# Name for critical lines in parameter space and plots thereof

Suppose I study a dynamical system as a function of some control parameters, and I find that the nature of the attractors changes discontinuously (or non-analytically) at certain critical values (or along critical lines) in the space of control parameters. These lines separate different basins of attraction. What do you call a plot of these basins in the space of control parameters? A phase diagram of the dynamical system? What is a proper terminology for the transition lines? (Any standard references for this kind of thing?)

Let’s first clarify some terminology:

• A basin of attraction is the set of states that converge to a given attractor (in a fixed dynamical system). For example, for a ball moving on some geography with high friction the lowest point of each valley is an attractor and the valleys are the basins of attraction.

• A bifurcation point is a point in a (one-dimensional) parameter space where the dynamics undergoes a sudden qualitative change, the bifurcation.. For example, the dynamics could start to oscillate or gain a new attractor (become bistable).

• A bifurcation diagram shows some observable of a dynamical system (e.g., the values of local maxima) depending on a single control parameter. This can usually not be reasonably extended to a two-dimensional parameter space (see this answer of mine).

• A dynamical regime denotes a maximal region in parameter space displaying the same kind of dynamics (or the dynamics occurring in this region). With other words, it’s a maximal region where no bifurcations happen.

I find that the nature of the attractors changes discontinuously (or non-analytically) at certain critical values (or along critical lines) in the space of control parameters.

Those would be bifurcations with the critical values being bifurcation points.

These lines separate different basins of attraction.

Basins of attractions are not regions in parameter space but in state space. What those lines separate is dynamical regimes.

What do you call a plot of these basins in the space of control parameters? A phase diagram of the dynamical system? What is a proper terminology for the transition lines? (Any standard references for this kind of thing?)

I would call the plot a map of dynamical regimes, but you can also find it called bifurcation diagram, although it is not a simple extension of a normal bifurcation diagram to a two-dimensional parameter space. A line of bifurcation points (i.e., your critical line or transition line) is usually called bifurcation line. You can find an illustrative example of bifurcation diagram and bifurcation line used for this in A. G. Balanov et al, Phys. Rev. E 71, 016222 (in particular Fig. 2).

What do you call a plot of these basins in the space of control parameters?

Such a plot is called "parameter space" (for example), but the specific regions are not usually called "basins" (a term more associated with the system's phase space), but simply "region" or "set" (e.g., here).

A phase diagram of the dynamical system?

When appropriately defined, it can of course be understood, but it might not be the best choice, since "phase" is most often understood as a synonym for "state", as in "phase space", "phase portrait", "phase plot" and even (though seldom) "phase diagram". That said, some dynamical systems papers (for instance this) do use the term "phase diagram" in the statistical sense, so this does remain an option.

What is a proper terminology for the transition lines? (Any standard references for this kind of thing?)

I'm not aware of any established nomenclature and usually generic descriptive terms such as "boundaries" or "lines" are used.