Composing the first frame…
The Order Beneath · No.04 · Live plates
Dynamical systems
Plates on differential equations drawn as the flows they make, each one computed in the browser as you watch: a linear flow whose matrix walks a loop through the trace–determinant plane, from a node through a spiral and a centre to a saddle, the Van der Pol oscillator growing a limit cycle out of a circle and stretching it into a relaxation oscillation, a pendulum whose conserved area shears a disc into a filament until, kicked, it opens into chaos, the Lorenz attractor, one orbit, two that part company and a cloud that covers both wings, and the logistic map's road from order to chaos, period doubling after period doubling, with the same picture at every scale. Every plate carries in its corner the same motion in time.
Plate I · 4 movements · 1:00 loop
A differential equation x′=Ax is a field of arrows, and its solutions are the curves that follow them. For a two-by-two matrix the whole picture is decided by its trace τ and its determinant Δ, because the eigenvalues solve λ2−τλ+Δ=0. Here the matrix walks a loop through the trace–determinant plane, drawn in the corner with the parabola Δ=τ2/4 that divides real eigenvalues from complex ones. Every pixel follows the flow upstream, the comets run along the streamlines, and beside the plane the gold orbit's first coordinate is plotted against time.
An ordinary differential equation x′=f(x) attaches an arrow to every point of the plane, and a solution is a curve that follows the arrows. The plates draw the arrows as streamlines, by walking every pixel upstream along them, and draw a few solutions as the bright orbits.
For a linear flow in the plane the trace and the determinant are all there is to know. The sign of the determinant separates saddles from the rest, the parabola Δ=τ2/4 separates nodes from spirals, and the sign of the trace says whether orbits go in or out.
In the plane an orbit that stays in a bounded region and keeps away from fixed points must wind onto a closed orbit: the theorem of Poincaré and Bendixson. Van der Pol's equation traps its orbits in a ring around the repelling origin, so a cycle has to be there.
A Hamiltonian flow preserves area in phase space (Liouville), so it can stretch and fold a region but never shrink it: there are no attracting cycles and no sinks. The standard map keeps the same property, which is why chaos there looks like a sea rather than a strange attractor.
Nothing in the Lorenz equations is random: the same starting point gives the same orbit every time. What makes them unpredictable is that any error, however small, is multiplied by about e0.9 every time unit, so a forecast can only look as far ahead as the logarithm of its precision.
Feigenbaum's δ≈4.669 and α≈−2.503 are the same for the logistic map, the sine map, a dripping tap or a fluid near the onset of turbulence: anything whose return map has a single smooth hump. Mitchell Feigenbaum found them in 1975 on a programmable pocket calculator.