11.3. Plotting of eigenfunctions
When investigating the stability of stationary solutions by bifurcation analysis, one is frequently interested in the eigenfunction corresponding to the critical eigenvalue which real value passes the zero. To plot these functions, we can directly use the very same plotting class, but just pass a few additional arguments to the constructor.
As illustration, let us consider the problem of Section 5.6.3, which depends on the scripts kuramoto_sivanshinsky_bifurcation.py, kuramoto_sivanshinsky_arclength_eigen.py and kuramoto_sivanshinsky.py. The plotting class can be rather short, since we only have to plot the height field. However, we directly add the functionality for the plots of the eigenfunction:
class KSEPlotter(MatplotlibPlotter):
def define_plot(self):
cb=self.add_colorbar("h",position="top center") # Add a color bar, but do not show it
cb.invisible=True
self.add_plot("domain/h",colorbar=cb) # plot the height field (or its eigenfunction)
# Parameters as text
self.add_text(r"$\gamma={:5.4f}, \delta={:5.4f}$".format(self.get_problem().param_gamma.value,self.get_problem().param_delta.value), position="bottom left",textsize=15)
# Information text, based on whether we plot the eigenfunction or the normal solution
if self.eigenvector is not None: # eigenfunction is plotted
self.add_text("eigenvalue={:5.4f}".format(self.get_eigenvalue()),position="bottom right",textsize=15) # eigenvalue (will be 0 anyhow)
if self.eigenmode=="abs":
self.add_text("eigenfunction magnitude",textsize=20,position="top center") # title
elif self.eigenmode=="real":
self.add_text("eigenfunction (real)", textsize=20, position="top center") # title
elif self.eigenmode == "imag":
self.add_text("eigenfunction (imag.)", textsize=20, position="top center") # title
else:
self.add_text("height field", textsize=20, position="top center") # title
The plotting is done as before, via a color bar (which is hidden here by setting invisible) and add_plot() of the height field. With the add_text(), the parameters are added. Depending on whether we want to plot an eigenfunction or the normal solution, we add additional text to the plot. If an eigenfunction is plotted, eigenvector will refer to the index of the eigenfunction, whereas it will be None if no eigenfunction (i.e. the normal solution) is about to be plotted. With get_eigenvalue() we can get the eigenvalue corresponding to the plotted eigenfunction. eigenmode can hold different modes to plot the eigenfunction. Since it is in general complex, we have to choose whether we want to plot the "real" part, the "imag" part, the "abs" magnitude or the "angle" (which is the phase).
To use this plotting class for the normal solution and different eigenfunction plots, we just have to create an instance of this plotter for each desired plot:
if __name__ == "__main__":
with KSEBifurcationProblem() as problem:
# Add a bunch of plotters
problem.plotter=[KSEPlotter(problem)] # Plot the normal solution
problem.plotter.append(KSEPlotter(problem,eigenvector=0,eigenmode="real",filetrunk="eigenreal_{:05d}")) # real part of the eigenfunction
problem.plotter.append(KSEPlotter(problem, eigenvector=0, eigenmode="imag", filetrunk="eigenimag_{:05d}")) # imag. part
problem.plotter.append(KSEPlotter(problem, eigenvector=0, eigenmode="abs", filetrunk="eigenabs_{:05d}")) # magnitude
for p in problem.plotter:
p.file_ext=["png","pdf"] # plot both png and pdf for all plotters
# ...
# the rest is the same
# ...
The plotter property is now a list containing multiple plotters. Without further arguments, the plotter will plot the normal solution. If eigenvector is set, it indexes the desired eigenfunction to be plotted and eigenmode selects the desired mode (see above). Since all plots would write to the same file, we also have to specify filetrunk, which takes a format string, which is formatted based on the output step. All plotters are instructed to plot both pdf and png files.
Some plots are depicted in Fig. 11.1. As expected, the critical eigenfunction is only real valued an similar to the solution itself, i.e. the collapse to the flat solution beyond the fold bifurcation is apparent.
Fig. 11.1 Critical solution and corresponding eigenfunction at the fold bifurcation of the damped Kuramoto-Sivashinsky equation (5.7).