Lecture 12: Fractal Microstructure of Strange Attractors, Fractal Geometry

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Lecture №12: Fractal microstructure of strange attractors, fractal geometry, fractal dimension, similarity and box dimensions, Cantor set, von Koch curve, 2-D maps, H´enonmap

Dmitri Kartofelev, PhD

Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of Solid Mechanics

D. Kartofelev YFX1520 1 / 36 Lecture outline

Geometry of strange attractors Analysis of stretch, fold and re-inject process in strange attractors Local microstructure of the Rösslerattractor trajectories Introduction to fractal geometry: non-integer dimensions The Cantor set and von Koch curve (von Koch star) Similarity dimension Box dimension Introduction to 2-D maps The Hénonmap as a simplified model of the Poincarésection of strange attractors Properties of the Hénonmap

D. Kartofelev YFX1520 2 / 36 Geometry of strange attractors

Microstructure of the Poincar´esection of R¨osslerattractor.

What happens inside the Lorenz section? What is the internal structure of the Lorenz section and how is it generated?

D. Kartofelev YFX1520 3 / 36 Stretching, folding and re-injecting

Cooking analogy — dough kneading

stretch fold and repeat

D. Kartofelev YFX1520 4 / 36 Stretching, folding and re-injecting

Cooking analogy — dough kneading

D. Kartofelev YFX1520 5 / 36 Stretching, folding and re-injecting

After repeated re-injection steps.

Resulting number of layers is 2n, where n is the number of iterations.

D. Kartofelev YFX1520 6 / 36 Stretching, folding and re-injecting

Puﬀ pastry dough (pˆatefeuill)

Resulting puff pastry (pâtefeuilletée)

Credit: CC BY-SA 3.0 Popo le Chien

D. Kartofelev YFX1520 7 / 36 Taﬀy pulling

Credit: Cubbies Fan #1, https://www.youtube.com/watch?v=XnndSkcjlBw

D. Kartofelev YFX1520 8 / 36 Lorenz section

Poincar´esection

Lorenz section

⇓ Internal structure of the Lorenz section

D. Kartofelev YFX1520 9 / 36 R¨osslerattractor

The R¨osslerattractor1 has the form (as mentioned in Lectures 9, 11):  x˙ = −y − x,  y˙ = x + ay, (1) z˙ = b + z(x − c).

Chaotic solution exists for a = 0.1, b = 0.1, c = 14.

1See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 10 / 36 Poincarésection dynamics in the Rösslerattractor

Credit: Timothy Jones, http://www.physics.drexel.edu/~tim/

D. Kartofelev YFX1520 11 / 36 Geometry of the R¨osslerattractor

D. Kartofelev YFX1520 12 / 36

in i i i

FH Geometry of the R¨osslerattractor

D. Kartofelev YFX1520 13 / 36

in i i i

Geometry of the R¨osslerattractor

D. Kartofelev YFX1520 14 / 36 in i i i

Geometry of the R¨osslerattractor

in i D. Kartofelev YFX1520 15 / 36 i i

Geometry of the R¨osslerattractor

Do not forget about the uniqueness of solutions.

D. Kartofelev YFX1520 16 / 36 in i i i

FH FH

FH EA 2 EA Geometry of the R¨osslerattractor

EA FH 1. 2. 3.

EA in 4. in 5. i 2See Mathematica .nb ﬁle uploaded to course webpage. i i D. Kartofelev YFX1520 17 / 36 i i i in in i i i i i i in i i i

Geometry of the R¨osslerattractor in

i Number of layers in a real attractor i i 2∞ = ∞.

D. Kartofelev YFX1520 18 / 36 Cantor set and the Lorenz section

Poincar´esection

Lorenz section

Internal inﬁnite ⇓ structure of the Lorenz section, 2∞ layers

D. Kartofelev YFX1520 19 / 36 Properties of the Cantor set

Cantor set3 shown to the third iteration (proto-fractal).

S0 = 1

2 S1 = 3

2 2 S2 = 3

2 3 S3 = 3

3See Mathematica .nb ﬁle uploaded to the course webpage. D. Kartofelev YFX1520 20 / 36 Properties of the von Koch curve von Koch curve4 shown to the fourth iteration.

S0 = 1

4 S1 = 3

4 2 S2 = 3

4 3 S3 = 3

4 4 S4 = 3

4See Mathematica .nb ﬁle uploaded to the course webpage. D. Kartofelev YFX1520 21 / 36 Properties of the von Koch curve

Selected properties of the von Koch curve: Measure (total length) of the curve is ∞ 4 42 43 |S | = 1, |S | = , |S | = , |S | = ,... 0 1 3 2 3 3 3 4n |S | = , where n ∈ + (2) n 3 Z

|S∞| = lim |Sn| = ∞ (3) n→∞ Distance between any two points is ∞, see (3) Self-similar — the curve is comprised of smaller copies (reduced by factor of 3) of itself Non-integer similarity and box dimension ln 4 d = ≈ 1.26 (4) ln 3

D. Kartofelev YFX1520 22 / 36 A way to think about dimensionality

Idea behind the similarity dimension.

