Rössler System

A minimal chaotic system — simpler than Lorenz, with a clean period-doubling route to chaos.

Origin and Equations

Otto Rössler introduced this system in 1976 as a deliberately engineered analogue of Lorenz — chaotic but with a single nonlinearity (the product \(zx\)). It is arguably the simplest 3D system that exhibits chaos:

\[\dot{x}=-y-z,\qquad \dot{y}=x+ay,\qquad \dot{z}=b+z(x-c).\]

In the \((x,y)\)-plane the first two equations describe a linear spiral that grows when \(a>0\). The third equation injects a threshold: when \(x\) exceeds \(c\), the term \(z(x-c)\) becomes positive and \(z\) grows rapidly, which bends the trajectory back down. The result is a flat spiral that occasionally "reinjects" itself upward through \(z\).

Parameters and the Period-Doubling Cascade

Fixing \(a=b=0.2\) and varying \(c\) gives the textbook period-doubling cascade to chaos:

\(c\lesssim 2.0\): stable equilibrium (no oscillation).
\(c\approx 2.3\): period-1 limit cycle.
\(c\approx 3.3\): period-2 — the cycle closes after two loops.
\(c\approx 4.3\): period-4.
\(c\approx 5.0\): period-8 and beyond — approach to chaos via the Feigenbaum cascade.
\(c=5.7\): fully developed chaotic attractor — the classical Rössler.
There are also periodic windows inside the chaotic range (e.g. period-3 near \(c\approx 5.3\)).

The Feigenbaum constant \(\delta\approx 4.6692\) governing the logistic map also governs this cascade — a deep universality result.

Geometry: Single-Scroll vs Two-Scroll

Unlike Lorenz (two wings, symmetry \(x\mapsto-x\)), the Rössler attractor has no symmetry. It is a single scroll: the trajectory spirals outward in the \((x,y)\)-plane, then occasionally is kicked upward in \(z\), then folds back to the spiral. Topologically it is a Möbius-like band — a stretch-and-fold map that is the 3D analogue of the logistic map's stretch-and-fold interval dynamics.

Numerical Method

All trajectories use 4th-order Runge–Kutta (RK4) with fixed step \(\Delta t=0.01\). Static plots use \(N=\lceil t_\text{end}/\Delta t\rceil\) steps — the default \(t_\text{end}=300\) gives \(N=30{,}000\), long enough for the attractor to be densely traced; \(t_\text{end}\) is adjustable in the Parameters panel. \(c\)-sweep animation frames use a fixed \(N=15{,}000\) (\(t=150\) per frame) so playback stays responsive regardless of the chosen \(t_\text{end}\). The Jet colorbar shows actual simulation time.

How to Use

Adjust \(a,b,c\) and initial conditions using the sliders or by typing in the value boxes.
Click one of the period presets (P1, P2, P4, Chaos) to jump to a classic value of \(c\).
Click Animate \(c\) to precompute the sweep \(c_\text{start}\to c_\text{end}\) and play it back. Click again at any time to stop.
Click Launch Particles to compare two trajectories separated by \(\varepsilon\) in \(x\) — watch the chaotic regime separate them exponentially.
The 3D plot is fully interactive: drag to rotate, scroll to zoom.

\[\begin{cases}\dot{x}=-y-z\\\dot{y}=x+ay\\\dot{z}=b+z(x-c)\end{cases}\]

Set \(c\) to:

Parameters & Initial Conditions

End time \(t_\text{end}\): \(N=t_\text{end}/\Delta t\)

Animate \(c\) (period-doubling sweep)

FPS

\(c_\text{start}\)

\(c_\text{end}\)

Sensitive Dependence

Traj 1: \(\mathbf{x}_1(0)=(x_0,y_0,z_0)\)
Traj 2: \(\mathbf{x}_2(0)=(x_0+\varepsilon,y_0,z_0)\)
In the chaotic regime: \(\|\mathbf{x}_1(t)-\mathbf{x}_2(t)\|\sim\varepsilon e^{\lambda_1 t}\) with \(\lambda_1\approx0.071\) for \(c=5.7\).

\(\varepsilon =\)

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Rössler Attractor