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

)1(1 nnn xxx

Function Notation

seed x0 orbit

)1()( xxxf

))),((()),((),( 000 xfffxffxf

lambda=2.5

lambda=3.1

lambda=3.8

lambda=3.8 histogram

Fixed Point and Periodic Point

Fixed point:

Periodic point:

xxF )(

xxFF ))((

xxFFF )))(((

Period-1

Period-2

Bifurcation Diagram

Period 3 Implies Chaos

Sarkovskii’s Theorem (1964)

1222

725232725232

725232753

23

233222

Filled Julia Set

Quadratic Map

Filled Julia Set

czzQc 2:

}|)(||{ zQCzJ ncc

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

}|)0(||{ ncQCcM

czzQc 2:

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)