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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
D. Kartofelev YFX1520 1 / 39
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
D. Kartofelev YFX1520 2 / 39
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.
D. Kartofelev YFX1520 4 / 39
Dynamics analysis methods
Discrete timeanalysis
Orbit diagram –long termbehaviour.
D. Kartofelev YFX1520 5 / 39
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
D. Kartofelev YFX1520 8 / 39
Mandelbrot set, self-similar properties (video)
No embedded video files in this pdf
D. Kartofelev YFX1520 9 / 39
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
D. Kartofelev YFX1520 11 / 39
Mandelbrot set and Julia sets
D. Kartofelev YFX1520 12 / 39
Fractal dimension of the edge d = 2.0
Credit: CC BY-SA 3.0 Adam Majewski, Wolf Jung, J.C. Sprott
D. Kartofelev YFX1520 13 / 39
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
D. Kartofelev YFX1520 16 / 39
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
D. Kartofelev YFX1520 18 / 39
Fractal geometry of nature
D. Kartofelev YFX1520 19 / 39
Fractal geometry of nature
Yarlung Tsangpo River, China. Credit: NASA/GSFC/LaRC/JPL, MISR Team.
D. Kartofelev YFX1520 20 / 39
Fractal geometry of nature
D. Kartofelev YFX1520 21 / 39
Fractal geometry of nature
No embedded video files in this pdf
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Fractal geometry of nature
D. Kartofelev YFX1520 23 / 39
Fractal geometry of nature (computer graphics)
D. Kartofelev YFX1520 24 / 39
Fractal geometry of nature (computer graphics)
Brownian noise with d = 2.0 → topography.
D. Kartofelev YFX1520 25 / 39
Fractal geometry and technology
D. Kartofelev YFX1520 26 / 39
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
D. Kartofelev YFX1520 29 / 39
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
D. Kartofelev YFX1520 31 / 39
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
D. Kartofelev YFX1520 33 / 39
Synchronisation: metronomes
No embedded video files in this pdf
D. Kartofelev YFX1520 34 / 39
Synchronisation: fireflies
No embedded video files in this pdf
D. Kartofelev YFX1520 35 / 39
Synchronisation: Millennium bridge (2000)
No embedded video files in this pdf
D. Kartofelev YFX1520 36 / 39
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.
D. Kartofelev YFX1520 39 / 39