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

1st 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 assumethat 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 asindicated

The Sierpinski gasketresults from an infinite number of suchoperations

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

Box-counting dimension of the Sierpinski gasket

- if d goes to zero, then n goes to infinity

d = lim r n 0 log (N(rn)) / log (1/rn) = lim n log (N(rn)) / log (1/rn) = lim n log (3n) / log (2n) = lim n n log (3) / n log (2) = log (3) / log (2) = 1.58996...

slope ≈ 1,59

Sierpinski carpet

d = log 8 / log 3 ≈ 1.8928

Menger sponge (1926)

d= log 20 / log 3 ≈ 2.7268

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

rn = (1/3)n We need N(rn) = 2n of such boxes

Box-counting dimension of the Cantor set

d = lim r n 0 log (N(rn)) / log (1/rn) = lim n log (N(rn)) / log (1/rn) = lim n log (2n) / log (3n) = 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(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.

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.

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

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

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

Length of the Koch’s curve

We start from a single unit length segment: L0= 1 (it is bigger!)

Now we approximate with 4 segments of length 1/3 each: L1= 4.(1/3)=4/3

Next:16 segments of length 1/9 each: L2 = 16.(1/9) = (4/3)2

The Koch’s curve has infinite length

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

Koch snowflakeKoch 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) ~ Ds , where s < 0• if the scale D goes to zero the length goes to infinity

s

Coastline of Norway

- 0.52

West coast of Britain

- 0.25

Land frontier Germany

- 0.15

Land frontier Portugal

- 0.14

Australian coastline

- 0.13

South African coastline

- 0.02

Any smooth curve 0

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) ~ D1-d

Richardson: N(D) ~ Ds hence d = 1 – s Ex.: Britain s = -0.25 d = 1 + 0.25 = 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.”

2nd 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 mapx → 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) = 1.2618

Strange attractors

Example: Hénon’s (discrete-time) map

xn+1 = yn + 1 - 1.4 xn2

yn+1 = 0.3 xn 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 (πn) and

unemployment rate (un) at time n = 0, 1, 2, … Nonlinear Phillips curve: πn – πn

e = a – d e-un with

a = -2.5 and d = 20 [A. S. Soliman: Chaos, Solitons & Fractals 7 (1996) 247]

Adaptive expectations: πn+1e = πn

e + c(πn – πne),

where 0 < c < 1 (speed of adjustment) Nominal monetary expansion rate is exogenous

(m) un+1 = un – b(m – πn), where b > 0: elasticity of

unemployment with respect to monetary expansion

Equilibrium points (stability analysis) Expected inflation rate: πe* = m Unemployment rate (NAIRU): u* = log(d/|a|) Equilibrium is asymptotically stable if

b < b1 = 4/(2 - c)|d| Stable node (exponential convergence to

equilibrium) if b > b2 = 4c/|d| < b1 Stable focus (damped oscillations towards

the equilibrium) if b < b2

We fix c = 0.75 and m = 2.0 Equilibrium point: u* = 2.079, πe* = 2.0

Bifurcation diagram

Tunable parameter: b b < b1 = 1.28: stable

equilibrium point b = b1: 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 bCR ≈ 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

smooth fractal

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) ~ εα 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 ≈ 0.615 f which is a proportional decrease of only

(f-f’)/f = 1 – 0.615 = 0.385 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) ~ ε0 = 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 basinsConsider the line segment at y = y0. Riddling implies that the line segment is intercepted by pieces of the basins of both attractors, no matter how small y0 is.

Conclusions