Chaotic Dynamical Systems

Experimental Approach Frank Wang

Striking the same key

Graphic Method

   Square root function f(x)=sqrt(x) Identity function y=x Vertical to the curve and horizontal to the line

Square Root Function

Logistic Difference Equation

x n

 1  

x n

( 1 

x n

)

Function Notation

f

(

x

)  

x

( 1 

x

)   seed x0 orbit

f

(

x

0 ),

f

(

f

(

x

0 )),

f

(

f

(

f

(

x

0 ))),

lambda=2.5

lambda=3.1

lambda=3.8

lambda=3.8 histogram

Fixed Point and Periodic Point

 Fixed point:

F

(

x

) 

x

 Periodic point:

F

(

F

(

x

)) 

x F

(

F

(

F

(

x

))) 

x

Period-1

Period-2

Bifurcation Diagram

Period 3 Implies Chaos

Sarkovskii’s Theorem (1964)

3  5  7    2  3  2  5  2  7    2 2  3  2 2  5  2 2  7    2 3  3  2 3  5  2 2  7    2 3  2 2  2  1

Filled Julia Set

 Quadratic Map

Q c

: 

z



z

2 

c

 Filled Julia Set

J c

 {

z



C

||

Q c n

(

z

) |   }

Sonya Kovalevskaya

 Introduction of i to a dynamical system.

Kovalevskaya Top

C=0.33+0.45 i

C=0.5+0.5 i

C=0.33+0.57 i

C=0.33+0.573 i

C=-0.122+0.745 i

C= i

C=0.360284+0.100376 i

C=-0.75+0.1 i

Mandelbrot set and bifurcation

 Mandelbrot set

M

 {

c



C

||

Q c n

( 0 ) |   }

Q c

: 

z



z

2 

c

Period 3 window

Magnification of the Mandelbrot set

Period 7 bulb (2/7)

Period 8 bulb (3/8)

Period 9 bulb (4/9)

Period 13 bulb (6/13)

Julia set for (1+2 i) exp(z)

Julia set for 2.96 cos(z)

Julia set for (1+0.2 i) sin(z)