Period doubling, the Feigenbaum constant, and the road to chaos
The logistic map is perhaps the simplest equation that produces chaos. Starting from a population model, it reveals how order can give way to complete unpredictability through a process called period doubling.
The equation is deceptively simple: xn+1 = r·xn(1 - xn). Starting with any x between 0 and 1, we iterate. What happens depends entirely on the parameter r.
This diagram shows all stable values of x for each value of r. Watch how a single stable point splits into two, then four, then eight... until chaos erupts:
Click to select an r value. The diagram shows stable values of x for each r in the logistic map x → rx(1-x). Watch order become chaos through period doubling.
The system settles to a single value. No matter where you start, you always end up at the same point.
The system oscillates between two values. It never settles, but it's still perfectly predictable.
The period keeps doubling. The gaps between doublings get smaller and smaller, following the Feigenbaum constant.
After the infinite cascade of period doublings, we enter chaos. The system never repeats - but windows of order still appear!
The cobweb plot shows exactly how iteration works. Starting at x₀ on the x-axis, go up to the parabola (apply f), then across to the diagonal (copy the y-value to x), and repeat:
The cobweb plot shows how iteration works: go up to the parabola, then across to the diagonal, then up again. Try r = 3.2 for a stable period-2 orbit, or r = 3.8 for chaos.
Mitchell Feigenbaum discovered something remarkable: the ratio of the gaps between successive period-doubling points approaches a universal constant:
This constant appears in a vast class of systems with period-doubling routes to chaos - not just the logistic map. Every map with a quadratic maximum shares the same δ, no matter its other details.
How do we measure chaos? The Lyapunov exponent (λ) quantifies how fast nearby trajectories diverge:
Nearby points converge. Small errors shrink. The system is predictable.
Nearby points diverge exponentially. Small errors grow. Prediction becomes impossible over time.
Look carefully at the bifurcation diagram around r = 3.83. You'll see a window where order suddenly returns - a period-3 cycle appears in the midst of chaos! These "periodic windows" are everywhere, infinitely many of them, creating a fractal structure within the chaos.
The period-3 window is special: mathematicians Tien-Yien Li and James Yorke proved that "Period 3 implies chaos" - if a system has a period-3 orbit, it must also have orbits of every other period, and chaotic behavior.