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Initial values, Range, boundary-value problems, 3D, animations and the Kinescope

Initial values

Workspace > Initial values lists each variable's value at the start of a run. Type a value, or set them all at once (XPPAUT's Initial conds menu):

  • From end of last run (I L) starts the next run where the last one ended. From end, time goes on (I S, Shift) also makes the start time T0 the time the run ended, so the clock carries on. It matters when the equations depend on t, as with a periodic forcing.
  • From equilibrium takes the last equilibrium found.
  • From file... (I F) reads one number per variable, in the order of the table, from a text file; spaces, new lines or commas separate them. Save to file... writes the current values exactly, one per line, so they can be read back later.
  • From a formula (I U, formUla) gives a variable the value of a formula in the parameters and the model's functions: V = vk + 0.1. For numbered variables write a range, and [j] in the formula stands for the index: u[0..199] = 0.8*exp(-([j]-100)^2/50) puts a bump in the middle of a 200-cell line (try the Amari model).

In the Phase Plane, Click to start begins a run wherever you click (Mouse). Tick Keep each beside it and every click adds a trajectory while the earlier ones stay (mIce): a few clicks sketch the phase portrait. Erase (E) clears them.

Commands: ic last, ic shift, ic read ics.dat, ic write ics.dat, ic u[0..199] = 0.8*exp(-([j]-100)^2/50), click -0.3 0 (a run from that point of the phase plane), click keep.

Range: many runs over a parameter

Workspace > Range is XPPAUT's Initial conds > Range. It runs the model once for each of Steps values of a parameter, or of a variable's initial value, from From to To:

  • Keep each trajectory overlays the runs on the plots (up to 16).
  • Max / min of a variable records the largest and smallest value of a variable after a transient: a brute-force bifurcation diagram. When the parameter matches the Bifurcation window's, the points appear on AUTO's diagram.
  • Liapunov exponent gives the largest exponent of each run. Where it rises above zero the system is chaotic; try the Lorenz model with r from 20 to 30.
  • Equilibria (no runs) is XPPAUT's Sing pts > Range: at each value Newton's method looks for an equilibrium, starting from the one found at the value before (the first time from the initial values). The plot shows the chosen variable, red where the equilibrium is stable and black where it is not: steady states and their stability without AUTO. When the parameter matches the Bifurcation window's, the dots appear on AUTO's diagram too, and should sit on its steady-state branch.

Start each run where the last ended is XPPAUT's "Use old ICs = No": each run continues from the end of the previous one, so a branch is followed as the parameter moves, instead of every run starting from the initial values. The parameter and initial values are put back afterwards.

Command: range iapp 0 0.2 40 V (max and min of V), range r 20 30 30 lyap continue, range V -0.5 0.5 10 (over V's initial value, keeping the runs), range iapp 0 0.3 31 eq V (equilibria). export range.csv range saves the table.

Two values at once

Vary a second value too is XPPAUT's Initial conds > 2par range. Pick the second parameter (or initial value) and its own From and To, then how the two move:

  • On a grid runs every pair: Steps values of the first times Steps values of the second (XPPAUT's "Array"). With Max / min or Liapunov exponent the result is a colour map over the two values, the first across and the second up. Show picks what the colour means for max and min: the swing (max - min, zero where the system settles to an equilibrium), the max or the min. For the Liapunov exponent the colours run from blue (negative) to red (positive), so chaotic regions stand out.
  • Together, along a line moves both at once, Steps runs from (From, From) to (To, To) (XPPAUT's "Crv"); the result is plotted against the first value, as with one.

Keep each trajectory works for up to 16 runs. Start each run where the last ended goes through the grid row by row: the first value changes fastest. Equilibria follow the first value only.

Command: range iapp 0 0.5 21 gca 0.8 1.4 13 V (a 21 by 13 grid), range iapp 0 0.5 30 gca 0.8 1.2 lyap (along a line: no second count), range iapp 0 0.3 3 V -0.2 0.2 2 (over a parameter and V's initial value, keeping the runs). export grid.csv range writes one row per run with both values.

Every equilibrium at once

All equilibria in the Phase Plane (or Run > Find all equilibria, or eq all) is XPPAUT's Sing pts > Monte Carlo. Newton's method starts from 100 random points and keeps every distinct equilibrium it reaches; each is marked in the phase plane, filled when stable, and listed in the command window with its kind. The points are spread over the phase plane's window and, for the other variables, over the range the last run covered, widened by a quarter. eq all 500 tries 500 points; export eqs.csv equilibria saves every equilibrium found with its eigenvalues.

In the Morris-Lecar model at its defaults this finds the three equilibria that the nullclines' crossings show: a stable node, a saddle and an unstable focus. The random points come from XPPAUT's generator, which repeats from one session to the next, so the same search finds the same equilibria every time.

Frozen nullclines

Freeze... in the Phase Plane is XPPAUT's Nullcline > Freeze: it keeps nullclines on the plot, drawn faded, while the live ones follow your changes. Freeze them, move a slider, and you see how they shift and where the equilibria go.

