Notebooks · posts/logistic-lyapunov

The logistic map at r = 4 has Lyapunov exponent ln 2

Adam Zettel · September 21, 2026 · written with an agent

Iterate $x \to r x (1 - x)$ . At $r = 3.2$ it settles on two values you can write down. At $r = 4$ it never settles, nearby starts separate by a factor of two per step on average, and that factor is exact: the Lyapunov exponent is $ln 2$ .

λ = n \to \infty lim \frac{1}{n} k < n \sum ln ∣ r (1 - 2 x_{k}) ∣

At $r = 4$ , substitute $x = sin^{2} θ$ and the map becomes $θ \to 2 θ$ : angle doubling, which is where $ln 2$ comes from. The two-cycle for $3 < r < 1 + 6$ solves $f (f (x)) = x$ with the fixed points divided out: $x_{\pm} = (r + 1 \pm (r - 3) (r + 1)) /2 r$ .

Look inside this program

Atlas
The program’s map: every function and which calls which.
There’s no recorded run of this program yet.

Shift+Enter in a cell runs it and moves to the next. Running a cell runs every cell above it too, as one program.

[1]lines 16–91

# iterations discarded before the orbit is on its attractor: the transient
TRANSIENT = 500.0

#| The Lyapunov exponent at r: the orbit average of ln |f'(x)| = ln |r (1 - 2x)|.
def lyapunov(r, steps) {
    x = 0.3
    for s to TRANSIENT {
        x = r * x * (1.0 - x)
    }
    total = 0.0
    for s to steps {
        x = r * x * (1.0 - x)
        total = total + log(abs(r * (1.0 - 2.0 * x)))
    }
    return total / steps
}

# the attractor: for each r (a column), discard a transient, then count where the
# orbit goes (the rows, x = 1 at the top). Counts are capped so the two-cycle's
# two hot cells do not wash out the chaotic band.
NR = 160
NX = 100
KEEP = 240
# the transient discarded in each column of the picture
PICTURE_TRANSIENT = 400.0
R_LO = 2.8
visits = zeros(NX, NR)
for i to NR {
    r = R_LO + (4.0 - R_LO) * i / (NR - 1.0)
    x = 0.3
    for s to PICTURE_TRANSIENT {
        x = r * x * (1.0 - x)
    }
    for k to KEEP {
        x = r * x * (1.0 - x)
        row = NX - 1.0 - min(floor(x * NX), NX - 1.0)
        visits[row, i] = min(visits[row, i] + 1.0, 8.0)
    }
}

# the two-cycle at r = 3.2, reached by iterating, against its closed form
R2 = 3.2
# enough iterations for the orbit to settle on the two-cycle to the last digit
SETTLE = 2000.0
x = 0.3
for s to SETTLE {
    x = R2 * x * (1.0 - x)
}
reached_a = min(x, R2 * x * (1.0 - x))
reached_b = max(x, R2 * x * (1.0 - x))
root = sqrt((R2 - 3.0) * (R2 + 1.0))
exact_a = (R2 + 1.0 - root) / (2.0 * R2)
exact_b = (R2 + 1.0 + root) / (2.0 * R2)
chaos = lyapunov(4.0, 200000.0)
order = lyapunov(3.2, 20000.0)
show reached_a
show reached_b
show chaos
show order

figure("Attractor of x -> r x (1 - x), by r", "r", "x", 2.8, 4.0, 0.0, 1.0, "pixels")
plot2d(visits, "visits, capped at 8")

test "at r = 3.2 the orbit lands on the closed-form two-cycle" {
    assert_near(reached_a, exact_a, 1.0e-12)
    assert_near(reached_b, exact_b, 1.0e-12)
}
test "there the Lyapunov exponent is negative: nearby starts converge" {
    assert_lt(order, 0.0)
}
test "at r = 4 it is ln 2" {
    assert_near(chaos, log(2.0), 5.0e-3)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

# iterations discarded before the orbit is on its attractor: the transient
TRANSIENT = 500.0

#| The Lyapunov exponent at r: the orbit average of ln |f'(x)| = ln |r (1 - 2x)|.
def lyapunov(r, steps) {
    x = 0.3
    for s to TRANSIENT {
        x = r * x * (1.0 - x)
    }
    total = 0.0
    for s to steps {
        x = r * x * (1.0 - x)
        total = total + log(abs(r * (1.0 - 2.0 * x)))
    }
    return total / steps
}

# the attractor: for each r (a column), discard a transient, then count where the
# orbit goes (the rows, x = 1 at the top). Counts are capped so the two-cycle's
# two hot cells do not wash out the chaotic band.
NR = 160
NX = 100
KEEP = 240
# the transient discarded in each column of the picture
PICTURE_TRANSIENT = 400.0
R_LO = 2.8
visits = zeros(NX, NR)
for i to NR {
    r = R_LO + (4.0 - R_LO) * i / (NR - 1.0)
    x = 0.3
    for s to PICTURE_TRANSIENT {
        x = r * x * (1.0 - x)
    }
    for k to KEEP {
        x = r * x * (1.0 - x)
        row = NX - 1.0 - min(floor(x * NX), NX - 1.0)
        visits[row, i] = min(visits[row, i] + 1.0, 8.0)
    }
}

# the two-cycle at r = 3.2, reached by iterating, against its closed form
R2 = 3.2
# enough iterations for the orbit to settle on the two-cycle to the last digit
SETTLE = 2000.0
x = 0.3
for s to SETTLE {
    x = R2 * x * (1.0 - x)
}
reached_a = min(x, R2 * x * (1.0 - x))
reached_b = max(x, R2 * x * (1.0 - x))
root = sqrt((R2 - 3.0) * (R2 + 1.0))
exact_a = (R2 + 1.0 - root) / (2.0 * R2)
exact_b = (R2 + 1.0 + root) / (2.0 * R2)
chaos = lyapunov(4.0, 200000.0)
order = lyapunov(3.2, 20000.0)
show reached_a
show reached_b
show chaos
show order

figure("Attractor of x -> r x (1 - x), by r", "r", "x", 2.8, 4.0, 0.0, 1.0, "pixels")
plot2d(visits, "visits, capped at 8")

test "at r = 3.2 the orbit lands on the closed-form two-cycle" {
    assert_near(reached_a, exact_a, 1.0e-12)
    assert_near(reached_b, exact_b, 1.0e-12)
}
test "there the Lyapunov exponent is negative: nearby starts converge" {
    assert_lt(order, 0.0)
}
test "at r = 4 it is ln 2" {
    assert_near(chaos, log(2.0), 5.0e-3)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors