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

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

1 st part: Fractals

Fractals Geometrical objects with two basic characteristics  self-similarity  fractional dimension

 The term fractal was coined in 1975 by the polish mathematician Benoit Mandelbrot  « Les objets fractals: forme, hasard, et dimension » 2e édition (1e édition, 1975) Paris: Flammarion,1984

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

In Nature we have many examples of self-similarity

Self-similarity is found in many daily situations (e.g., advertisement)

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

Fractals exist both in Nature...

...and in mathematical models

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?

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.

Box-counting dimension We cover the figure which we want to know the dimension with identical boxes of sidelength r In practice we use a mesh with sidelength r ): N(r): minimum number of boxes of sidelength r necessary to cover the figure completely

Straight line segment How does the number of boxes depend on their sidelength? N(r) = 1/r (one- dimensional) The smaller are the boxes, the more boxes are necessary to cover the straight line segment

A unit square  Two- dimensional object  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

Box-counting dimension - for fractal objects, in general, the relation between N(r) and 1/r is a power-law N(r) = k (1/r) d where d = box-counting dimension, k = constant - On applying logarithms log N(r) = log k + d log (1/r) - Is the equation of a straight line in a log-log diagram with slope d

Formal definition - we hope that the expression N(r) = k (1/r) d improves as the length of the boxes r becomes increasingly small - we have seen that, for finite r, log N(r) = log k + d log (1/r) - taking the limit as the box sidelength r goes to zero the box- counting dimension is defined as d = lim r  0 log (N(r)) / log (1/r) Obs.: if r goes to zero, 1/r goes to infinity, but we assume that log(k)/log(1/r)  0

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

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

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

Box-counting dimension of the Sierpinski gasket - if d goes to zero, then n goes to infinity d = lim r n  0 log (N(r n )) / log (1/r n ) = lim n   log (N(r n )) / log (1/r n ) = lim n   log (3 n ) / log (2 n ) = lim n   n log (3) / n log (2) = log (3) / log (2) = slope ≈ 1,59

Sierpinski carpet d = log 8 / log 3 ≈

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

Cantor set  A fractal set created by the german mathematician Georg Cantor (1872)  It is also obtained from the infinite limit of a sequence of steps  We start from a unit interval and remove the middle third in each step

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

Box-counting dimension of the Cantor set d = lim r n  0 log (N(r n )) / log (1/r n ) = lim n   log (N(r n )) / log (1/r n ) = lim n   log (2 n ) / log (3 n ) = lim n   n log 2 / n log 3 = log 2/ log 3 = 0,67

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(r n )=2 n intervals of length r n /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.

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 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.

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)

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

Construction of the Koch’s curve  We start from a unit segment and divide it in three parts  On the middle third we construct an equilateral triangle and remove its base  We repeat the procedure for each resulting segment

Box-counting for the Koch’s curve  r 1 = 1/3 = 1/3 1 N(r 1 ) = 3 = 3.1 =  r 2 = 1/9 = 1/3 2 N(r 2 ) = 12 = 3.4 =  r 3 = 1/27 = 1/3 3 N(r 3 ) = 48 =  r n = 1/3 n N(r n ) = 3.4 n-1

Box-counting dimension of the Koch’s curve d = lim n   log (N(r n )) / log (1/r n ) = lim n   log (3.4 n-1 ) / log (3 n ) = 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, slope = 1,26

Length of the Koch’s curve  We start from a single unit length segment: L 0 = 1 (it is bigger!)  Now we approximate with 4 segments of length 1/3 each: L 1 = 4.(1/3)=4/3  Next:16 segments of length 1/9 each: L 2 = 16.(1/9) = (4/3) 2

The Koch’s curve has infinite length  At the n-th step we approximate with a polygonal with 4 n segments of length 1/3 n  Total length of the polygonal: L n = (4/3) n  letting n go to infinity the total length is likewise infinite, since 4/3 > 1

Koch snowflake the total length is infinite the area enclosed by the snowflake is finite (there exists a finite R such that the snowflake is contained in a circle of radius R) is an example of a “fractal island”

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

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

Measuring the length of a curve We approximate a curve by a polygonal with N segments of equal length D (“yardstick”) The total length of the polygonal is L(D) = N D Example: length of a circle of radius 1 2  N L

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!

Lewis Fry Richardson (1961) he found that L(D) ~ D s, where s < 0 if the scale D goes to zero the length goes to infinity s Coastline of Norway West coast of Britain Land frontier Germany Land frontier Portugal Australian coastline South African coastline Any smooth curve0

B. Mandelbrot: “How long is the coast of Britain?” Science 156 (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) ~ D 1-d  Richardson: N(D) ~ D s hence d = 1 – s  Ex.: Britain s = d = = 1.25

Quarrel between Portugal-Spain  Measurements of the borderline between Portugal and Spain present differences of more than 20 % !  Portugal: 1214 km  Spain: 987 km  Portugal used a scale D half the value used by Spain for measuring the borderline

The geometry of the Nature: Cézanne versus Mandelbrot  Conventional view: Nature is described by Euclidean geometry with random perturbations  Alternative view: Nature is intrinsecally described by fractal geometry S. Botticelli: “Nascita de Venere” (1486)

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]

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.”

2 nd part: Fractals and Dynamics

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)

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)

Chaotic behavior in a nutshell  Irregular (but deterministic!) fluctuations  Absence of periodicity  Sensitivity to initial conditions: positive Lyapunov exponent Logistic map x → r x (1 - x)

Attractors in phase space  Subsets of the phase space to which converge (for large times) the trajectories stemming from initial conditions  The set of initial conditions converging to a given attractor is its basin of attraction

Box-counting dimensions of attractors  d = 0: stable equilibrium point  d = 1: closed curve (limit cycle)  fractional d: “strange” attractor  Chaotic attractors are typically fractal (but not always!)

