Separatrix

What you can do

The View control in the right-hand panel switches between views. A flow offers Phase, Series, Section, 3D, λ, Branch and Parameter plane. A map offers Phase, Series, Cobweb, λ, Orbit, Branch and Parameter plane. A view that doesn't fit the system is grayed out, and the panel says why. Each guide page below links an example project that opens in that view.

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  Phase plane

  The phase plane draws trajectories of a system with two or more states against two of them, with the direction field, the nullclines and the fixed points.

  Example: Rabbits vs sheep, Strogatz §6.4

  Read the guideDownload phase-plane.sepx

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  Time series

  The time series view plots a state against time, t for a flow or n for a map, for every trajectory.

  Example: Van der Pol oscillator, Strogatz §7.1

  Read the guideDownload time-series.sepx

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  Cobweb

  The cobweb view draws the graph of a one-dimensional map x_n+1 = f(x_n), the diagonal x_n+1 = x_n, and the staircase of each orbit between them (Strogatz §10.1).

  Example: Logistic map, Strogatz §10.2

  Read the guideDownload cobweb.sepx

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  3D view

  The 3D view draws trajectories of a flow against three of its states inside a box you can turn and zoom (Strogatz chapter 9).

  Example: Lorenz equations, Strogatz §9.3

  Read the guideDownload 3d-view.sepx

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  Poincaré section

  A Poincaré section shows where a flow's orbits cross a plane, one dot per crossing (Strogatz §8.7).

  Example: Lorenz equations, Strogatz §9.3

  Read the guideDownload poincare-section.sepx

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  Lyapunov spectrum

  The Lyapunov view measures every Lyapunov exponent of a flow or a map along one orbit, and plots the running estimate of each exponent against t or n (Strogatz §9.3 and §10.5).

  Example: Lorenz equations, Strogatz §9.3

  Read the guideDownload lyapunov.sepx

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  Orbit diagram

  The orbit diagram iterates a map at many values of one parameter and draws, above each value, the iterates left after a transient (Strogatz §10.2, Figure 10.2.7).

  Example: Logistic map, Strogatz §10.2

  Read the guideDownload orbit-diagram.sepx

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  Continuation

  The Branch view follows a fixed point, of a flow or of a map, as one parameter changes, and marks the bifurcations it passes.

  Example: Logistic map, Strogatz §10.2

  Read the guideDownload continuation.sepx

Examples

Download a project and open it with Open…, or drop it on the Separatrix window. Each file is a few kilobytes: it holds the equations, the start points and the view, and Separatrix recomputes the rest.

The app also opens on a gallery of 33 examples from chapters 2 to 10 and 12 of Strogatz's Nonlinear Dynamics and Chaos (2nd edition), grouped by chapter. Click a card to start from that system.

Why Separatrix

• Works offline

  Separatrix needs no internet connection after you install it. The solvers and the plots run on your computer.

• LaTeX first

  Copy a pgfplots figure, the system in an align environment, or a booktabs table of fixed points, Lyapunov exponents or bifurcations, ready for your notes. SVG, PNG and CSV too.

• Strogatz's notation

  Type \dot{x} and x_{n+1} as on the board in LaTeX, or x' in plain text. Filled dots are stable fixed points and open circles unstable ones, named in his words: stable spiral, saddle, center.

• Projector mode

  A light theme with type 15% larger and plot lines 1.75 times as thick, so a plot reads from the back of a lecture hall.

• Small project files

  A .sepx file holds what you typed and chose, not computed data, so it is a few kilobytes and travels by email or a course platform.

• Checked against the book

  Three gates check the numbers on every change: Strogatz's chapter 2 to 8 results, published Lyapunov exponents and closed-form bifurcation points. For Lorenz, λ₁ = 0.9051 ± 0.0013 against Sprott's 0.9056.

Start the ten-minute tutorial or read the user guide.