11.1. The interactive bifurcation GUI

Everything in the sections on arclength continuation, bifurcation tracking and branch switching is done from a script: you write down the continuation, the bifurcation tracking and the branch switching in advance, run it, and look at the result afterwards.

If you do not know anything about a system yet, i.e. when you try to explore the bifurcation diagram of an unfamiliar problem, you might want to have dynamic control over arclength step sizes, refine branches by adding more points, investigate the spectrum at points, plot different observables against the continuation parameter or continue in multiple parameters. Moreover, you also want to select the direction when switching branches, etc, without having to restart your script-based investigation over and over again.

That is what pyoomph’s bifurcation GUI is for. It drives exactly the same machinery the scripted route uses - arclength_continuation(), activate_bifurcation_tracking(), switch_branch(), switch_to_hopf_orbit(), deflated_continuation() - but you can interactively decide when to use which from the data shown on the screen. Already computed data is fully stored: every point is a full state dump, the diagram is written to disk as you go, and restarting the same script picks it up where you left it. When the diagram is finished it can be exported as plain columns for a plotting tool, and the state dumps are there to restart a scripted run from any point you tagged.

A typical boot script for the GUI is just this:

from pyoomph import *
from pyoomph.utils.bifurcation_gui import BifurcationGUI

with MyProblem() as problem:
        problem.setup_for_stability_analysis(analytic_hessian=True)
        gui=BifurcationGUI(problem,"my_parameter")   # which parameter is continued
        gui.neigen=20                                 # eigenvalues per point
        gui.set_initial_observable("domain/my_obs")   # what goes on the vertical axis
        if gui.must_init():        # False when a stored diagram was found, which is loaded instead
                problem.solve()        # whatever it takes to reach the first stationary solution
        gui.start(0.001)           # opens the window; the argument is the initial arclength step

must_init() is what makes the script restartable: on the first run it returns True and the block below it produces the starting solution, on every later run it returns False because the diagram written last time is on disk and is loaded instead. To restart the diagram from scratch, just wipe the output directory.

analytic_hessian=True is required if you intend to classify bifurcations, switch branches or step onto a periodic orbit - all three need the second derivative.

The window that opens has five parts:

• the bifurcation diagram itself, an ordinary matplotlib canvas with its navigation toolbar, in which points and branches can be clicked;

• the field plots beside it, one per plotter the problem carries (plotter, which also accepts a list), redrawn as you move along a branch, so you can see what the solution actually looks like without leaving the window;

• the toolbar at the top with the handful of commands used constantly (Step, Multi, ds up/down/reverse, Find bif, Switch, Delete) and the Abort button on the right, which stops a running sweep between steps;

• the notebook of tabs on the right, which holds the settings and the tables - the step size, the eigenvalue settings, the list of parameters, the spectrum at the current point, the branch tree, and the deflation and orbit settings (see The tabs on the right);

• the log along the bottom and a status bar, which is where a refused command says why.

Every command is reachable three ways - from the menu bar (The menu bar), from a keyboard shortcut (Default keyboard shortcuts), and, for the frequent ones, from the toolbar or a button in a tab. The menus show each command’s shortcut next to it, so the keys can be learned by using the menus.

We will start by constructing the bifurcation diagram of a representative example on the next page.