Is multifurcation possible?

Question

I have recently read on bifurcation in chaos theory. Now, I have a simple question (which hopefully has a simple answer): is this possible to have a family of functions that can model multifurcation (rather than bifurcation)? If yes, is there a general rule to determine what pattern would a type of function produce? for example, if I have $f(x) = rx(1-x)$ can I provisionally say that it would make a bifurcation pattern rather than, let's say, trifurcation? (if it is possible to have a trifurcation).

Answer 1

Is multifurcation possible?

Yes. An example is the "Mandelbrot iteration" $z_{n+1}=z_n+c$ with $z_0=0$: When you transition $c$ from the main cardioid of $M$ to the "arm" bulb labeled "3" in this image (or to its complex conjugate mirror), the period will triple, i.e. transition 1→3. Image source

Or you could transition $c$ from the "head" (the circle around −1 with label "2") to one of the attached "ears" with label "6" and you get two "trifurcations" 2→6.

When you transition from the main body cardioid to a bulb with label "5", you'll get period 1→5.

For the main cardioid, the two "arms" of order 3 are sprouting at $$c=c(\mu)=\frac\mu2\left(1-\frac\mu2\right)$$ with $\mu=\pm\mu_3$, $\mu_3=e^{2\pi i/3}$, and where the main cardioid are the points for $|\mu| \leqslant 1.$ Hence you can use the family of parameters $c(\lambda\cdot\mu_3)$ with $\lambda\in\mathbb R$ where the transitions occurs at $\lambda=\pm 1$.

We can also give explicit values where "$n$-furcation" 1→$n$ is happening, namely at $c(e^{2\pi i/n})$.

Answer 2

Here are 2D images of multifurcations :

These are images of the limit cycles for c along escape routes : 1/3, 1/4 i 2/5 so we have : trifurcation, 4-furcation and 5-furcations.

More description by Claude ( author of the code ):

For non-real C you can plot all the limit-cycle Z on one image, chances of overlap are small.