  • Freeze these nullclines (N F) keeps the ones for the current values.
  • For several values of a parameter (Freeze > Range) gives the parameter each of Values values from From to To in turn and keeps the nullclines for each; the parameter is put back afterwards. In the Morris-Lecar model, iapp from 0 to 0.4 lifts the V nullcline step by step while the W nullcline stays put.
  • Show picks one set to show alone, with its value in the legend, and Play steps through them (Freeze > Animate). Delete them clears the list. Choosing other variables for the axes clears it too: frozen nullclines belong to their plane.

Export... > Nullclines as CSV... (XPPAUT's Nullcline > Save) writes one row per line segment: the set (0 for the nullclines shown, then 1, 2, ... for those frozen), which nullcline (1 for the horizontal variable's, 2 for the vertical one's, as XPPAUT numbers them) and the segment's ends x1, y1, x2, y2. SVG figures include the frozen nullclines.

Commands: nullclines freeze, nullclines freeze iapp 0 0.4 6, nullclines show 3 (0: all), nullclines play, nullclines unfreeze, export nc.csv nullclines.

Boundary-value problems

A model with bndry statements is a boundary-value problem: instead of all the initial values, it gives as many conditions as there are equations, at the start (plain names) or at the end (primed names) of the interval from T0 to T0 + Total. For example, the cable equation

v'=vx
vx'=v/lam2
b v-1        # v = 1 at the start
b v'         # v = 0 at the end

Run > Boundary-value problem, the Solve BVP toolbar button, or the command bvp solve it by XPPAUT's shooting method. Starting from the current initial values as a guess, it adjusts them until the conditions hold. The solution becomes the run, and its initial values replace the old ones, so the next solve starts from it. Open the cable model and press Solve BVP to try it.

If it does not converge, give a better guess for the initial values. From Python, m.bvp(max_iterations=..., tolerance=...) also sets XPPAUT's shooting limits.

The 3D view

View > 3D View shows three columns of the run (and any kept curves) in a box. Drag to turn it, scroll to zoom, double-click to reset the zoom. Tick Spin to have it turn slowly by itself. Choose the columns with x, y and z.

The angles are XPPAUT's: theta turns the box about the vertical axis, and phi tilts it, with 90 for a side view and 0 looking straight down. A model that asks for a 3D plot (@ xplot=x, yplot=y, zplot=z, axes=3d, phi=60, and xmin= ... zmax= for the box) opens with its view, as in XPPAUT; Reset returns to it. Try the Lorenz model.

Commands: view3d X Y Z, view3d spin (view3d spin off), view3d angles THETA PHI.

Animations

XPPAUT animations are .ani files: lines, circles, rectangles, text and comets (points with a trail) whose positions and colours are formulas of the variables. The Animator (View > Animator) plays one over the run: Play, the row slider, speed in rows per second and Loop. When a model has an animation of the same name next to it, XPP Lab loads it and shows an Animate button; open another with Open .ani....

Open the double pendulum, run it (run) and press Animate. Other models with animations include pend, invpend, elaspen, duck and the roller coasters.

A small example, pend.ani:

# the rod and the bob of a pendulum of angle x
line .5;.5;.5+.4*sin(x);.5-.4*cos(x);$BLACK;3
fcircle .5+.4*sin(x);.5-.4*cos(x);.05;$RED
end

Coordinates run from 0 to 1 unless a dimension x1;y1;x2;y2 line says otherwise. Commands are evaluated at every frame, except those after a permanent line, which are drawn once. Colours are $BLACK, $RED, $ORANGE, $YELLOW, $GREEN, $BLUE, $PURPLE and the others in between, or a formula from 0 to 1 along the colour scale.

Commands: ani pend.ani, ani play, ani row 200.

The Kinescope

The Kinescope (View > Kinescope, or Run > Kinescope...) is XPPAUT's film strip: it keeps pictures of a figure exactly as it is on the screen, plays them back and saves them as an animated GIF for a talk, a web page or a paper's supplement.

  • Choose the figure (Phase Plane, Time Series, Bifurcation or the two-parameter diagram) and press Capture (K C) whenever it shows something worth keeping: change a parameter, run, capture again.
  • Play (K A) steps through the frames at the speed set beside it, in a loop if Loop is ticked; the arrows step by hand, as XPPAUT's Playback does. Delete all (K R) empties the strip.
  • Save animated GIF... writes every frame into one GIF (XPPAUT's Make AniGif), with the delay set by the speed. Save frames as PNG... writes NAME_001.png, NAME_002.png, ... (XPPAUT's Save).

Workspace > Range > Film it in the Kinescope is XPPAUT's Range with Movie set to Yes: the model runs for each value in turn, each run drawn alone, and the Kinescope captures its figure after every run. Keep that figure in view while it films. In the Morris-Lecar model, a film of iapp from 0 to 0.3 with the nullclines on shows the V nullcline rising, and the trajectory going from rest to a limit cycle and back to rest as the current grows.

A capture is a picture of the screen, so the figure must be in view: a plot in a tab behind another cannot be captured (nor saved as a PNG). The session's Python script writes a film's runs as m.sweep(...); the pictures themselves are not in the script. In Python, matplotlib's animation module makes the same kind of film from those runs.

Commands: kinescope capture phase, kinescope play 200 (ms a frame), kinescope show 3, kinescope stop, kinescope clear, kinescope save film.gif 200, kinescope frames frame, and range iapp 0 0.3 10 film (or kinescope film, the Range tab's runs).