Bifurcations with AUTO¶

model.auto runs XPPAUT's AUTO continuation. It is the same code XPPAUT's AUTO window uses, driven by function calls. For the concepts, see Bifurcation analysis with AUTO.

A Morris-Lecar bifurcation diagram

Steady states¶

m = xs.load("lecar.ode")
m.apply_set("hopf")
d = m.auto.equilibria("iapp", parmax=0.6)

equilibria() first moves the initial values onto a nearby equilibrium with Newton's method, then follows it as iapp changes. It returns the new points as a Diagram.

Special points¶

d.special()          # every labelled HB, LP, BP, PD, TR point
d.special("HB")      # only Hopf points
hb = d.first("HB")
hb.value, hb.parameter, hb.stable
d.label(3)           # the point with label 3

Each point carries multipliers, as XPPAUT's AUTO window stores them: exp(λ) for steady states and the Floquet multipliers for periodic orbits. Either way a point is stable when they lie inside the unit circle. For steady states, point.eigenvalues recovers λ itself. Its real parts are exact, and its imaginary parts are known modulo 2π.

Continuing from a point¶

orbits = m.auto.continue_from(hb, "periodic")        # limit cycles
hopf2 = m.auto.continue_from(hb, "two-parameter", second="gca")
# from a fold (LP) of a model that has one:
# m.auto.continue_from(d.first("LP"), "two-parameter", second="gca")

The actions available depend on the point, as in XPPAUT's menus:

Point	Actions
HB	`periodic`, `two-parameter`, `extend`, `new`
LP	`two-parameter`
BP	`switch`, `extend`, `new`, `two-parameter`
PD	`doubling`, `two-parameter`, `fixed-period`, `extend`
TR	`two-parameter`, `fixed-period`, `extend`
periodic point	`extend`, `fixed-period`
steady-state point	`extend`

An action that does not apply raises a ValueError that lists the ones that do.

Settings¶

m.auto.set(ds=0.01, dsmax=0.05, nmax=500, npr=50, parmin=0, parmax=2, ntst=30)
m.auto.settings          # all of them, with xs.auto.SETTINGS explaining each

A negative ds continues towards smaller parameter values. Settings can also be passed to equilibria() and continue_from() as keyword arguments.

The whole diagram¶

full = m.auto.diagram       # every point computed so far
full.plot("V")              # max/min of V; full.plot("norm") also works
full.to_pandas()            # one row per point
full.branches()             # {branch number: Diagram}
m.auto.reset()              # start over

Plots use XPPAUT's colours: red and black for stable and unstable steady states, green and blue for stable and unstable periodic orbits. Unstable branches are dashed, and special points are labelled.

Saving a diagram, and its orbits¶

m.auto.save("lecar.auto")        # settings, points and AUTO's restart data
m.auto.load("lecar.auto")        # later, or in another session: replaces the diagram
m.auto.continue_from(17, "extend")   # its labelled points carry on as before

sol = m.auto.orbit(6)            # the periodic orbit at point 6, one period
sol.plot("V")

The file is XPPAUT's .auto format (AUTO > File > Save diagram), so XPPAUT and the desktop app open it too. Numbers are written in full, so a loaded diagram is the one saved. orbit() is XPPAUT's Import orbit: the orbit becomes the model's last run, and the analysis methods use it.