Dynamics and chaos in Logos 3
Maps and flows: the road to chaos and Feigenbaum’s δ, a strange attractor, the Lorenz butterfly cut by a Poincaré section, and predator–prey cycles — each built live, each number computed.
Logos 3 steps maps and integrates flows, and then reads what they do: where they settle, whether they repeat, how fast nearby starts separate. This page is 5 examples, each with the words that ask for it, what the engine builds, what to look at, and the numbers it computes — from the road to chaos to the Lorenz butterfly cut open.
How to read these#
- Ask
- The words to type into the Logos 3 conversation. Logos reads them and proposes a model; the engine checks it, and refuses it — naming what is missing — rather than inventing a value.
- Views
- A map offers its Iterates, a Cobweb when it has one state, and a Bifurcation diagram across the range of a control. A flow offers its Path, Phase portrait and Against time, and — with three or more states and no clock in its rates — a Poincaré section.
- Computed
- The engine’s own arithmetic on the model as this page was built. Periods are read off the orbit; Lyapunov exponents come from products of Jacobians along it; Feigenbaum’s δ comes from the map’s own superstable cycles, found one by one. Each is checked in Socria’s tests against what is known.
Maps and chaos#
Systems that step — once a generation, once a cycle. Their pictures are the cobweb and the bifurcation diagram, and the numbers that say what they do are a period, a Lyapunov exponent, and Feigenbaum’s δ.
The road to chaos
AskA population grows by a factor r each generation but is held back by crowding: x′ = r·x(1 − x). What happens to it as r rises from 2.5 to 4?
- What it builds
- A map — x at the next step from x now — stepped from x = 0.2, with r as a control. Its bifurcation diagram sweeps r across the control’s range: at each of 400 values, 400 steps to settle and 96 kept.
- What to look at
- The bifurcation diagram: one value, then two, four, eight — the period doubling faster and faster — and then chaos, broken by windows of order. The dashed line is r now; the Cobweb shows the same orbit as a walk between the graph and the diagonal.
- Computed
- At r = 3.5 it settles into a cycle of period 4; largest Lyapunov exponent −0.8725 per step. Its cycles of period 1, 2, 4, 8 are superstable at r = 2, 3.23607, 3.49856, 3.55464 — the second is 1 + √5 — and the ratio of successive gaps reaches 4.6692: Feigenbaum’s δ, 4.6692.
The same cascade in another map
AskTake a different hump — x′ = μ·sin(πx) — and sweep μ. Does it go to chaos the same way?
- What it builds
- The sine map, stepped and swept exactly as the logistic map was. Nothing about the logistic map is in it.
- What to look at
- Its bifurcation diagram looks like the logistic map’s, and the ratio of its gaps is the same number. That sameness — universality — is why δ is a constant of nature rather than a property of one equation.
- Computed
- Its superstable cycles sit at μ = 0.5, 0.77773, 0.84638, 0.86145, …, and the ratio of the gaps between them reaches 4.669 — the logistic map’s δ, from a map that shares nothing with it but a single hump.
A strange attractor
AskShow the Hénon map, x′ = 1 − a·x² + y, y′ = b·x, at a = 1.4, b = 0.3.
- What it builds
- A two-state map stepped 4000 times from the origin; its Lyapunov exponents from products of Jacobians along the orbit.
- What to look at
- The iterates trace a curved, folded band — zoom in and it is bands within bands. Nearby starts separate (one exponent positive) while areas shrink (their sum is ln b).
- Computed
- Its Lyapunov exponents are 0.418 and −1.62 per step: one positive, so nearby starts separate; their sum, −1.204, is ln b = −1.204, because every step multiplies area by b.
Flows and attractors#
Systems that flow, and the maps hidden inside them: fixed points with their eigenvalues, closed orbits, and the Poincaré section of a strange attractor.
The Lorenz butterfly, cut
AskIntegrate the Lorenz system — σ = 10, ρ = 28, β = 8/3 — and show where its path crosses a plane.
- What it builds
- Three states and their rates, integrated by RK4; its fixed points and their eigenvalues; and the Poincaré section — every upward crossing of z through its mean.
- What to look at
- The Path is the butterfly in three dimensions. The Poincaré section turns the flow into a map: the crossings fall on thin curves, never repeating. The Inspector lists the three fixed points and their eigenvalues.
- Computed
- The engine finds 3 fixed points; the two wings’ centres sit at (±√(β(ρ − 1)), ±√(β(ρ − 1)), ρ − 1) = (±8.485, ±8.485, 27), found at (8.485, 8.485, 27) with eigenvalues 0.094 + 10.2i, 0.094 − 10.2i, −13.9. The path pierces its plane 597 times in the section.
Predator and prey
AskRabbits grow at 0.6 a year and are eaten at 0.3 per fox; foxes die at 1 a year and grow at a third per rabbit eaten. Show the cycles.
- What it builds
- Lotka–Volterra as two states and their rates, integrated by RK4; its phase portrait with nullclines and its fixed points.
- What to look at
- The Phase portrait: closed orbits around a centre, every one a cycle of boom and bust. The Inspector finds the centre at (d/c, a/b) with eigenvalues ±i√(ad).
- Computed
- The engine finds a centre at (3, 2) with eigenvalues 0 + 0.775i, 0 − 0.775i; theory puts it at (d/c, a/b) = (3, 2) with ±i√(ad) = ±0.7746i.
Where it stops#
- Everything is numerical. Fixed points are found by Newton’s method from many starts in the region the orbit visits; one outside it is not looked for.
- A bifurcation diagram sweeps one control across its own range, carrying each column’s end into the next. Near a bifurcation the orbit settles slowly, and a column there can look undecided — the period is only counted where its values are visibly apart.
- A Poincaré section is taken at one plane: the third state crossing its mean, upward. Another plane is not chosen for you yet.
Fields — heat along a rod, a reaction front, a shock, patterns on a plane — are on Engineering in Logos 3.