Re(c)

Im(c)

-2 -1 0 1 2-2

-1

0

1

2-2 -1 0 1 2

-2

-1

0

1

2

Re(c)

Im(c)

p = 5.5 p = 10 p = 45

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Fractal geometry of nature

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Fractal geometry of nature

Yarlung Tsangpo River, China. Credit: NASA/GSFC/LaRC/JPL, MISR Team.

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Fractal geometry of nature

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Fractal geometry of nature

No embedded video files in this pdf

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Fractal geometry of nature

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Fractal geometry of nature (computer graphics)

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Fractal geometry of nature (computer graphics)

Brownian noise with d = 2.0 → topography.

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Fractal geometry and technology

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Coastline paradox

The coastline paradox5 is the counterintuitive observation that thecoastline of a landmass does not have a well-defined length. Thisresults from the fractal-like properties of coastlines. The firstrecorded observation of this phenomenon was by Lewis FryRichardson and it was expanded by Benoit Mandelbrot.

Read: B. Mandelbrot, “How long is the coast of Britain? Statisticalself-similarity and fractional dimension,” Science, New Series,156(3775), 1967, pp. 636–638.

5See Mathematica .nb file uploaded to course webpage.D. Kartofelev YFX1520 27 / 39

Coastline paradox

∆x = b ∆x = a

Slope of the resulting graph

d = − ln(L(∆x))

ln(∆x)=

ln(L(∆x))

ln(1/∆x), (9)

where L is the resulting measurement and ∆x is the measurementresolution (accuracy).

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Coastline paradox

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Coastline paradox, Estonia6

Resulting length = 809 km

6See Mathematica .nb file uploaded to course webpage.D. Kartofelev YFX1520 30 / 39

Coastline paradox, Estonia

Resulting length = 1473 km

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Coastline paradox, von Kock snowflake7

Measured length 3.16879 Measured length 4.91986

7See Mathematica .nb file uploaded to course webpage.D. Kartofelev YFX1520 32 / 39

Coastline paradox

Great Britain d = 1.25; Norway d = 1.52; Estonia∗ d = 1.2; SouthAfrican coast d = 1.0

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Synchronisation: metronomes

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Synchronisation: fireflies

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Synchronisation: Millennium bridge (2000)

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Synchronisation

Aperiodicity of chaos → bifurcation/s → periodic solution

Conceptual modelφ = µ− sinφ, (10)

where µ ≥ 0 is the system parameter and φ is the phase difference/s.

-3 -2 -1 1 2 3ϕ

-1.0

-0.5

0.5

1.0

1.5

2.0

ϕ′

-3 -2 -1 1 2 3ϕ

-1.0

-0.5

0.5

1.0

1.5

2.0

ϕ′

-3 -2 -1 1 2 3ϕ

-1.0

-0.5

0.5

1.0

1.5

2.0

ϕ′

µ > 1 0 < µ < 1 µ = 0D. Kartofelev YFX1520 37 / 39

Conclusions

Definition of fractal

Spectral characteristics of periodic, quasi-periodic and chaoticsystems

Second look at 1-D and 2-D maps, complex valued maps

Mandelbrot set and Julia sets, connection to nonlinear dynamicalsystems

Generation of Mandelbrot set and 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 paradoxSynchronisation

D. Kartofelev YFX1520 38 / 39

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.

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