Fractal structures in nonlinear dynamics and prospective applications in economics

1. Fractal structures in nonlinear dynamics and prospective applications in economics Ricardo Viana Departamento de Física, Setor de Ciências Exatas & Pós-Graduação em Desenvolvimento Econômico, Setor de Ciências Sociais Aplicadas Universidade Federal do Paraná, Curitiba, Brasil

2. Outline • Fractals and self-similarity • The concept of dimension • Cantor sets and Koch curves • Chaos and strange attractors • Fractal basin boundaries • Riddled basins: extreme fractal sets • Consequences for models in economic dynamics

3. 1st part: Fractals

4. Fractals Geometrical objects with two basic characteristics • self-similarity • fractional dimension

5. The term fractal was coined in 1975bythepolishmathematician Benoit Mandelbrot • « Les objets fractals: forme, hasard, et dimension »2e édition (1e édition, 1975)Paris: Flammarion,1984

6. Self-similarity Fractals have the same aspect when observed in different scales (scale-invariance)

7. In Naturewehavemanyexamplesofself-similarity

8. Self-similarity is found in manydailysituations (e.g., advertisement)

9. Self-similarity in art Salvador Dali: “The face of war”, 1940 M. C. Escher: “Smaller and smaller”, 1956

10. Fractals exist both in Nature...

11. ...and in mathematical models

12. The concept of dimension • A point has dimension 0, • A straight line has dimension 1, • A curve has also dimension 1, • A plane surface has dimension 2, • The surface of a sphere has dimension 2, • The sphere itself has dimension 3, • Are there other possibilities?

13. Our strategy • To give meaning to a fractional dimension, such as 1.75, it is necessary an operational definition of dimension. • There are many definitions. • We will use the box-counting dimension proposed by the german mathematician Felix Hausdorff in1916.

14. Box-counting dimension • Wecoverthe figure whichwewant to knowthedimension • withidentical boxes ofsidelength r • In practicewe use a meshwithsidelength r N(r):minimumnumberof boxes ofsidelength r necessary to coverthe figure completely

15. Straight line segment • How does thenumberof boxes depend on their sidelength? • N(r) = 1/r (one-dimensional) • Thesmaller are the boxes, themore boxes are necessary to cover the straight line segment

16. A unit square • Two-dimensionalobject • N(1) = 1 • N(1/2) = 4 = 1 / (1/2)2 • N(1/4) = 16 = 1 / (1/2)4 • etc...etc.... • N(r) = (1/r)2

17. Box-counting dimension • for fractal objects, in general, therelationbetween • N(r) and1/r is a power-law N(r) = k (1/r)d whered = box-countingdimension, k = constant • Onapplyinglogarithms • log N(r) = log k + d log (1/r) • - Is theequationof a straightline in a log-logdiagramwithslope d

18. Formal definition • wehopethattheexpression N(r) = k (1/r)d improves as • thelengthofthe boxes r becomesincreasingly small • wehaveseenthat, for finite r, • log N(r) = log k + d log (1/r) • takingthelimit as theboxsidelength r goes to zero thebox- • counting dimension is defined as d = limr  0 log (N(r)) / log (1/r) Obs.: ifr goes to zero, 1/r goes to infinity, butwe assume that log(k)/log(1/r)  0

19. Sierpinski gasket • A fractal object created by the polish mathematician Waclaw Sierpinski em 1915 • It presents self-similarity • Its box-counting dimension is d  1.59

20. The Sierpinski gasket is constructed by a sequence of steps We start from a filled square and remove an “arrow-like” triangle as indicated The Sierpinski gasket results from an infinite number of such operations

21. Box-counting for the Sierpinski gasket • For each step n, let rn be the box sidelength rn = (1/2)n • The minimum number of boxes in each step is given by N(rn) = 3n

22. Box-counting dimension of the Sierpinski gasket • ifd goes to zero, thenn goes to infinity • d = limr n 0 log (N(rn)) / log (1/rn) • = limn   log (N(rn)) / log (1/rn) • = limn   log (3n) / log (2n) • = limn   n log (3) / n log (2) • = log (3) / log (2) • = 1.58996... slope ≈ 1,59

23. Sierpinski carpet d = log 8 / log 3 ≈ 1.8928

24. Menger sponge (1926) d= log 20 / log 3 ≈ 2.7268

25. Cantor set • A fractal set createdbythegermanmathematician Georg Cantor (1872) • It is alsoobtainedfromtheinfinitelimitof a sequenceofsteps • We start from a unitintervaland remove themiddlethird in eachstep

26. Box-counting for the Cantor set • At each step the box sidelength is given by rn = (1/3)n • We need N(rn) = 2n of such boxes

27. Box-counting dimension of the Cantor set d = limr n 0 log (N(rn)) / log (1/rn) = limn   log (N(rn)) / log (1/rn) = limn   log (2n) / log (3n) = limn   n log2 / n log3 = log2/ log3 = 0,67

28. The length of the Cantor set is zero • The total length L of the Cantor set is 1 – (sum of all the subtracted middle third intervals). Since in the nth step we remove N(rn)=2n intervals of length rn/3, the total length subtracted is • The total removed length, after an infinite number of steps, is the infinite sum (geometrical series) • The Cantor set cannot contain intervals of nonzero length. In other words, the Cantor set is a closed set (since it is the complement of a union of open sets) of zero Lebesgue measure.

