1. Theory of nonlinear dynamic systems Practice 9
Juhász János
Szélig Ádám
Goda Márton

2. Introduction– fractals I.
What does fractal mean?
Some kind of shape
It is self-similar (smaller parts resembles to the original shape)
Its mathematical description is usually simply
The borders of fractals are „infinitely creased”

3. Introduction– fractals II.
Koch-curve
Mandelbrot-set
Julia-set
Fern shape

4. Introduction– fractals III.
Why are they useful?
Simple mathematical equation
Few memory => graphical applications
„natural shapes”
Base of compression algorithms
Tool development: e.g.: antenna
maximizing the surface/volume ratio

5. Cantor set
formula for Cantor set construction:

6. Sierpinski triangle
Similar to the Cantor set, but the we cut middle ¼ of a triangle
Method:
Start with an equilateral triangle.
Subdivide it into four smaller congruent equilateral triangles and remove the central one.
Repeat step 2 with each of the remaining smaller triangles
The area tends to zero.

7. Sierpinski triangles in 3D – optional homework (+3 points)
Draw the Siepinski tetraeder based on the 2D triangle. Send your code and a figure about the tetraeder to the instructors.
Deadline: 22th of november Tuesday eveningA

8. Newton's method For finding the roots of equations
A generalization of Newton's iteration is
Equations to try: e.g.: x3+1, x3-1, x3-2x+2
The colors show the sets initial points (in the complex plain) from which the system converges to the same root
It can „get stuck in” in special cases
Fractal develops around from the fate of the initial points around the roots

9. Fractal dimension
Dim=log(number of self-similar pieces in the next iteration)/log(magnification factor)
E.g.:
DimCantor=log(2)/log(3)
DimSiepr=log(3)/log(2)
Box-counting dimension:
ε :resolution
n=1/ε :magnification
b :number of boxes with the feature (boundary)

10. Thank you for your attention!

11. Matlab® supplement
[X,Y] = meshgrid(x,y) replicates the grid vectors x and y to produce a full grid.
[y(2); y(1)]; [1. equ of the system; 2. equ of the system]
[t,y]=ode45(equation, [t0,tmax][Xinit,Yinit]); there are other solvers as well, first we try this
figure creates figure graphics objects. Figure objects are the individual windows on the screen in which the MATLAB software displays graphical output.
plot(x,y,how…) drawing, has many options
subplot(m,n,p) (divide the figure m*n parts, draws in the p th region)
contour(X,Y,Z), contour(X,Y,Z,n), and contour(X,Y,Z,v) draw contour plots of Z using X and Y to determine the x- and y-axis limits.