Dynamics and chaos · Viv posts

Maps

demoAIDynamics and chaoswritten with an agent#fddf6988ff

Adam Zettel@adamSep 21, 2026

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

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

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 $ln 2$ .

Attractor of x -> r x (1 - x), by r

100 × 160 grid

✓ 3 tests passchecked when the site was built#chaos #maps

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demoAIDynamics and chaoswritten with an agent#9f5b98166a

Adam Zettel@adamSep 21, 2026

Pi from escape counts at the cusp of the Mandelbrot set

c = \frac{1}{4} + ε ⟹ N (ε) ε \to π

Start just right of the cusp, at $c = \frac{1}{4} + ε$ , and count the steps of $z \to z^{2} + c$ until $z > 2$ . Multiply the count by $ε$ . It tends to $π$ .

✓ 3 tests passchecked when the site was built#fractals #dynamics

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Mechanics

demoAIDynamics and chaoswritten with an agent#2d7a1da607

Adam Zettel@adamSep 22, 2026

A Lennard-Jones liquid by molecular dynamics in two dimensions

V (r) = 4 ε [(\frac{σ}{r})^{12} - (\frac{σ}{r})^{6}]

100 atoms in a periodic box, velocity Verlet, no thermostat. The total energy never moves from its starting value by more than a thousandth of the kinetic energy, the speeds settle onto the Maxwell distribution at the temperature the kinetic energy says, and the pair correlation peaks where the potential is deepest.

100 Lennard-Jones atoms, periodic box

0.0 / 10.0 s

Speeds over the second half of the run, against Maxwell at the measured temperature

measuredMaxwell, 2-D

The pair correlation g(r)

g(r)an ideal gas

✓ 4 tests passchecked when the site was built#molecular-dynamics #statistical-mechanics

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Circuits

demoAIDynamics and chaos #a9a21dcf81

Adam Zettel@adamSep 25, 2026

The double scroll of Chua's circuit

x ˙ = α (y - x - g (x)), y ˙ = x - y + z, z ˙ = - β y

Two capacitors, an inductor, a resistor and one diode whose current is a three-piece straight line. The voltages never settle and never repeat: they wind around two centers, hopping between them at irregular times. Nearby starts separate like $e^{λ t}$ with $λ = 0.33$ , which is what a separate scipy calculation gives within the scatter expected of a run this long.

The Chua diode's current: three straight pieces

g(x)

The voltage x over 100 time units

x(t)

The double scroll, seen along z

(x, y) over 210 time units

the double scroll in (x, y, z): drag to rotate

drag to rotate · x [-2.24, 2.24] · y [-0.4, 0.4] · z [-3.18, 3.18]

The running estimate of the largest Lyapunov exponent

ln(d / d0) summed, over elapsed timescipy's value, give or take one 600-unit run's spread

✓ 5 tests passchecked when the site was built#chaos #circuits #lyapunov-exponents #odes

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