The Rössler Attractor: One of the Simplest Chaotic Flows

The most famous chaotic system, the Lorenz equations, came from a model of atmospheric convection Lorenz system, Wikipedia^[1] Lorenz 1963^[2]. In 1976 the German scientist Otto Rössler set out to build something simpler: a set of equations that would behave like the Lorenz system but be easier to analyse Rössler attractor, Wikipedia^[3] Rössler 1976^[4]. The result has only one nonlinear term, and it traces one of the clearest pictures of chaos there is. That band is the image on our Rössler Band poster.

What are the Rössler equations?

The Rössler system is three first-order ordinary differential equations for a point (x, y, z) Rössler attractor, Wikipedia^[3]:

dx/dt = −y − z

dy/dt = x + ay

dz/dt = b + z(x − c)

All quantities are dimensionless: this is an abstract model, not a particular machine, so x, y, z and time t carry no units. The constants a, b and c set its behaviour. Rössler studied the chaotic attractor with a = 0.2, b = 0.2 and c = 5.7, and a = b = 0.1, c = 14 has been widely used since Rössler attractor, Wikipedia^[3]. The only nonlinear term is the product zx in the last equation Rössler attractor, Wikipedia^[3].

How does the Rössler attractor work?

Set z = 0 and the first two equations describe a point spiralling outward in the x–y plane, because a > 0 makes the spiral unstable Rössler attractor, Wikipedia^[3]. While x stays below c, z stays small. Once the spiral grows past x ≈ c, the term z(x − c) makes z shoot up. The −z in the first equation then pulls x back down, and the orbit drops near the centre to start spiralling out again Rössler attractor, Wikipedia^[3].

That cycle of spiral out, lift, fold and drop is a stretch-and-fold machine. Neighbouring orbits are pulled apart during the spiral, then folded back on top of each other during the lift. Rössler himself compared it to a taffy-pulling machine Rössler attractor, Wikipedia^[3]. Repeated stretching and folding produces layered bands, similar to a Cantor set, and Rössler showed the attractor combines a normal band and a Möbius strip Rössler attractor, Wikipedia^[3].

Where are the fixed points?

Setting all three derivatives to zero gives two fixed points, with x-coordinates Rössler attractor, Wikipedia^[3]

x_± = (c ± √(c² − 4ab))/2

and y = −x/a, z = x/a. For Rössler’s original parameters, the central fixed point has eigenvalues of about 0.097 ± 1.0i and −5.69 Rössler attractor, Wikipedia^[3]. The complex pair with a positive real part makes orbits spiral outward; the large negative eigenvalue squashes them strongly onto a nearly flat sheet. This mix of instability in some directions and strong attraction in another is typical of strange attractors. Fixed points and their stability also decide where spacecraft can park, as in our guide to Lagrange points and JWST.

How do we know the Rössler attractor is chaotic?

The signature of chaos is sensitive dependence on initial conditions: nearby trajectories separate exponentially, at a rate set by the largest Lyapunov exponent Lyapunov exponent, Wikipedia^[5]. For a = b = 0.2, c = 5.7, our own numerical integration gives a largest exponent of about 0.07 per unit time. That is small, so predictability is lost slowly, over tens of loops rather than one.

The return map makes the order inside the chaos visible. Plot each maximum of x against the next one and the points fall almost on a single curve, so one maximum nearly predicts the next Rössler attractor, Wikipedia^[3]. For the maxima of z, Rössler’s parameters give a single-humped curve Rössler attractor, Wikipedia^[3], the kind of one-dimensional map Feigenbaum studied Feigenbaum constants, Wikipedia^[6].

How does the Rössler system become chaotic?

Hold a = b = 0.1 and increase c. At low c the orbit is periodic, then it quickly becomes chaotic, with windows of periodicity interspersed with chaos Rössler attractor, Wikipedia^[3]. A period-one orbit appears around c = 4 and is never found again, and period-three orbits are found until c = 12 Rössler attractor, Wikipedia^[3]. Our integration shows the path in between: one loop at c = 4, two at c = 6, four at c = 8.5, and chaos soon after. This period-doubling cascade is the route described by Feigenbaum’s universal constant Feigenbaum constants, Wikipedia^[6] Feigenbaum 1978^[7]. The same cascade appears in the forced oscillator on our Duffing's Strange Attractor poster.

The attractor is also packed with unstable periodic orbits: closed loops that the chaotic orbit visits again and again but never settles on Rössler attractor, Wikipedia^[3].

Quick FAQ

Is the Rössler system a model of anything real?

It was designed as a mathematical model, but the equations were later found useful in modelling chemical reactions Rössler attractor, Wikipedia^[3]. Turbulent fluids are much harder, as in our piece on vortex streets and hydraulic jumps.

Rössler or Lorenz: which is simpler?

Rössler’s, by design: one nonlinear term instead of two, and a single band instead of two wings Rössler attractor, Wikipedia^[3] Lorenz system, Wikipedia^[1]. More chaos prints are in our physics posters.

References

1. Wikipedia, “Lorenz system”. https://en.wikipedia.org/wiki/Lorenz_system
2. Edward N. Lorenz (1963), “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences 20, 130–141 (DOI 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2; metadata verified via Crossref). https://doi.org/10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2
3. Wikipedia, “Rössler attractor”. https://en.wikipedia.org/wiki/R%C3%B6ssler_attractor
4. O.E. Rössler (1976), “An equation for continuous chaos,” Physics Letters A 57, 397–398 (DOI 10.1016/0375-9601(76)90101-8; metadata verified via Crossref). https://doi.org/10.1016/0375-9601(76)90101-8
5. Wikipedia, “Lyapunov exponent”. https://en.wikipedia.org/wiki/Lyapunov_exponent
6. Wikipedia, “Feigenbaum constants”. https://en.wikipedia.org/wiki/Feigenbaum_constants
7. Mitchell J. Feigenbaum (1978), “Quantitative universality for a class of nonlinear transformations,” Journal of Statistical Physics 19, 25–52 (DOI 10.1007/BF01020332; metadata verified via Crossref). https://doi.org/10.1007/BF01020332

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.