Strange Attractors

Comments on Homework #4 (Lorenz Attractor)

• Everyone did a good job
• To get smooth graphs, make h smaller or connect the dots

Review (last week) - Lyapunov Exponents

• Lyapunov Exponents are a dynamical measure of chaos
• There are as many exponents as the system has dimensions
• dV/dt / V = l₁ + l₂ + l₃ + ...
• = <log |det J|> for maps
• = <trace J> = <f_x + g_y + h_z + ...> for flows
• Where J is the Jacobian matrix
• Sl must be negative for an attractor
• Sl must be zero for a conservative (Hamiltonian) system
• For chaos we require l₁ > 0  (at least one positive LE)
• For 1-D Maps, l = <log |df/dX|>
• 2-D example, Hénon map:
• X_n+1 = 1 - CX_n² + BY_n   [= f(X, Y)]
• Y_n+1 = X_n   [= g(X, Y)]
• Usual parameters for chaos: B = 0.3, C = 1.4
• l₁ + l₂ = <log |f_xg_y - f_yg_x|> = log |-B| = -1.204 (base-e)
• Numerical calculation gives l₁ = 0.419 (base-e)
• Hence l₂ = -1.204 - 0.419 = -1.623 (base-e)
• Fixed points at x* = y* = -1.1313445 and x* = y* = 0.63133545
• General character of Lyapunov exponents in flows:
• (-, -, -, -, ...)  fixed point  (0-D)
• (0, -, -, -, ...)  limit cycle  (1-D)
• (0,0, -, -, ...)  2-torus  (2-D)
• (0, 0, 0, -, ...)  3-torus, etc.  (3-D, etc.)
• (+, 0, -, -, ...)  strange (chaotic)  (2⁺-D)
• (+, +, 0, -, ...)  hyperchaos, etc.  (3⁺-D)
• Numerical Calculation of Largest Lyapunov Exponent
• Start with any initial condition in the basin of attraction
• Iterate until the orbit is on the attractor
• Select (almost any) nearby point (separated by d₀)
• Advance both orbits one iteration and calculate new separation d₁
• Evaluate log |d₁/d₀| in any convenient base
• Readjust one orbit so its separation is d₀ in same direction as d₁
• Repeat steps 4-6 many times and calculate average of step 5
• The largest Lyapunov exponent is l₁ = <log |d₁/d₀|>
• If map approximates an ODE, then l₁ = <log |d₁/d₀|> / h
• A positive value of l₁ indicates chaos


• Shadowing lemma: The computed orbit shadows some possible orbit
• Kaplan-Yorke (Lyapunov) Dimension
• Attractor dimension is a geometrical measure of complexity
• Random noise is infinite dimensional (infinitely complex)
• How do we calculate the dimension of an attractor?  (many ways)
• Suppose system has dimension N  (hence N Lyapunov exponents)
• Suppose the first D of these sum to zero
• Then the attractor would have dimension D
• (in D dimensions there would be neither expansion nor contraction)
• In general, find the largest D for which l₁ + l₂ + ... + l_D > 0
• (The integer D is sometimes called the topological dimension)
• The attractor dimension would be between D and D + 1
• However, we can do better by interpolating:
• D_KY = D + (l₁ + l₂ + ... + l_D) / |l_D+1|
• The Kaplan-Yorke conjecture is that D_KY agrees with other methods
• 2-D Map Example: Hénon map  (B = 0.3, C = 1.4)
• l₁ = 0.419 and l₂ = -1.623
• D = 1 and D_KY = 1 + l₁ / |l₂| = 1 + 0.419 / 1.623 = 1.258
• Agrees with intuition and other calculations
• 3-D Flow Example:  Lorenz Attractor  (p = 10, r = 28,  b = 8/3)
• Numerical calculation gives l₁ = 0.906
• Since it is a flow, l₂ = 0
• l₁ + l₂ + l₃ = <f_x + g_y + h_z> = -p - 1 - b = -13.667
• Therefore, l₃ = -14.572
• D = 2 and D_KY = 2 + l₁ / |l₃| = 2 + 0.906 / 14.572 = 2.062
• Chaotic flows always have D_KY > 2
• Chaotic maps always have D_KY > 1
• Higher order interpolations are possible
• Precautions
• Be sure orbit is bounded and looks chaotic
• Be sure orbit has adequately sampled the attractor
• Watch for contraction to zero within machine precision
• Test with different initial conditions, step size, etc.
• Supplement with other tests (Poincaré section, Power spectrum, etc.)

Strange Attractors

• This is a "lecture within a lecture"
• Sample strange attactors and their properties will be discussed.
• Typical experimental data often lacks obvious structure.
• One goal is to find evidence of determinism in such data.
• An Example is a 2-D quadratic map.
• Models developed to fit chaotic data are seldom chaotic.
• This raises the question of how common is chaos?
• The Hénon map is about 6% chaotic over its bounded range.
• The Hénon map is a strange attractor with fractal structure.
• The Mandelbrot set is chaotic almost nowhere.
• It has a very complicated basin boundary.
• General 2-D quadratic maps are 10-20% chaotic.
• Maps become less and flows more chaotic as dimension increases.
• Artificial neural networks are another kind of iterated map.
• Neural networks become more chaotic as dimension increases.
• There are many types of attractors.
• Strange attractors have many interesting properties:
• Limit set as t --> infinity
• Set of measure zero
• Basin of attraction
• Fractal structure
• non-integer dimension
• self-similarity
• infinite detail
• Chaotic dynamics
• sensitivity to initial conditions
• topological transitivity
• dense periodic orbits
• Aesthetic appeal
• Strange attractors are produced by a stretching and folding.
• Attractor dimension increases with system dimension.
• Lyapunov exponent decreases with system dimension.
• Attractor search turned up the simplest chaotic flow.
• Simplest flow has a strange attractor that's a Mobius strip.
• There are also conservative chaotic system but not attractors.
• Strange attractors can be used as general approximators.
• They can also be used to generate computer art.