Proposed in 1976 by Otto Rössler as a simpler analogue of the Lorenz system, the Rössler attractor requires only one quadratic nonlinearity yet produces full deterministic chaos — an infinite spiral that never retraces itself.
Three differential equations (ẋ = −y−z, ẏ = x+ay, ż = b+z(x−c)) generate the attractor. For a = 0.2, b = 0.2, c = 5.7 the system is chaotic. The attractor has a characteristic band-shaped topology different from Lorenz's double-lobed butterfly.
Rotate the 3D attractor by dragging. Adjust parameters a, b, c to move through period-doubling to chaos. Launch multiple trajectories from nearby points to watch exponential divergence — the hallmark of chaos.
Rössler designed the system while studying chemical reaction oscillators. It later appeared in models of cardiac arrhythmia and the circadian clock. The Rössler band is topologically equivalent to a Möbius strip — one-sided and non-orientable.
This simulation traces the Rössler attractor, a chaotic solution of three coupled ordinary differential equations introduced by Otto Rössler in 1976: dx/dt = −y − z, dy/dt = x + ay, and dz/dt = b + z(x − c). With just a single quadratic nonlinearity, the system produces deterministic chaos. The trajectory is advanced numerically using the forward Euler method with a fixed time step of dt = 0.005, plotting each successive point as a continuous coloured line in 3D.
Sliders set the three parameters a, b and c, while the Trails slider launches up to twelve trajectories from nearly identical starting points so you can watch them diverge — the signature of chaos. Reset restarts the curves, and Auto-rotate spins the camera; dragging orbits the view. The Rössler band illustrates how simple, fully deterministic rules can yield long-term unpredictability, a principle that underpins weather modelling, chemical oscillators and nonlinear dynamics.
What is the Rössler attractor?
It is a strange attractor produced by a set of three differential equations devised by Otto Rössler in 1976. As the equations are integrated forward in time, the trajectory settles onto a folded, band-like surface in three dimensions that it never exactly retraces, making it a classic example of chaos.
What equations does this simulation solve?
It solves dx/dt = −y − z, dy/dt = x + ay, and dz/dt = b + z(x − c). Only the final term, z(x − c), is nonlinear. This minimal structure is why the Rössler system is often used as a simpler companion to the Lorenz equations when teaching chaos theory.
What do the a, b and c sliders do?
They set the three constants in the equations. Here a ranges from 0.05 to 0.5, b from 0.05 to 0.5, and c from 2 to 10. The classic chaotic values are a = 0.2, b = 0.2 and c = 5.7. Changing them moves the system through periodic cycles, period-doubling and chaos.
The Trails slider sets how many trajectories run at once, from 1 to 12. Each starts from very slightly different initial conditions, offset by about 0.01 in x and y. Because nearby paths separate exponentially in a chaotic system, the trails fan apart over time, visually demonstrating sensitive dependence on initial conditions.
The simulation uses the forward Euler method: at each step the derivatives are evaluated and multiplied by a fixed time step dt = 0.005, then added to the current position. Ten such steps are taken per animation frame, so the curve grows smoothly while keeping the integration cheap enough to run in real time in your browser.
Forward Euler is a first-order method, so it introduces some numerical error and is less accurate than Runge–Kutta schemes. For visualisation the small step of 0.005 keeps the attractor's overall shape faithful, but because chaotic systems amplify any error, the exact point-by-point path here will diverge from a higher-precision solution over long runs.
Both are three-variable chaotic systems, but Lorenz has two quadratic nonlinearities and forms a symmetric two-lobed butterfly. Rössler has only one nonlinear term and folds into a single spiral band, making it structurally simpler. Rössler designed it specifically to be the minimal system capable of producing Lorenz-like chaos.
In the x–y plane the trajectory spirals outward as the linear terms push it away from the origin. When x grows large enough, the z(x − c) term switches on, lifting the path up in z and folding it back toward the centre. This repeated stretch-and-fold action is the geometric mechanism that generates chaos.
A strange attractor is a bounded region of state space that nearby trajectories are drawn towards, yet within which motion never settles into a fixed point or simple loop. It typically has a fractal structure, meaning detail at every scale. The Rössler band is one of the most studied examples.
Otto Rössler originally formulated it while studying chemical reaction oscillators. Similar dynamics have since been used to model cardiac rhythms, electronic circuits and the kinetics of certain reactions. More broadly, it serves as a teaching benchmark for chaos, bifurcation analysis and nonlinear control.