René Thomas's 1999 attractor is the mathematical equivalent of a marble rolling on a bumpy 3D landscape: driven by the sine of each coordinate, it wanders forever in a ghostly lattice pattern, never repeating, never escaping.
The system ẋ = sin(y) − bx, ẏ = sin(z) − by, ż = sin(x) − bz has cyclic symmetry: it maps to itself under x→y→z→x. For b ≈ 0.208 the attractor is chaotic. The parameter b controls dissipation — near zero the attractor fills 3D space; larger b compresses it to lower dimension.
Drag to rotate the 3D attractor. Adjust b near 0.208 for peak chaos, below 0.1 for hyperchaos, above 0.32 for period-3 orbit. Launch many particles simultaneously from a tight cluster to visualise the Lyapunov divergence.
Thomas designed the system to model the minimal conditions for chaos: a 3D autonomous system needs at least one cubic term or two quadratic terms — or, as he showed, three sine terms. His interest was not physics but the minimal grammar required for biological regulatory networks.
This simulation traces the Thomas attractor, a three-dimensional dissipative dynamical system introduced by René Thomas in 1999. Its evolution is governed by the symmetric equations dx/dt = sin(y) − b·x, dy/dt = sin(z) − b·y, dz/dt = sin(x) − b·z. The page integrates these coupled ordinary differential equations using the fourth-order Runge–Kutta (RK4) method with a fixed time step of 0.005, drawing each trajectory as a coloured ribbon in WebGL.
The single parameter b sets the damping, and its slider spans 0.1 to 0.3; lower values weaken dissipation and push the system towards space-filling chaos, while higher values collapse it onto simpler orbits. A live readout reports the regime, while the Trails slider launches up to twenty independent particles and the Speed slider sets integration steps per frame. Because the equations are invariant under the cyclic swap x→y→z→x, the attractor reveals the minimal grammar of chaos that Thomas studied in biological regulatory networks.
What is the Thomas attractor?
It is a three-dimensional strange attractor proposed by René Thomas in 1999. It belongs to a class of dissipative chaotic systems built entirely from sine functions, producing an intricate, lattice-like trajectory that never repeats and never escapes to infinity.
What equations does this simulation solve?
The system is dx/dt = sin(y) − b·x, dy/dt = sin(z) − b·y, dz/dt = sin(x) − b·z. Each coordinate is driven by the sine of the next, which gives the cyclic symmetry, while the −b·x, −b·y and −b·z terms supply the linear damping that makes the system dissipative.
What does the parameter b control?
The parameter b is the damping or dissipation strength, adjustable here from 0.1 to 0.3. Small values let the trajectory spread out and fill more of three-dimensional space, while larger values compress the motion onto simpler structures such as limit cycles or fixed points.
The page advances each particle with the classic fourth-order Runge–Kutta integrator and a fixed step of 0.005 units. RK4 evaluates the derivative four times per step and combines the results, giving far better accuracy than a simple Euler method for the same step size, which keeps the chaotic orbit faithful over many iterations.
Trails sets how many independent particles are launched at once, from one up to twenty, each starting from a slightly different point and drawn in its own colour. Speed controls how many RK4 integration steps are performed per animation frame, so raising it makes the curves grow and evolve more quickly on screen.
Starting many trails from a tight cluster lets you watch sensitive dependence on initial conditions, the hallmark of chaos. Trajectories that begin almost identical drift apart exponentially over time, a visual demonstration of the positive Lyapunov exponent that defines a chaotic attractor.
The equations are unchanged if you relabel the coordinates in the cycle x→y→z→x. This rotational symmetry means no axis is special, so the attractor looks statistically the same when viewed from the three equivalent directions and tends to form a balanced, threefold structure in space.
It faithfully integrates the published Thomas equations rather than approximating them loosely, so the geometry and the transition between regimes are mathematically correct. The Thomas attractor is an abstract dynamical model rather than a physical object, so accuracy here means fidelity to the differential equations, not to any laboratory experiment.
Chaos appears around b ≈ 0.208, where the attractor displays its richest, space-filling behaviour. As b is lowered the dynamics approach a near-conservative, almost space-filling regime, while increasing b past this range gradually settles the motion into limit cycles and eventually fixed points.
Thomas devised the system to capture the minimal feedback needed for chaos in biological regulatory networks, where genes and proteins activate and inhibit one another in loops. More broadly, such cyclically symmetric attractors are studied in nonlinear dynamics, secure communications and as benchmark problems for testing numerical chaos and synchronisation methods.