chaotic dynamical systems experimental approach frank wang
TRANSCRIPT
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Chaotic Dynamical Systems
Experimental Approach
Frank Wang
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Striking the same key
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Graphic Method
Square root function f(x)=sqrt(x)
Identity function y=x Vertical to the curve
and horizontal to the line
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Square Root Function
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Logistic Difference Equation
)1(1 nnn xxx
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Function Notation
seed x0 orbit
)1()( xxxf
))),((()),((),( 000 xfffxffxf
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lambda=2.5
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lambda=3.1
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lambda=3.8
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lambda=3.8 histogram
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Fixed Point and Periodic Point
Fixed point:
Periodic point:
xxF )(
xxFF ))((
xxFFF )))(((
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Period-1
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Period-2
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Bifurcation Diagram
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Period 3 Implies Chaos
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Sarkovskii’s Theorem (1964)
1222
725232725232
725232753
23
233222
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Filled Julia Set
Quadratic Map
Filled Julia Set
czzQc 2:
}|)(||{ zQCzJ ncc
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Sonya Kovalevskaya
Introduction of i to a dynamical system.
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Kovalevskaya Top
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C=0.33+0.45 i
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C=0.5+0.5 i
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C=0.33+0.57 i
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C=0.33+0.573 i
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C=-0.122+0.745 i
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C= i
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C=0.360284+0.100376 i
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C=-0.75+0.1 i
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Mandelbrot set and bifurcation
Mandelbrot set
}|)0(||{ ncQCcM
czzQc 2:
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Period 3 window
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Magnification of the Mandelbrot set
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Period 7 bulb (2/7)
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Period 8 bulb (3/8)
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Period 9 bulb (4/9)
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Period 13 bulb (6/13)
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Julia set for (1+2 i) exp(z)
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Julia set for 2.96 cos(z)
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Julia set for (1+0.2 i) sin(z)