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: X n+1 = μ ( 1-2 |X n -1/2 |)

Examples of 1-D Chaotic Maps (II) 2 X Modulo 1 Map: M(X) = 2 X modulo 1

Examples of 1-D Chaotic Maps (III) Logistic Map: X n+1 = r X n ( 1- X n )

Forced Duffing Equation (I) mx” + cx’ + kx + β x 3 = F 0 cos ω t m = c = β = 1, k = -1, F 0 = 0.80

Forced Duffing Equation (II) m = c = β = 1, k = -1, F 0 = 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,