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Chaos in Dynamical Systems
Baoqing ZhouSummer 2006
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Dynamical Systems•Deterministic Mathematical Models•Evolving State of Systems (changes as time goes on)
Chaos•Extreme Sensitive Dependence on Initial Conditions•Topologically Mixing•Periodic Orbits are Dense•Evolve to Attractors as Time Approaches Infinity
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Examples of 1-D Chaotic Maps (I)Tent Map: Xn+1 = μ(1-2|Xn-1/2|)
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Examples of 1-D Chaotic Maps (II)2X Modulo 1 Map: M(X) = 2X modulo 1
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Examples of 1-D Chaotic Maps (III)Logistic Map: Xn+1 = rXn(1-Xn)
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Forced Duffing Equation (I)mx” + cx’ + kx + βx3 = F0 cos ωt
m = c = β = 1, k = -1, F0 = 0.80
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Forced Duffing Equation (II)
m = c = β = 1, k = -1, F0 = 1.10
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Lorenz System (I)dx/dt = -sx + sydy/dt = -xz + rx – ydz/dt = xy – bz
b = 8/3, s = 10, r =28x(0) = -8, y(0) = 8, z(0) =27
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Lorenz System (II)
b = 8/3 s = 10 r =70
x(0) = -4 y(0) = 8.73 z(0) =64
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BibliographyOtt, Edward. Chaos in Dynamical Systems. Cambridge: Cambridge University Press, 2002. http://local.wasp.uwa.edu.au/~pbourke/fractals/http://mathworld.wolfram.com/images/eps-gif/TentMapIterations_900.gifhttp://mathworld.wolfram.com/LogisticMap.html