Plotting

• Plotting
• • Standard plots using the plot recipe from Plots.jl
  • Specific plotting keyword arguments
  • Choosing Variables
  • Plotting bifurcation diagrams
  • Plotting without the plot recipe
  • Standard plots using the Makie.jl [Experimental]
  • • Example

Standard plots using the plot recipe from Plots.jl

Plotting is provided by calling recipes to Plots.jl. It means that to plot a branch br, you just need to call

#]add Plots # You need to install Plots.jl before your first time using it!
using Plots
plot(br)

where br is a branch computed after a call to br = continuation(...). Plots can be customized using all the keyword arguments provided by Plots.jl. For example, we can change the plotting backend to the GR package and put a title on the plot by doing:

gr()
plot!(br, title = "I have a branch!")

or you can use a scatter plot

scatter(br)

Then to save the plot, use savefig, for example:

savefig("myplot.png")

Specific plotting keyword arguments

The available arguments specific to our plotting methods are

• plotfold = true: plot the fold points with black dots
• putspecialptlegend = true: display the legend corresponding to the bifurcation points
• vars = nothing: see below
• plotstability = true: display the stability of the branch
• plotspecialpoints = true: plot the special (bifurcation) points on the branch
• branchlabel = "fold branch": assign label to a branch which is printed in the legend
• linewidthunstable: set the linewidth for the unstable part of the branch
• linewidthstable: set the linewidth for the stable part of the branch
• plotcirclesbif = false use circles to plot bifurcation points
• applytoX = identity apply transformation applytoX to x-axis
• applytoY = identity apply transformation applytoY to y-axis

If you have several branches br1, br2, you can plot them in the same figure by doing

plot(br1, br2)

in place of

plot(br1)
plot!(br2)

Note that the plot recipes use the parameter axis as xlabel, and the passed variable as ylabel.

Choosing Variables

You can select which variables to plot using the keyword argument vars, for example:

plot(br, vars = (:param, :x))

The available symbols are :x, :param, :itnewton, :itlinear, :ds, :θ, :n_unstable, :n_imag, :stable, :step,... and:

• x if record_from_solution (see continuation) returns a Number.
• x1, x2,... if record_from_solution returns a Tuple.
• the keys of the NamedTuple returned by record_from_solution.

The available symbols are provided by calling propertynames(br.branch).

Plotting bifurcation diagrams

To do this, you just need to call

plot(diagram)

where diagram is a branch computed after a call to diagram = bifurcationdiagram(...). You can use the keywords provided by Plots.jl and the different backends. You can thus call scatter(diagram). In addition to the options for plotting branches (see above), there are specific arguments available for bifurcation diagrams

• code specify the part of the bifurcation diagram to plot. For example code = (1,1,) plots the part after the first branch of the first branch of the root branch.
• level = (-Inf, Inf) restrict the branching level for plotting.

Plotting without the plot recipe

What if you don't want to use Plots.jl? You can define your own plotting functions using the internal fields of br which is of type ContResult. For example, in PyPlot, Gadfly, GR, etc., you can do the following to plot the branch (like the plot recipe plot(br, vars = (:param, :x))):

plot(br.branch.param, br.branch.x)

You can also have access to the stability of the points by using br.stable. More information concerning the fields can be found in ContResult. For example, you can change the color depending on the stability:

col = [stb ? :green : :red for stb in br.stable]
plot(br.param, br.x, color=col)

You can also plot the spectrum at a specific continuation step::Int by calling

# get the eigenvalues
eigvals = br.eig[step].eigenvals

# plot them in the complex plane
scatter(real.(eigvals), imag.(eigvals))

Standard plots using the Makie.jl [Experimental]

Plotting is also provided by calling recipes to Makie.jl. It means that to plot a branch br, you just need to call

#]add GLMakie # You need to install GLMakie.jl before your first time using it!
using GLMakie
BifurcationKit.plot(br)

The keyword arguments to BifurcationKit.plot are the same as decribed above in the page. You can also combine diagrams with BifurcationKit.plot(br1, br2) or use BifurcationKit.plot!(ax, br) to add a branch to an existing plot.

Example

using Revise, GLMakie, BifurcationKit
Makie.inline!(true)
q = 1/0
k = 2
F(x, p) = (@. p + x - x^(k+1)/(k+1))
prob = BifurcationProblem(F, [0.8], 1., (@lens _); record_from_solution = (x,p; k...) -> x[1])
opts = ContinuationPar(dsmax = 0.1, dsmin = 1e-3, ds = -0.001, p_min = -1., p_max = 1.)
br = continuation(prob, PALC(), opts)
BifurcationKit.plot(br)