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 R¨osslerattractor 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 H´enonmap as a simplified model of the Poincar´esection of strange attractors Properties of the H´enonmap 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 (p^atefeuill) Resulting puff pastry (p^atefeuillet´ee) 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): 8 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 Poincar´esection dynamics in the R¨osslerattractor Credit: Timothy Jones, http://www.physics.drexel.edu/~tim/ D. Kartofelev YFX1520 11 / 36 Geometry of the R¨osslerattractor 1. FH D. Kartofelev YFX1520 12 / 36 EA in i i i FH Geometry of the R¨osslerattractor 2. EA D. Kartofelev YFX1520 13 / 36 in i i i FH EA Geometry of the R¨osslerattractor 3. D. Kartofelev YFX1520 14 / 36 in i i i FH EA Geometry of the R¨osslerattractor 4. in i D. Kartofelev YFX1520 15 / 36 i i FH Geometry of the R¨osslerattractor EA 5. Do not forget about the uniqueness of solutions. D. Kartofelev YFX1520 16 / 36 in i i i FH FH EA FH EA FH Geometry of the R¨osslerattractor2 EA 1. 2. 3. EAFH in 4. 5. i 2See Mathematica .nb file uploaded to course webpage. D. Kartofelev YFX1520 17 / 36 i i EA in in i i i i i i in i i i in i i i FH EA Geometry of the R¨osslerattractor in i Number of layers in a real attractor i i 21 = 1. D. Kartofelev YFX1520 18 / 36 Cantor set and the Lorenz section Poincar´esection Lorenz section Internal infinite + structure of the Lorenz section, 21 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 1 4 42 43 jS j = 1; jS j = ; jS j = ; jS j = ;::: 0 1 3 2 3 3 3 4n jS j = ; where n 2 + (2) n 3 Z jS1j = lim jSnj = 1 (3) n!1 Distance between any two points is 1, 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, " is the box size and N is the number of boxes. lazy better best L N(") ∼ (6) " " = "1 " = "2 < "1 D. Kartofelev YFX1520 25 / 36 Box dimension, 2-D A is the area of the rectangle, " is the box size, and N is the number of covering boxes. A N(") ∼ (7) "2 " = "1 " = "2 < "1 The scaling law also holds for Euclidean D-dimensional subset regions R where D > 2 (the covering Euclidean cubes are D-dimensional). R N(") ∼ (8) "d D. Kartofelev YFX1520 26 / 36 Box dimension, power law R 1 N(") ∼ ) N(") ∼ ; (9) "d "d where R is the (unit) volume. Solving for the power d gives 1 N(") ∼ · "d; (10) "d d " N(") ∼ 1 ln(·); (11) d ln " + ln N(") ∼ 0; (12) 2 3 ln N(") since ln N(") d ∼ − ∼ 4ln(1=") = ln 1 − ln "5 ∼ : (13) ln " |{z} ln(1=") =0 D. Kartofelev YFX1520 27 / 36 Box dimension, power law 1 N(") ∼ "d Conclusion: the power of " carries information about the dimensionality of an object. Slope of the log-log plot ln N(") d = lim : (14) "!0 ln(1=") If limit " ! 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 Geometry of strange attractors Analysis of stretch, fold and re-inject process in strange attractors Local microstructure of the R¨osslerattractor 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 H´enonmap as a simplified model of the Poincar´esection of strange attractors Properties of the H´enonmap 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 H´enonmap, its significance? D. Kartofelev YFX1520 36 / 36.