This question is related to my other recent question on MO (although I am not confident that the dynamical system described in that other question is actually "difficult," in the sense that I will describe here).

For a parameterized family of functions on a metric space $M$,

$$f_\lambda:M\mapsto M$$

(where $\lambda$ is either also from $M$ or is from some other metric space, as long as it's possible to talk about "nearby" parameters) I would like to visualize the parameters $\lambda$ for which the dynamics of the iterates of $f_\lambda$ are unstable.

Some families are not very difficult to analyze because their global dynamics happen to be governed by the dynamics of one, or a few, orbits. In particular, when $M$ is the Riemann Sphere and the $f_\lambda$ are rational functions it is frequently (always?) possible to find the bifurcation locus by investigating the orbits of the critical points.

My question is:

How are the bifurcations visualized in the more general case?

For example, what if $M$ is a less convenient metric space, or the global dynamics are not governed by a finite number of orbits, or $f_\lambda$ is given numerically?

In the linked question I suggested a technique for estimating stability by essentially running thousands of image comparisons; but it's a naive method, really nothing more than a brute force way to detect when the dynamics are sensitive to perturbation of the parameter. I think better methods must exist I would like to see references for those methods.

As an aside: I am taking for granted that the parameter space is amenable to visualization *at all*, but there are plenty of interesting parameter spaces for which visualization is not trivial. That might be an interesting problem in and of itself, but I am trying to restrain the scope of this MO question.

functionson arbitrary metric spaces. Basically, anything could happen at any time. There is no stability of any form. Perhaps you want a more restricted context than that? $\endgroup$ – Ryan Budney Jan 15 '15 at 4:27toorestricted. Even the system in the linked question operates in the broader context of $\mathbb{C}^2$, and yet appears to have inherited some structure from one-variable complex dynamics. I would be happy with references to restricted contexts that admit better tools. $\endgroup$ – Aaron Golden Jan 15 '15 at 19:16