Two-parameter bifurcation diagram

Question

In an assignment I am currently doing, we are considering a system of the form $$\dot{x}=f(x,\mu,\delta).$$ In class we have been confronted with systems of the form $$\dot{x}=f(x,\mu),$$ where $\mu,\delta$ are the bifurcation parameters. Usually for $\dot{x}=f(x,\mu)$ this is a 2D bifurcation diagram where you plot the critical points of the system as $\mu$ varies. Would the bifurcation diagram of $\dot{x}=f(x,\mu,\delta)$ be a surface in $\mathbb{R}^3$ as $\mu,\delta$ vary?

Answer 1

Quick googling shows that people prefer to either do 2D heatmaps where the color denotes the asymptotic value of X, or straight up 2D bifurcation diagrams where one parameter is fixed to some value and the other varies. Generally, there is no perfect way to display high dimensional data on a piece of paper. If you present on a computer, then 3D is great because you can rotate it real time.

Answer 2

If anything interesting is going on (such as bifurcations), a normal bifurcation diagram already has multiple observable values per parameter value – infinitely many in case of chaos. This makes it impossible to translate it to a colour map as these would only work with one observable value per pair pair of parameter values. In geometrical 3D plots, it is impossible to avoid parts of the diagram will be obscured by others, as you have multiple surfaces above each other (not that geometrical 3D plots ever are a good idea).

So, you need to boil down your information somehow, e.g., by plotting the regions of different dynamical regimes (fixed point, period-two oscillation, period-four oscillation, chaos, bistability, …) in your 2D parameter space or only the locations of bifurcations.