this is why
a) i think cubic map is maybe a tiny bit more useful than logistic for music - it always oscillates
b) you always want to constrain the params for the behavior you find interesting - iād suggest r in [3.2, 4.0) for both of these if you want them to always be āobviously chaoticā
hereās my ālow-resā bifurcation plot for cubic map; you can see that it is exactly like a symmetrically oscillating logistic
matlab
% cubic
Npre = 200;
Nplot = 100;
x = zeros(Nplot,1);
rmin = 2.5;
rmax = 4.0;
rstep = 0.005;
i = 0;
for r = rmin:rstep:rmax
% fake it til you make it...
% using same initial value each time won't capture sign flipping
i = i + 1;
x(1) = 0.5 * (-1)^i;
for n = 1:Npre
x(1) = r*x(1)^3 + (1 - r)*x(1);
end
for n = 1:Nplot-1
x(n+1) = r*x(n)^3 + (1 - r)*x(n);
end
plot(r*ones(Nplot,1), x, '.', 'markersize', 1, 'Color', [0, 0, 0]);
hold on;
end
title('cubic map bifurcation');
xlabel('r'); ylabel('x_n');
set(gca, 'xlim', [rmin rmax]);
hold off;
iād like to do one for henon too, but itās trickier. 2 params so itād have to be a ābifurcation surfaceā ā¦ hard to readā¦ might as well just say that it looks much ānoisierā throughout - periods of different lengths are interspersed across the range of parameters. but IIRC the period is always >=2.
[ed] heh, here is someoneās henon bifurcation from wikipedia, with one parameter fixed b=0.3 (a reasonable value)
the notes on the image give an idea of how non-trivial it is to create a meaningful bifurcation plot in this case (and many others)
also, in practical terms (@sliderule) iād say if you want 1 output, 1 param, that henon oscillator is still useful and interesting if you just fix b=0.3 and take the newest value as output.
oh! for all these deterministic, iterative things, i also think itās kinda important for musical purposes to be able to arbitrarily set the running state at any time. this lets you take advantage of the coolest thing about these processes - which is that they are repeatable.
pedantic point: itās not strictly correct to say that some outputs of these functions are āchaoticā and some are ānot chaotic.ā being āchaoticā is a (somewhat subjectively-defined) property of the system, meaning just that it is a) sensitive to initial conditions, and b) orbits of different lengths lie close together in parameter space. if you wanna refer to output sequences that are ānoisierā you can say that they have a longer period or orbit.
however, I guess itād be reasonable to say, for instance, that above R=3.4 or something the logistic map āis chaoticā (orbits are closely spaced) and below that it is ānot chaoticā cause the orbits are all basically the same. iām still not super comfy with this because it emphasizes the subjectivity too much; your perception of āclosely-spacedā is dependent on scale.
</pedantry>