Rule: m = r2

r = 3 m = 9

Rule: m = r3

Power of r carries information about the r = 3 m = 27 dimensionality.

D. Kartofelev YFX1520 23 / 36 Similarity dimension

If exists a scaling relation

ln m m = rd ⇒ d = , (5) ln r where r is the reduction factor of the self-similar substructures and m is the number of the self-similar substructures that comprise the original object. Then we have d — a similarity dimension.

Example: The Cantor set, a scaling law for iterates of the fractal exist and one can ﬁnd that ln m ln 2 d = = ≈ 0.63. ln r ln 3

D. Kartofelev YFX1520 24 / 36 Box dimension

Generalisation of similarity dimension. Below: L is the line length, ε is the box size and N is the number of boxes.

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better

best

L N(ε) ∼ (6) ε ε = ε1 ε = ε2 < ε1

D. Kartofelev YFX1520 25 / 36 Box dimension, 2-D

A is the area of the rectangle, ε is the box size, and N is the number of covering boxes.

A N(ε) ∼ (7) ε2

ε = ε1 ε = ε2 < ε1 The scaling law also holds for Euclidean D-dimensional subset regions R where D > 2 (the covering Euclidean cubes are D-dimensional). R N(ε) ∼ (8) εd

D. Kartofelev YFX1520 26 / 36 Box dimension, power law

R 1 N(ε) ∼ ⇒ N(ε) ∼ , (9) εd εd where R is the (unit) volume. Solving for the power d gives 1 N(ε) ∼ · εd, (10) εd

d ε N(ε) ∼ 1 ln(·), (11) d ln ε + ln N(ε) ∼ 0, (12)   ln N(ε) since ln N(ε) d ∼ − ∼ ln(1/ε) = ln 1 − ln ε ∼ . (13) ln ε |{z} ln(1/ε) =0

D. Kartofelev YFX1520 27 / 36 Box dimension, power law

1 N(ε) ∼ εd Conclusion: the power of ε carries information about the dimensionality of an object. Slope of the log-log plot

ln N(ε) d = lim . (14) ε→0 ln(1/ε)

If limit ε → 0 of d exist we have a box dimension (power law).

This approach allows also to estimate non-Euclidean non-integer dimensions of fractals and fractal-like objects.

D. Kartofelev YFX1520 28 / 36 2-D maps: H´enonmap

H´enoncreated a simple fractal attractor in order to study the general properties of strange attractors.

2-D H´enonmap5 has the following form: ( x = y + 1 − ax2 , n+1 n n (15) yn+1 = bxn, where a and b are the control parameters. Chaotic solution exists for a = 1.4, b = 0.3. Read: M. H´enon,“A two-dimensional mapping with a strange attractor,” Communications in Mathematical Physics, Vol. 50, No. 1 (1976), pp. 69–77.

5See Mathematica .nb ﬁle uploaded to the course webpage. D. Kartofelev YFX1520 29 / 36 H´enonmap

Modeling the stretch–fold–re-inject dynamics of the Poincar´esection of a 3-D attractor:

T 0 T 00 T 000

D. Kartofelev YFX1520 30 / 36 H´enonmap, local trapping region R

D. Kartofelev YFX1520 31 / 36 H´enonmap, orbit diagram6

Period doubling and chaotic dynamics are present for b = 0.3.

6See Mathematica .nb ﬁle uploaded to the course webpage. D. Kartofelev YFX1520 32 / 36 Gingerbreadman map or fractal7

4 Showing 4 · 10 iterates for initial conditions x0 = −0.3 and y0 = 0. 7See Mathematica .nb ﬁle uploaded to the course webpage. D. Kartofelev YFX1520 33 / 36 Conclusions

D. Kartofelev YFX1520 34 / 36 Revision questions

How is it possible for two trajectories with almost equal initial conditions to deviate exponentially and remain attracted to a strange attractor (remain in basin of attraction)? Explain fractal microstructure of strange attractors. Deﬁne fractal. Construct a simple fractal (general idea). What is self-similarity? What is scale-invariance? Are all fractals self-similar? What is fractal geometry? What is fractal dimension? What are similarity and box dimensions?

D. Kartofelev YFX1520 35 / 36 Revision questions

What is a power law? What is the Cantor set? What is the von Koch curve? What is a 2-D map? How to find fixed points of 2-D maps (period-1 point)? What is the Hénonmap, its significance?

D. Kartofelev YFX1520 36 / 36