Dynamical Systems 4
Deterministic chaos, fractals
Ing. Jaroslav Jíra, CSc.
One-dimensional Maps
One-dimensional maps (sometimes called difference equations or iterated
maps or recursion relations) are mathematical systems that model a single
variable as it evolves over discrete steps in time.
The general form of such map is
xn1  f ( xn )
We already know an example of
linear map (bank account):
xn1  rxn
and we also know an example of
nonlinear map (logistic map):
xn1  rxn (1  xn )
In the following we will learn a new technique for solving these difference
equations. The technique is called Cobwebbing or Cobweb diagram.
Cobweb diagram
As an example we take our well known logistic equation (or logistic map), this
time in a shape y=rxn(1-xn) .
The basic step is drawing a graph of the function y=f(x) and a line
corresponding to y=x.
a) Then we choose on the x-axis initial
value, in our example x0=0.08 .
b) From this point we draw a vertical
line to the function f(x), obtaining
f(x0)=x1.
c) From this point we draw a horizontal
line to the line y=x and we are at the
point, which is right above the x1 value
on the x-axis.
d) From this point we again draw a
vertical line to the function f(x),
obtaining f(x1)=x2
We repeat the procedure until we finish
at the fixed point.
Examples of Cobweb diagrams for various values of the parameter r
For r=2.0 there is just one fixed
point at 0.5
For r=2.93 there is still one fixed
point at 0.6587, but the
converging is very slow
Examples of Cobweb diagrams for various values of the parameter r
For r=3.39 there are two cycling
values 0.45 and 0.84
For r=3.45 there are four cycling
values 0.846, 0.4495, 0.8537
and 0.4309
Examples of Cobweb diagrams for various values of the parameter r
For r=3.57 there are sixteen
various cycling values
For r=3.97 we are already in the
chaos area
Feigenbaum constants
Feigenbaum constants are two
constants, that are named after the
mathematician Michel Fiegenbaum
and they are related to bifurcation
diagrams.
These constants are universal for
any period doubling bifurcations.
They can also be observed in the
Mandelbrot set, for example.
If we try to zoom grayed area on
the bifurcation diagram of the
logistic equation, we obtain a
graph shown below.
Vertical blue lines mark values,
where bifurcations occur.
If we try to zoom again into the
grayed area, we obtain similar
pattern.
Feigenbaum constant delta
Evaluating bifurcating values numerically will
result in the following table, where n is order
number of bifurcation, Period is number of
cycling values after this bifurcation and rn is
corresponding bifurcating value. The last value
marked by the ∞ is called an accumulation point.
Behind this point begins chaos.
Ratio is evaluated
according to this
formula:
rn  rn 1
ratio 
rn 1  rn
For example
ratio 
3.44948974  3
 4.751446
3.54409036  3.44948974
The ratio limits to a value 4.669201609…, which is called
Feigenbaum constant delta
rn  rn1
  lim
 4.669201609
n  r
n 1  rn
Feigenbaum constant alpha
If we measure vertical distances between tines of successive forks related to the
value x=0.5 in the bifurcation diagram, and mark them a1, a2, a3 …, we can
observe another ratio, that converges to a constant value.
an
  lim
 2.502907875
n  a
n 1
The value 2.502907875 is called Feigenbaum constant alpha
An Attractor of Dynamical System
An attractor is a set towards which a dynamical system evolves over time. The
basic types ot attractors are:
1. Fixed point. The system evolves towards a single state and remains there. An
example is damped pendulum or a sphere at the bottom of a spheric bowl.
2. Periodic or quasiperiodic attractor. The system evolves towards a limit cycle. An
example is undamped pendulum or a planet orbiting around the Sun.
3. Chaotic attractor. The system is very sensitive to initial conditions and we are
not able to simply predict its behavior. Chaotic behavior does not mean random
behavior. We obtain always the same solution for the same initial conditions. An
example is the Lorenz attractor.
4. Strange attractor. The system is also very sensitive to initial conditions and we
are not able to simply predict its behavior, but in this case the system has the
same properties like fractals. In another words, the strange attractor represents a
fractal. An example is the Mandelbrot set.
Lorenz Attractor
This strange attractor is named after a meteorologist Edward Lorenz, who tried to
create a mathematical model of the atmosphere for the weather prediction. The
model consisted of a cylindrical box filled by air, which was heated at the bottom
and cooled from above, while the side was kept at the constant temperature. The
original set of 12 differential equations was simplified to the set of three ones:
dx
  ( y  x)
dt
dy
 r x  y  xz
dt
dz
 xy   z
dt
Where δ is ratio of the viscosity of the substance to
its thermal conductivity, r represents the temperature
difference between the top and the bottom of the
box, β is the width to height ratio of the box, x
represents rate of rotation, y represents the
difference between temperatures of rising and falling
fluid and z represents the deviation from linear
temperature gradient in vertical direction.
The most frequently used values of parameters:
δ= 10, r= 28 and β= 8/3
When calculating this model, Lorenz encountered a strange phenomenon. After
entering slightly different input values for two successive attempts he obtained
completely different outputs.
This effect was later named a butterfly effect, which should express the sensitivity
to initial condition by a metaphor that „a single flap of butterfly wings in South
America can change weather in Texas“.
The following graphs show time dependence of functions x(t) and z(t) in the
Lorenz attractor for the recommended parameter values and for initial conditions
x(0)=1; y(0)=1; z(0)=10
Butterfly effect demonstration
The following graphs show time dependence of functions x(t) and z(t) in the
Lorenz attractor for the recommended parameter values, while
blue curves are related to initial conditions x(0)=1; y(0)=1; z(0)=10
and red curves are related to initial conditions x(0)=1; y(0)=1; z(0)=10.01
Very slight change of initial condition results in large change in solution of the
function.
3D plot of the Lorenz attractor from various angles
3D comparison of various initial conditions
x(0)=1; y(0)=1; z(0)=10
x(0)=1.01; y(0)=1; z(0)=10
Fixed points and stability of the Lorenz attractor
Set of equations
Fixed point conditions
x   ( y  x)
y  r x  y  x z
 ( y  x)  0
yx
r x  y  xz  0
r x  x  xz
z  x y   z
xy   z  0
x2   z
Solution leads to three fixed points
z  r 1
x 2   z   (r  1)
For r=28 and β=8/3
  (r  1) 
  (r  1) 
0




