Rössler Attractor - Simpler Than Lorenz, Still Chaotic

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Parameters

a 0.20

b 0.20

c 5.70

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The Equations

dx/dt = -y - z

dy/dt = x + ay

dz/dt = b + z(x - c)

Only ONE nonlinearity: z×x in the third equation. That's all it takes for chaos!

Current State

x 0.1

y 0.1

z 0.1

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Otto Rössler (1976)

German biochemist Otto Rössler created this system while modeling chemical reactions. His goal: find the simplest possible equations that produce chaos. The Lorenz system has THREE nonlinearities—Rössler found ONE was enough! Published as "An Equation for Continuous Chaos."

The Spiral and the Kick

Watch the trajectory: it spirals outward in the x-y plane, then suddenly gets "kicked" up in z, reinjected closer to the center, and spirals out again. This mechanism—spiral, kick, reinject—is simpler and easier to analyze than Lorenz's butterfly wings.

Period-Doubling Route to Chaos

Change parameter c: at c≈4 you get a simple periodic orbit. As c increases, the period doubles (2→4→8→16...) until chaos erupts around c≈5.7. This is the same route to chaos as the Logistic Map—Feigenbaum's universal route!

Why "Simpler" Matters

With only one nonlinearity, Rössler's system became a testbed for chaos theory. It helped prove that complex behavior doesn't require complex equations. One product of two variables—that's all you need to generate deterministic chaos!

The Rössler Attractor