Chaos in Dynamical Systems

Baoqing Zhou

Summer 2006

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

Examples of 1-D Chaotic Maps (I)

Tent Map: Xn+1 = μ(1-2|Xn-1/2|)

Examples of 1-D Chaotic Maps (II)

2X Modulo 1 Map: M(X) = 2X modulo 1

Examples of 1-D Chaotic Maps (III)

Logistic Map: Xn+1 = rXn(1-Xn)

Forced Duffing Equation (I)

mx” + cx’ + kx + βx3 = F0 cos ωt

m = c = β = 1, k = -1, F0 = 0.80

Forced Duffing Equation (II)

m = c = β = 1, k = -1, F0 = 1.10

Lorenz System (I)

dx/dt = -sx + sy

dy/dt = -xz + rx – y

dz/dt = xy – bz

b = 8/3, s = 10, r =28

x(0) = -8, y(0) = 8, z(0) =27

Lorenz System (II)

b = 8/3 s = 10 r =70

x(0) = -4 y(0) = 8.73 z(0) =64

Bibliography

Ott, 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.gif

http://mathworld.wolfram.com/LogisticMap.html