Chaotic Dynamical Systems Experimental Approach Frank Wang.

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Presentation transcript:

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

Function Notation seed x0 orbit

lambda=2.5

lambda=3.1

lambda=3.8

lambda=3.8 histogram

Fixed Point and Periodic Point Fixed point: Periodic point:

Period-1

Period-2

Bifurcation Diagram

Period 3 Implies Chaos

Sarkovskii’s Theorem (1964)

Filled Julia Set Quadratic Map Filled Julia Set

Sonya Kovalevskaya Introduction of i to a dynamical system.

Kovalevskaya Top

C= i

C= i

C= i

C= i

C= i

C= i

C= i

C= i

Mandelbrot set and bifurcation Mandelbrot set

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)