~
x A  0; ~
xB    (r  1) ; ~
xC    (r  1) ;
 r 1 
 r 1 
0




 72
 72

 ~ 

~
xB   72; xC   72;
 27 
 27 




Jacobian matrix of the original system
x   ( y  x)
y  r x  y  x z
z  x y   z

0 
 
Df  r  z  1  x 
 y
x   
Linearized Jacobian matrices for the fixed points and corresponding eigenvalues
0 
 10 10
Df ( ~
x A )   28  1
0 
 0
0  8 / 3
λ1=-22.8, λ2=11.8, λ3=-2.67,
  10
Df ( ~
xB )   1
 72
λ1=-13.8, λ2=0.09+10.2i, λ3=0.09-10.2i

 1  72 
72  8 / 3 
10
0
10
0 
  10
Df ( ~
xC )   1
1
72 
 72  72  8 / 3
λ1=-13.8, λ2=0.09+10.2i, λ3=0.09-10.2i
Conclusion concerning fixed points: all three fixed points are unstable, because
they all include an eigenvalue with positive real part.
A program in Mathematica, which draws the Lorenz attractor.
The picture on the left shows an output of the Mathematica for one of the fixed
points, while the right picture shows an output for slightly different initial point.
A  ( 72, 72, 27)
B  ( 72, 72, 27.0001)
Fractals
The fractal is a geometric shape, that has the following features:
• It is self-similar, which means, that observing the shape in various zooms
results in the same characteristic shapes (or at least approximately the same)
• It has a simple and recursive definition
• It has a fine structure at arbitrarily small scales
• It is too irregular to be easily described in traditional Euclidean geometric
language.
• It has a Hausdorff dimension of its border higher than the topological dimension
of the border.
Topological dimension – a point has topological dimension 0, a line has
topological dimension 1, a surface has topological dimension 2, etc.
Hausdorff dimension – if an object contains n copies of itself reduced to one k-th
of the original dimension, the Hausdorff dimension can be calculated as
log(n)/log(k)
Example – a Cantor set
Procedure of creation – the original line is divided into three parts while the
middle part is erased. The same procedure is applied to newly created lines etc.
Repeating this procedure to the infinity, we obtain an infinite number of points
with topological dimension 0.
The set contains n=2 copies of itself reduced to 1/3 of the original dimension
(k=3). Hausdorff dimension is log(2)/log(3)=0.6309…, which is greater than 0.
The Mandelbrot set
The Mandelbrot set M is the set of values of c in the complex plane for which the
orbit of 0 under iteration of the complex quadratic polynomial zn+1=zn2+c remains
bounded.
It is a set of complex numbers, for which
lim z n  
n 
where the sequence z0,z1,z2… is
defined by a recursive formula
z n 1  z n  c ; z0  0
2
Constant c represents the coordinates of each examined point.
Basic properties:
• the entire set lies inside a circle with radius 2 around the origin
• the set is connected
• Hausdorff dimension of the set is 2
• the area of the Mandelbrot set is estimated to 1.50659177
• if the absolute value of any zn is larger than 2, then the sequence escapes to
infinity
• the intersection of M with the real axis is precisely the interval [-2, 0.25]
Examples of iterations according to the formula zn+1=zn2+c for various values of
the constant c.
The Mandelbrot set calculated by Matlab
Black areas represent points, which did not escape to the infinity after 500
iterations. Colors on the HSV colorbar represent number of iterations needed for
the value to escape to the infinity (Matlab’s infinity is about 10308). Red areas
represent points, which escaped to the infinity after less than 35 iterations. Each
color higher means 30 iterations more, e.g. yellow color represents points, which
escaped to the infinity between 95th and 125th iteration.
Some nice areas in the Mandelbrot set
The Julia set
Like the Mandelbrot set, also Julia set uses the complex quadratic polynomial
zn+1=zn2+c for its creation.
The difference is in initial values. In case of the Mandelbrot set the initial value
for z0 was 0 and the constant c represented coordinates of the examined point.
In case of Julia set the initial value z0 represents coordinates of the examined
point and the constant c characterizes the Julia set.
The Julia set is a set of complex
numbers, for which
lim z n  
where the sequence z0,z1,z2… is
defined by a recursive formula
z n 1  z n  c ; z0  ( x0 , y0 )
n 
2
Since there is an infinite number of possible values of c, there is also an infinite
number of possible Julia sets.
The Julia set for c= -0.70176 -0.3842i
The Julia set for c= 0.285 + 0.01*i
The Julia set for c= -0.8 + 0.156*i
The Julia set for c= -0.4 + 0.6*i