Rössler Attractor: Spiral Chaos and the Simplest Route to a Strange Attractor

The simplest chaotic system

The Rössler system has a single nonlinearity — the product z·x — yet still produces a full strange attractor:

dx/dt = −y − z
dy/dt = x + a·y
dz/dt = b + z·(x − c)
canonical spiral chaos: a = 0.2, b = 0.2, c = 5.7

The mechanism is a three-phase stretch-and-fold: the trajectory spirals outward in the xy-plane (driven by a·y), then "lifts off" once x grows large enough for z·(x−c) to turn positive, and finally the large z pushes dx/dt = −y−z negative, collapsing x and y back near the origin for reinjection. Unlike the Lorenz system's two-lobe butterfly, the Rössler attractor is a single twisted band — topologically simpler and easier to simulate, since its contraction (λ₃ ≈ −5.4) is far less numerically stiff than Lorenz's (λ₃ ≈ −14.6).

live demo · a single spiralling band folding on itself● LIVE

Period-doubling and the Feigenbaum constant

Fixing a = b = 0.2 and sweeping c reveals a textbook period-doubling cascade: a simple closed orbit at c = 2.5 becomes period-2 near c = 3.5, period-4 near c = 4.0, and chaos sets in around c ≈ 4.2, reaching well-developed spiral chaos at the canonical c = 5.7. The ratio between successive bifurcation intervals converges to the Feigenbaum constant δ ≈ 4.669 — the same universal number that governs period-doubling in the logistic map and countless unrelated systems, one of the most striking discoveries in chaos theory (Feigenbaum, 1978).

λ₁ ≈ +0.09   λ₂ ≈ 0.00   λ₃ ≈ −5.39      (canonical parameters)
D_KY = 2 + λ₁/|λ₃| ≈ 2.017   ← barely above 2, thin band-like structure

Push c higher, past ≈6.5, and the attractor undergoes a band-merging crisis into "screw chaos," where the orbit winds multiple times before reinjection and the Poincaré return map becomes multi-modal instead of the simple unimodal map seen at c = 5.7 — a map that closely resembles the logistic map, tying this continuous flow directly back to 1D chaos theory.

Frequently asked questions

What are the Rössler equations?

dx/dt = −y−z, dy/dt = x+a·y, dz/dt = b+z(x−c), with the canonical spiral-chaos parameters a=0.2, b=0.2, c=5.7. The only nonlinearity is the single z·x product in the third equation — Rössler designed it as the simplest possible 3D chaotic system.

How does the Rössler attractor differ from the Lorenz attractor?

The Rössler system has one nonlinear term and a single-lobe twisted-band attractor reached via period-doubling; the Lorenz system has two nonlinear terms and a two-lobe butterfly reached via an abrupt subcritical bifurcation. Rössler's largest Lyapunov exponent (λ₁ ≈ 0.09) is far smaller than Lorenz's (λ₁ ≈ 0.91), so it mixes more slowly and predictably.

What is the Feigenbaum constant and why does it appear here?

As parameter c increases, the Rössler system undergoes a period-doubling cascade toward chaos, and the ratio between successive bifurcation intervals converges to the Feigenbaum constant δ ≈ 4.669. The same universal constant governs period-doubling in the logistic map and countless unrelated chaotic systems.

Try it live

Everything above runs in your browser — open Rössler Attractor and adjust a, b and c in real time to watch the spiral fold, double its period, and tip into chaos. Nothing is installed, nothing is uploaded.

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