Feigenbaum Constant

Turn the control knob of a Logistic Map and its steady state period-doubles: a 1-cycle becomes a 2-cycle, then 4, 8, 16, the doublings arriving faster and faster until the period is infinite and the motion is chaotic. The remarkable thing is not that this happens but how fast. The parameter windows between successive doublings shrink geometrically, and the ratio of one gap to the next homes in on a single number,

This is the Feigenbaum constant. Mitchell Feigenbaum discovered in the 1970s that the same δ governs the period-doubling cascade in an enormous range of systems — dripping faucets, convecting fluids, nonlinear circuits, cardiac cells — provided only that the underlying map has a smooth quadratic maximum. The details of the system wash out; the rate at which order dissolves into chaos is universal, a constant of nature in the same spirit as \pi or e. It is one of the cleanest pieces of evidence that chaos has a rigid, quantitative structure rather than being mere messiness.

The cascade, located numerically

We can pin the cascade down from first principles, with no data and no fitting — just the arithmetic of x \mapsto r\,x(1-x). The trick is that a period-2^{\,n-1} cycle loses stability exactly where its multiplier — the product of the slope f'(x)=r(1-2x) taken around the cycle — passes through -1. That r is the bifurcation r_n at which the period doubles. Solving the cycle equations with Newton's method and driving the multiplier to -1 with a secant step locates each r_n to machine precision:

#!/usr/bin/env python3
# A period-2^(n-1) cycle loses stability where its multiplier (product of
# f'(x_i) around the cycle) reaches -1; that r is the bifurcation b_n. Pin
# each b_n with a Newton-polished cycle + a secant on the multiplier, then
# form delta_n = (b_n - b_{n-1}) / (b_{n+1} - b_n) -> 4.6692016...
def f(r, x):  return r * x * (1.0 - x)
def fp(r, x): return r * (1.0 - 2.0 * x)
def cycle_mult(r, p, x):
    for _ in range(200):
        y, deriv = x, 1.0
        for _ in range(p):
            deriv *= fp(r, y); y = f(r, y)
        if deriv == 1.0: break
        dx = (y - x) / (deriv - 1.0); x -= dx
        if abs(dx) < 1e-15: break
    y, M = x, 1.0
    for _ in range(p):
        M *= fp(r, y); y = f(r, y)
    return x, M
def bifurcation(p, r_seed):
    x = 0.5
    for _ in range(200000): x = f(r_seed, x)          # land on the cycle
    r0, (x0, M0) = r_seed, cycle_mult(r_seed, p, x)
    r1, (x1, M1) = r_seed + 1e-4, cycle_mult(r_seed + 1e-4, p, x)
    for _ in range(100):                              # secant: multiplier -> -1
        if M1 == M0: break
        r2 = r1 - (M1 + 1.0) * (r1 - r0) / (M1 - M0)
        x2, M2 = cycle_mult(r2, p, x1)
        r0, M0, x0, r1, M1, x1 = r1, M1, x1, r2, M2, x2
        if abs(r1 - r0) < 1e-15: break
    return r1
periods = [1, 2, 4, 8, 16, 32]
b = [bifurcation(p, s) for p, s in zip(periods, [2.9,3.4,3.52,3.560,3.5675,3.5694])]
print("n  period  r_n (bifurcation)      gap             delta_n")
for i in range(len(b)):
    gap   = f"{b[i]-b[i-1]:.10f}" if i >= 1 else ""
    delta = f"{(b[i-1]-b[i-2])/(b[i]-b[i-1]):.5f}" if i >= 2 else ""
    print(f"{i+1}  {periods[i]:6d}  {b[i]:.10f}   {gap:>13}   {delta:>9}")
best = (b[-2] - b[-3]) / (b[-1] - b[-2])
print(f"best ratio delta_5 = {best:.5f}  ->  delta ~ 4.669  (Feigenbaum 4.6692016...)")

The convergence is not monotone-from-one-side or accidental: it is the geometric signature of the cascade. Each gap is very nearly \delta^{-1} times the one before, so the doublings accumulate at a finite parameter rather than running off to r=4.

Where the doublings run out

Because the gaps shrink by a constant factor, the infinite tally of bifurcations converges. Summing the remaining geometric tail of gaps past the ones we computed gives the accumulation point r_\infty — the exact parameter where the period first becomes infinite and the orbit, no longer settling onto any finite cycle, lands on a strange attractor of fractal (Cantor-set) structure:

#!/usr/bin/env python3
# The bifurcations b_n pile up geometrically (gaps shrink by ~delta each step).
# Locate the last two we can resolve, then sum the remaining geometric tail of
# gaps to reach the accumulation point r_inf, where the period goes infinite.
# (bifurcation() as in the delta check)
b5 = bifurcation(16, 3.5675)
b6 = bifurcation(32, 3.5694)
delta = 4.669201609102990
last_gap = b6 - b5
tail = (last_gap / delta) / (1.0 - 1.0 / delta)   # b7-b6 + b8-b7 + ...
r_inf = b6 + tail
print(f"b5 (period 16 -> 32) = {b5:.10f}")
print(f"b6 (period 32 -> 64) = {b6:.10f}")
print(f"last gap b6-b5       = {last_gap:.10f}")
print(f"geometric tail / delta = {tail:.10f}")
print(f"accumulation r_inf   = {r_inf:.5f}  (true 3.5699456...)")

Beyond r_\infty lies the chaotic regime, shot through with narrow periodic windows — but the universal number \delta has already done its work by the time we arrive.

Why universal?

The depth of Feigenbaum's discovery is that \delta does not depend on the logistic map at all. Any smooth one-dimensional map with a single quadratic hump — \sin, a cubic, a tent rounded at the tip — period-doubles its way to chaos with the same δ and the same spatial scaling constant \alpha\approx2.5029. The explanation is a renormalization argument borrowed from statistical physics: zoom in on the map near its peak after each doubling and rescale, and the rescaled maps converge to a fixed shape, a fixed point of a "doubling operator." δ is simply the rate at which that operator stretches parameter space — an eigenvalue, blind to which particular map you started from. That is why a bifurcation cascade in a dripping tap and one in the Logistic Map share a digit-for-digit constant.

For all this precision, δ's basic arithmetic nature is unknown — see the open problem Is the Feigenbaum Constant Transcendental?.

See also

• Logistic Map
• Bifurcation
• Strange Attractor
• Sensitive Dependence on Initial Conditions
• Chaos