I think you'd have to write a sorting routine for the computed eigenvalues because for each gain they may not be in the same order as the previous gain. For large steps between gains, I'm not sure if you can sort them, but for small gain steps you can for most cases. It isn't trivial for all bifurcation points though.

I think this issue is related with this line:

# Now plot the loci
for index, col in enumerate(mymat.T):
    ax.plot(real(col), imag(col), plotstr, label='rootlocus')`

where col seems to be each branch. Indeed, if plotstr input is modified when running the command, all the branches are plotted similarly. Its default behaviour introduces:

# Default values for module parameters
_rlocus_defaults = {
    'rlocus.grid':True,
    'rlocus.plotstr':'b' if int(matplotlib.__version__[0]) == 1 else 'C0',
    'rlocus.PrintGain':True,
    'rlocus.Plot':True
}

I think setting that parameter free would color each branch separatelly. I will check later on.

I just tried this and it seems to work.

Replaced:

    'rlocus.plotstr': 'b' if int(mpl.__version__[0]) == 1 else 'C0',

with:

    'rlocus.plotstr': '' if int(mpl.__version__[0]) == 1 else '',

Test script:

system = control.tf([1], [1, 8, 30, 56, 65])
control.root_locus(system)

Result before:

Result after:

Would be nice to change the colour of the crosses too.

I'm not sure what the benefit of the different colours is though. Without a legend or other way of knowing what each line is, what is the benefit?

Here's a better example which illustrates the benefit of the coloured lines.

One time when different colours per locus might be a bad idea, is when plotting the root loci of more than one system on the same plot. Is this actually useful or something that we want to support? If so, we could offer both options.