The Thomas Attractor: Chaos From Three Sine Waves

One parameter, three symmetric equations

The Thomas attractor, introduced by physical chemist Rene Thomas in 1999, is one of the simplest-looking chaotic systems ever written down. Its three coupled ODEs are cyclic in x, y, z -- rotate the three variables and the equations map onto themselves -- and every nonlinearity is a single sine function:

x' = sin(y) - b*x
y' = sin(z) - b*y
z' = sin(x) - b*z

There is exactly one free parameter, b, which plays the role of a linear damping/dissipation coefficient. Because the system is invariant under cyclically permuting (x, y, z), the resulting attractor -- when it exists -- has a threefold rotational symmetry: it looks the same after relabelling the axes, which gives Thomas-attractor renders their distinctive braided, lattice-like structure rather than the two-winged look of the Lorenz attractor.

Dissipation and why trajectories don't fly apart

Whether a flow like this contracts phase-space volume, and hence can support a bounded attractor at all, is governed by the divergence of the vector field. Here div(F) = -3b, a constant, independent of position. For b > 0 the divergence is uniformly negative: every infinitesimal blob of nearby initial conditions shrinks in volume as it evolves, so trajectories cannot spread out to infinity, but they also cannot generically converge to a single point unless a fixed point is actually stable there. That combination -- volume-contracting overall, yet not collapsing to a point -- is exactly the geometric signature of a strange attractor: an infinitely folded, fractal object of zero volume that trajectories are drawn onto and wander across forever without ever exactly repeating.

From fixed point to chaos as b shrinks

The single knob b sweeps the system through a textbook route into chaos. At large b (strong damping, roughly b > 1), the origin and any nearby fixed points are stable and everything spirals down to rest -- boring, but a useful sanity check. As b is lowered, the fixed points lose stability through a Hopf bifurcation and a stable limit cycle appears -- a single closed periodic loop. Lower b further, past roughly b ~ 0.32-0.20 depending on exactly where you look, and the periodic orbit undergoes a cascade of period-doubling bifurcations, each doubling the number of loops before the trajectory repeats, accumulating faster and faster until the period becomes effectively infinite and the motion never exactly repeats -- full chaos. The commonly cited "classic" chaotic regime sits around b ~ 0.19-0.20, which is the default most renders (including this site's) use because it produces the fullest, most recognisable lattice structure.

b ~ 1.0    -> stable fixed point, trajectory spirals to rest
b ~ 0.5    -> stable limit cycle, one closed loop
b ~ 0.32   -> period-doubling cascade begins
b ~ 0.19   -> fully chaotic, fractal strange attractor

Rendering it: RK4 and slow accumulation

Because the vector field only involves sine and a linear term, it is cheap to evaluate, so a fourth-order Runge-Kutta step at a small fixed dt (0.01-0.05) tracks the trajectory accurately over the tens of thousands of steps needed for the attractor's fractal structure to fill in visually. Unlike a map like the Barnsley fern, there is no closed-form shortcut -- every point on the attractor is only reachable by actually integrating the ODE forward from a starting point, which is why these renders always show one continuously growing trail rather than a scattering of independently placed dots. Colour is typically mapped to instantaneous speed or to time-since-start, which is what turns the otherwise monochrome lattice into a visually readable braid showing which regions the trajectory revisits often.

Frequently asked questions

Try it live

Everything above runs in your browser — open Thomas Attractor and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.