29. Strange properties of the Cantor set • In each step we remove open intervals, such that the end points like 1/3 and 2/3 are not subtracted. In the further steps these endpoints are likely not removed, and they belong to the Cantor set even after infinite steps, since the subtracted intervals are always in the interior of the remaining intervals. • However, not only endpoints but also other points like ¼ and 3/10 belong to the Cantor set. For example, ¼ < 1/3 belongs to the “bottom” third of the first step and it is thus not removed. Since ¼ > 2/9 it is in the “top” third of the “bottom” third and it is not removed in the second step, and so on, alternating between “top” and “bottom” thirds in successive steps.

30. What is the Cantor set made of? • There are infinite points in the Cantor set which are not endpoints of removed intervals. • It can be proved that the set of numbers belonging to the Cantor set may be represented in base 3 entirely with digits 0s and 2s (whereas any real number in [0,1] can be represented in base 3 with digits 0, 1 and 2). • Hence the Cantor set is uncountable, i.e. it contains as many points as the interval [0,1] from which it is taken, but it does not contain any interval! • Paradigmatic example of Cantor’s theory of transfinite numbers (raised a strong debate at that time)

31. Koch’s curve • Created by the swedish mathematician Helge von Koch em 1904 • It is a fractal object of box-counting dimension d  1,26 • Koch snowflake: a closed curve

32. Construction of the Koch’s curve • We start from a unitsegmentand divide it in threeparts • Onthemiddlethirdweconstructanequilateraltriangleand remove its base • We repeat the procedure for each resulting segment

33. Box-counting for the Koch’s curve • r1 = 1/3 = 1/31 N(r1) = 3 = 3.1 = 3.40 • r2 = 1/9 = 1/32 N(r2) = 12 = 3.4 = 3.41 • r3 = 1/27 = 1/33 N(r3) = 48 = 3.42 • rn = 1/3n N(rn) = 3.4n-1

34. Box-counting dimension of the Koch’s curve d = lim n   log (N(rn)) / log (1/rn) = lim n   log (3.4n-1) / log (3n) = lim n   [(n-1) log (4) + log(3)] / n log (3) = lim n   [n log (4) – log(4) + log(3)] / n log (3) = lim n   [n log (4)/n log(3)] + lim n   [-log(4) + log(3)] / n log (3) = log (4) / log (3) = 1,26186... slope= 1,26

35. Length of the Koch’s curve • We start from a singleunitlengthsegment: L0= 1 (it is bigger!) • Nowweapproximatewith 4 segmentsoflength1/3 each: L1= 4.(1/3)=4/3 • Next:16 segmentsoflength 1/9 each: L2 = 16.(1/9) = (4/3)2

36. TheKoch’s curve hasinfinitelength • At the n-th step we approximate with a polygonal with 4n segments of length 1/3n • Total length of the polygonal: Ln = (4/3)n • letting n go to infinity the total length is likewise infinite, since 4/3 > 1

37. Kochsnowflake • the total length is infinite • theareaenclosedbythesnowflake is finite (thereexists a finite R suchthatthesnowflake is contained in a circleofradius R) • is anexampleof a “fractal island”

38. Coastlines and fractal islands • Coastlines are typically fractal • They present self-similarity and fractionary dimension • They have infinite length even though containing a finite area

39. Paranagua Bay (satellite photo provided by “Centro de Estudos do Mar” – UFPR)

40. Measuring the length of a curve • Weapproximate a curve by a polygonalwith N segmentsof • equal length D (“yardstick”) • The total lengthofthepolygonal is L(D) = N D Example: lengthof a circleofradius 1 2  L N

41. For a smooth curve the process converges. What about a fractal coastline? (Britain, for example) • The length of the coastline depends on the scale D. • The length increases if D decreases • If D goes to zero, the length goes to infinity!

42. Lewis Fry Richardson (1961) • he found that • L(D) ~ Ds , where s < 0 • if the scale D goes to zero the length goes to infinity

43. B. Mandelbrot: “Howlong is thecoastofBritain?”Science156(1967) 636 • Interpreted Richardson’s results as a consequence of the fractality of the coastlines and border lines • The number of sides of the polygonal is given by N(D) ~ D-d, where d is the box-counting dimension of the coastline • The total length of the coastline is thus L(D) = D N(D) ~ D1-d • Richardson: N(D) ~ Ds hence d = 1 – s • Ex.: Britain s = -0.25 d = 1 + 0.25 = 1.25

44. Quarrel between Portugal-Spain • Measurementsoftheborderlinebetween Portugal andSpainpresentdifferencesof more than 20 % ! • Portugal: 1214 km • Spain: 987 km • Portugal used a scale D half the value used by Spain for measuring the borderline

45. ThegeometryoftheNature: Cézanne versusMandelbrot • Conventionalview: Nature is describedbyEuclideangeometrywithrandomperturbations • Alternativeview: Nature is intrinsecallydescribedby fractal geometry S. Botticelli: “Nascita de Venere” (1486)

46. Paul Cézanne “Everything in nature is modeled according to the sphere, the cone, and the cylinder. You have to learn to paint with reference to these simple shapes; then you can do anything." [excerpt of a 1904 letter to Emile Bernard]

47. Benoit Mandelbrot “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

48. 2nd part: Fractals and Dynamics

49. Dynamical systems • Deterministic equations giving the value of the variables of interest as a function of time • Continuous-time models: systems of differential equations (vector fields, flows) • Discrete-time models: systems of difference equations (maps)

50. Phase space description of dynamics • phase space: variables describing the dynamical system • each point represents a state of the system at a given time t • Initial condition: state at time t = 0 • trajectory in phase space: time evolution of the system (according to its governing equations)