Stephen Smale “horseshoe” (1967)  It is the building block of chaotic attractors  Results from an infinite sequence of smooth stretch and fold operations  It is a non-attracting invariant set with fractal geometry

Properties of the Smale horseshoe It is an invariant Cantor- like set and contains: 1. a countable set of periodic orbits 2. an uncountable set of bounded non-periodic orbits 3. a dense orbit

Smale = Cantor times Cantor  Cantor dust: Cartesian product of two Cantor sets  The Smale horseshoe has the topology of a Cantor dust  Geometry and dynamics relation: fractality leads to self-similarity and vice-versa d = 2 (log 2/log 3) =

Strange attractors  Example: Hénon’s (discrete-time) map x n+1 = y n x n 2 y n+1 = 0.3 x n  Chaotic attractor with box-counting dimension d = 1.26  Self-similarity revealed by zooming

Strange attractors and Smale horseshoe  The Smale horseshoe map is the set of basic topological operations for constructing a strange attractor  This set consists of stretching (which gives sensitivity to initial conditions) and folding (which gives the attraction).

Example: an inflation-unemployment nonlinear model  Discrete-time model for inflation ( π n ) and unemployment rate (u n ) at time n = 0, 1, 2, …  Nonlinear Phillips curve: π n – π n e = a – d e -u n with a = -2.5 and d = 20 [A. S. Soliman: Chaos, Solitons & Fractals 7 (1996) 247]  Adaptive expectations: π n+1 e = π n e + c( π n – π n e ), where 0 < c < 1 (speed of adjustment)  Nominal monetary expansion rate is exogenous (m) u n+1 = u n – b(m – π n ), where b > 0: elasticity of unemployment with respect to monetary expansion

Equilibrium points (stability analysis)  Expected inflation rate: π e * = m  Unemployment rate (NAIRU): u* = log(d/|a|)  Equilibrium is asymptotically stable if b < b 1 = 4/(2 - c)|d|  Stable node (exponential convergence to equilibrium) if b > b 2 = 4c/|d| < b 1  Stable focus (damped oscillations towards the equilibrium) if b < b 2  We fix c = 0.75 and m = 2.0  Equilibrium point: u* = 2.079, π e * = 2.0

Bifurcation diagram  Tunable parameter: b  b < b 1 = 1.28: stable equilibrium point  b = b 1 : period-doubling bifurcation leads to a stable period-2 orbit  Period-doubling cascade leading to chaos (strange attractor) after b ∞ ≈ 1.38  Strange attractor disappears after b CR ≈ 1.49

Basin boundaries  When the system has more than one attractor, their correspondent basins are separated by a basin boundary  An initial condition is always determined up to a given uncertainty ε  If the uncertainty disk intercepts the basin boundary the initial condition is uncertain

Smooth and fractal basin boundaries smoothfractal

Fractal basin boundaries  Uncertain fraction f( ε): fraction of initial conditions with uncertainty ε  When the basin boundary is smooth the uncertain fraction scales linearly with the uncertainty f( ε) ~ ε  When the basin boundary is fractal the scaling is a power-law: f( ε ) ~ ε α where α = d – D, with d = box-counting dimension of the basin boundary and D = dimension of the phase space

Final-state sensitivity  Fractal basin boundaries: 0 < α < 1  Even a large improvement in the accuracy with which the initial conditions are determined do not imply in a proportional reduction of the uncertain fraction

Fractal basin boundaries in the inflation-unemployment model  b = 1.47: attractor is chaotic (white points)  Dark points: basin of the chaotic attractor  Gray points: basin of another attractor at infinity  Fractal boundary: α ≈ 0.7, d = 2 - α ≈1.4

Final-state sensitivity in the inflation- unemployment model  Fractal basin boundary with exponent α ≈ 0.7  If we improve the determination of initial condition such that the uncertainty is cut by half (ε → ε’= ε/2, or 50 %), the uncertain fraction will be reduced to f’ ~ (ε’) 0.7 = (ε/2) 0.7 ≈ f  which is a proportional decrease of only (f-f’)/f = 1 – = x 100 % = 38.5 %

Riddled basins: extreme fractal sets  A dynamical system is said to present riddled basins when it has a chaotic attractor A whose basin of attraction is riddled with holes belonging to the basin of another (non necessarily chaotic) attractor C  Every point in the basin of attractor A has pieces of the basin of attractor C arbitrarily nearby

Consequences of riddling  For riddled basins: α = 0  Uncertain fraction f( ε ) ~ ε 0 = 1: does not depend on the uncertainty radius  No improvement in the initial condition accuracy may decrease the uncertain fraction of initial conditions  No matter how small is the uncertainty with which an initial condition is determined, the asymptotic state of the system remains virtually unpredictable

Riddled basins in a simple two- commodity price model  pn: price of commodity 1 at discrete time n  qn: price of commodity 2 (e.g. corn-hog cycle)  Suppose pn undergoing a chaotic evolution pn+1 = f(pn) [e.g. logistic or tent map]: independent of qn  Evolution of price of commodity 2 is influenced by price of commodity 1: qn+1=g(qn,pn) where g is an odd function of q: g(-q)=-g(q)  q = 0 is an invariant manifold  q infinite: a second attractor (unbounded growth)  g(q,p) = r e-b(p-p*) q + q3 + higher odd powers of q  r: bifurcation parameter, b: convergence parameter

Characterization of riddled basins Consider the line segment at y = y 0. Riddling implies that the line segment is intercepted by pieces of the basins of both attractors, no matter how small y 0 is.

Conclusions