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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