One-Dimensional Maps

Review (last week)

• Dynamical systems
• Random (stochastic) versus deterministic
• Linear versus nonlinear
• Simple (few variables) versus complex (many variables)
• Examples (solar system, stock market, ecology, ...)
• Some Properties of chaotic dynamical systems
• Deterministic, nonlinear dynamics (necessary but not sufficient)
• Aperiodic behavior (never repeats - infinite period)
• Sensitive dependence on initial conditions (exponential)
• Dependence on a control parameter (bifurcation, phase transition)
• Period-doubling route to chaos (common,  but not universal)
• Demonstrations
• Computer animations (3-body problem, driven pendulum)
• Chaotic pendulums
• Ball on oscillating floor
• Falling leaf (or piece of paper)
• Fluids (mixing, air hose, dripping faucet)
• Chaotic water bucket
• Chaotic electrical circuits

Logistic Equation - Motivation

• Exhibits many aspects of chaotic systems (prototype)
• Mathematically simple
• Involves only a single variable
• Doesn't require calculus
• Exact solutions can be obtained
• Can model many different phenomena
• Ecology
• Cancer growth
• Finance
• Etc...
• Can be understood graphically

Exponential Growth (Discrete Time)

• X_n+1 = AX_n  (example: compound interest)
• Example of linear deterministic dynamics
• Example of an iterated map (involves feedback)
• Exhibits stretching (A > 1) or shrinking (A < 1)
• Attracts to X = 0 (for A < 1) or X = infinity (for A > 1)
• Solution is X_n = X₀Aⁿ  (exponential growth or decay)
• A is the control parameter (the "knob")
• A = 1 is a bifurcation point.

Logistic Equation

• X_n+1 = AX_n(1 - X_n)
• Quadratic nonlinearity (X²)
• Graph of X_n+1 versus X_n is a parabola
• Equivalent form: Y_n+1 = B - Y_n²  (quadratic map)
• Y = A(X - 0.5)
• B = A²/4 - A/2
• Solutions: X^* = 0, 1 - 1/A  (fixed point)
• Graphical solution  (reflection from 45° line - "cobweb diagram")
• Computer simulation of logistic map

Bifurcations

• 0 < A < 1 Case:
• Only non-negative solution is X^* = 0
• All X₀ in the interval 0 < X₀ < 1 attract to X^*
• They lie in the basin of attraction
• The nonlinearity doesn't matter
• 1 < A < 3 Case:
• Solution at X^* = 0 becomes a repellor
• Solution at X^* = 1 - 1/A appears
• It is a point attractor (also called "period-1 cycle")
• Basin of attraction is 0 < X₀ < 1
• 3 < A < 3.449... Case:
• Attractor at 1 - 1/A becomes unstable (repellor)
• This happens when df/dX < -1  (==> A > 3)
• This bifurcation is called a flip
• Growing oscillation occurs
• Oscillation nonlinearly saturates (period-2 cycle)
• X_n+2 = f(f(X_n)) = f⁽²⁾(X_n) = X_n
• Quartic equation has four roots
• Two are the original unstable fixed points
• The other two are are the new 2-cycle
• 3.449... < A < 3.5699... Case:
• Period-2 becomes unstable when df⁽²⁾(X)/dX < -1
• At this value (A = 3.440...) a stable period-4 cycle is born
• The process continues with successive period doublings
• Infinite period is reached at A = 3.5699... (Feigenbaum point)
• This is period-doubling route to chaos
• Bifurcation plot is self-similar (a fractal)
• Feigenvalues: delta = 4.6692..., alpha = 2.5029...
• Feigenvalues are universal (for all smooth 1-D unimodal maps)
• 3.5699... < A < 4 Case:
• Most values of A in this range produce chaos (infinite period)
• There are infinitely many periodic windows
• Each periodic window displays period doubling
• All periods are present somewhere for 3 < A < 4
• A = 4 Case:
• This value of A is special
• It maps the interval 0 < X < 1 back onto itself  (endomorphism)
• Notice the fold at X_n = 0.5
• Thus we have stretching and folding  (silly putty demo)
• Stretching is not uniform  (cf: tent map)
• Each X_n+1 has two possible values of X_n  (preimages)
• Error in initial condition doubles (on average) with each iteration
• We lose 1 bit of precision with each time step
• A > 4 Case:
• Transient chaos for A slightly above 4 for most X₀
• Orbit eventually escapes to infinity for most X₀

Other Properties of the Logistic Map (A = 4)

• Eventually fixed points
• X₀ = 0 and X₀ = 1 - 1/A = 0.75 are (unstable) fixed points
• X₀ = 0.5 --> 1 --> 0 is an eventually fixed point
• There are infinitely many such eventually fixed points
• Each fixed point has two preimages, etc..., all eventually fixed
• Although infinite in number they are a set of measure zero
• They constitute a Cantor set (Georg Cantor)
• Compare with rational and irrational numbers
• Eventually periodic points
• If X_n+2 = X_n orbit is (unstable) period-2 cycle
• Solution (A = 4):  X^* = 0, 0.345491, 0.75, 0.904508
• 0 and 0.75 are (unstable) fixed points (as above)
• 0.345491 and 0.904508 are (unstable) period-2 cycle
• All periods are present and all are unstable
• (Unstable) period-3 orbit implies chaos (Li and Yorke)
• Each period has infinitely many preimages
• Still, most points are aperiodic (100%)
• Periodic orbits are dense on the set
• Probability density (also called invariant measure)
• Many X_n values map to X_n+1 close to 1.0
• These in turn map to X_n+2 close to 0.0
• Thus the probability density peaks at 0 and 1
• Actual form: P = 1 / pi[X(1 - X)]^1/2
• Ergodic hypothesis: the average over all starting points is the same as the average over time for a single starting point
• Nonrecursive representation
• X_n = (1 - cos(2ⁿcos^-1(1 - 2X₀)))/2
• Ref: H. G. Schuster, Deterministic Chaos, (VCH, Weinheim, 1989)

Other One-Dimensional Maps

• Sine map
• X_n+1 = A sin(pi X_n)
• Properties similar to logistic map (except A = 1 corresponds to A = 4)
• Tent map
• X_n+1 = A min(X_n, 1 - X_n)
• Piecewise linear
• Uniform stretching
• All orbits become unstable at A = 1
• Uniform (constant) probability density at A = 2
• Numerical difficulties
• General symmetric map
• X_n+1 = A(1 - |2X_n - 1|^alpha)
• alpha = 1 gives the tent map
• alpha = 2 gives the logistic map
• alpha is a measure of the smoothness of the map
• Binary shift map
• X_n+1 = 2X_n (mod 1)
• Stretching, cutting, and reattaching
• Resembles tent map
• Chaotic only for irrational initial conditions
• Can be used to generate pseudo-random numbers