Dynamical Systems 4

Deterministic chaos, fractals

Ing. Jaroslav Jíra, CSc.

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.

One-dimensional Maps

)(1 nn xfx The general form of such map is

We already know an example of linear map (bank account): nn rxx 1

)1(1 nnn xrxx and we also know an example of nonlinear map (logistic map):

In the following we will learn a new technique for solving these difference equations. The technique is called Cobwebbing or Cobweb diagram.

As an example we take our well known logistic equation (or logistic map), this time in a shape y=rxn(1-xn) .

Cobweb diagram

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.

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.

Feigenbaum constant delta

Ratio is evaluated according to this formula: nn

nn

rr

rrratio

1

1

669201609.4lim1

1

nn

nn

n rr

rr

The ratio limits to a value 4.669201609…, which is called Feigenbaum constant delta

751446.444948974.354409036.3

344948974.3

ratioFor example

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.

Feigenbaum constant alpha

The value 2.502907875 is called Feigenbaum constant alpha

502907875.2lim1

n

n

n a

a

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:

zyxdt

dz

zxyxrdt

dy

xydt

dx

)(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

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

Butterfly effect demonstration

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

zyxz

zxyxry

xyx

)(

0

0

0)(

zyx

zxyxr

xy

zx

xzxxr

xy

2

)1(

12

rzx

rz

;

1

)1(

)1(~;

1

)1(

)1(~;

0

0

0~

r

r

r

x

r

r

r

xx CBA

For r=28 and β=8/3 ;

27

72

72~;

27

72

72~

CB xx

Set of equations Fixed point conditions

Solution leads to three fixed points

xy

xzrf 1

0

D

3/800

0128

01010

)~( AxfD

zyxz

zxyxry

xyx

)(

3/87272

7211

01010

)~( BxfD

3/87272

7211

01010

)~( CxfD

Jacobian matrix of the original system

λ1=-22.8, λ2=11.8, λ3=-2.67,

λ1=-13.8, λ2=0.09+10.2i, λ3=0.09-10.2i

λ1=-13.8, λ2=0.09+10.2i, λ3=0.09-10.2i

Linearized Jacobian matrices for the fixed points and corresponding eigenvalues

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.

)27,72,72(A )0001.27,72,72(B

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=zn

2+c remains bounded.

It is a set of complex numbers, for which

0;

lim

02

1

zczz

z

nn

nn

where the sequence z0,z1,z2… is defined by a recursive formula

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]

Constant c represents the coordinates of each examined point.

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=zn

2+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

0002

1 yxzczz

z

nn

nn

where the sequence z0,z1,z2… is defined by a recursive formula

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

Deterministic chaos – double pendulum