Chaotic Dynamical Systems
description
Transcript of Chaotic Dynamical Systems
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)