Separatrix

An iPad app

Separatrix

An iPad app for exploring dynamical systems written as equations: flows and maps, Lorenz, Rössler, the logistic map and many more. Watch them on scopes, turn their parameters on knobs, compare two runs, analyse them, write your own.

For a system, the app can generate its analog computer patch as a PDF for The Analog Thing (THAT), anabrid’s analog computer.

A scope showing the Lorenz attractor (sigma 10, rho 28, beta 8/3), x against z: an orange trace in two wings, like a butterfly, on a graticule. Lorenz · σ 10 · ρ 28 · β 8/3

Run

Watch a system run, and turn its parameters while it does.

Up to three scopes show it at once: a time scope with a lane for each state variable, and phase plots in 2D and 3D. Every parameter has a knob you turn by touch, with the Pencil or with a MIDI controller, and so do the speed and the window.

Sound is off until you switch it on; then the system modulates the pitch or the amplitude of a plain oscillator.

A time scope with two lanes showing a Van der Pol oscillator against time: x in orange, a relaxation oscillation with sharp jumps, and y in teal, a smoother wave. Van der Pol · μ 4 · x and y against t

Compare

Run the same system twice, side by side.

Compare shows two traces of one system at once, each with its own knob values, its own initial state, or both. In a chaotic system a tiny difference between the two starts grows until the runs part: the butterfly effect, which you can set up and watch.

A time scope showing the Lorenz system's x against time for two runs whose starts differ by one part in a billion: the orange and teal traces lie on top of each other at first, then part and go different ways. Lorenz · two starts 1e-9 apart

Analysis

See what a system does as a parameter changes.

The bifurcation diagram sweeps a parameter and shows where the behaviour splits, settles or turns chaotic; tap it to set the parameter there. The Poincaré section and the Lyapunov exponents sit beside it, and every view recomputes by itself when you change something.

A bifurcation diagram of the logistic map: the long-run values of x against the parameter r from 2.8 to 4, a single line that splits into two, four, and more branches, and then into a band of chaos with narrow windows of order. Logistic map · r from 2.8 to 4

Build

Write your own system as equations.

Pick the symbols from keys and type numbers on the app’s own keypad. A system is a flow, written with a dot over the variable, or a map, xₙ₊₁ = f(xₙ), with up to 16 state variables and 16 parameters.

Your systems are kept in My systems on the iPad.

Built-in systems

39 systems come with the app.

Oscillators, driven systems, attractors and maps, from the classics of the field. Their Ref keys open the original article by its DOI, or Strogatz’s Nonlinear Dynamics and Chaos.

Each has a tour: steps with a key that loads a preset or sets a parameter and starts the run again, so its behaviours come one at a time.

The Rössler attractor, x against y: an orange spiral that loops outwards and folds back.
Rössler
The Hénon map's attractor, x against y: points along folded, curved bands.
Hénon map
Thomas' cyclically symmetric attractor, x against y: an orange tangle of loops filling a square-ish region.
Thomas

Discover

More systems, from the catalog online.

Discover, in the gallery, lists 28 rarer systems. Tap Download and one is downloaded into My systems, where it is an ordinary system you can run and edit. Discover is part of the trial and the Full Version. You can browse the catalog in a browser too.

The Tinkerbell map's attractor, x against y: a tangle of curved bands of points.
Tinkerbell map
The gingerbreadman map, x against y: points scattered over a gingerbread-man-shaped region.
Gingerbreadman map
The Lü system, x against z: an orange double-scroll trace.
Lü system

Analog computer patch

Take a system to an analog computer.

For a system, the app can generate its analog computer patch as a PDF for The Analog Thing (THAT), anabrid’s analog computer, and open it in the system’s viewer.

A block diagram of the equation x double dot equals minus x: a first integrator turns the second derivative into the first derivative, a second integrator turns that into x, and x returns through a sign change to the start. ẍẋx ∫∫ −1 ẍ = −x as integrators in a loop

Try it

Fourteen days to try everything.

The trial starts when you first install the app. After it, the logistic map, the Van der Pol oscillator and the Lorenz system stay open; the Full Version, a one-time purchase through the App Store, unlocks every system, your own and Discover.

Questions

Which iPads does it run on?

iPads with iPadOS 27 or later. Separatrix is an iPad app.

Do I need an account?

No. There is no account and no sign-in, and the app collects no data; see the privacy policy.

Does it need the internet?

Only Discover does. Everything else works without a connection.

Where are my systems kept?

On your iPad, in My systems. A system is one file with the extension .separatrix, a plain text file you can share through the Files app, AirDrop or Mail.

Can I use a MIDI controller?

Yes, one with endless encoders: tap MIDI learn, tap a control on screen and move a knob. It works while Separatrix is on screen.