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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.

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    $\begingroup$ I'm not sure there's much to be said about arbitrary families of functions on 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 4:27
  • $\begingroup$ @RyanBudney I see what you mean, but I'm not sure what level of restriction is appropriate, which is why I tried to phrase the question in a (perhaps excessively) general way. Certainly there is a great deal of structure if we restrict to meromorphic functions on $\mathbb{C}$ (or on the Riemann sphere) but that's probably too restricted. 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

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