Why is this RegionFunction not working in my StreamPlot?

Question

I am trying to bound the plotting region inside a StreamPlot using RegionFunctionwith a piecewise equation. Here's the code:

mm = 2;
(*Boundary on state space*)
f1 = Sqrt[-2 *x1 - x1^2];
f2 = -Sqrt[2 *x1 - x1^2];
x2f = Piecewise[{{f1, x1 <= 0}, {f2, x1 > 0}}];
x2ff = Function[x1, Evaluate@x2f];
plot1 = Plot[x2ff[x1], {x1, -mm, mm}];

(*Dynamical system*)
A = {{0, 1}, {-1, 0}};
b = {{0}, {1}};
eqdyn = Flatten[A . {{x1}, {x2}} + b];

StreamPlot[eqdyn, {x1, -mm, mm}, {x2, -mm, mm},
 RegionFunction -> Function[{x1, x2}, x2 < x2ff[x1]]]

The bounded region is correctly highlighted in the plot, but it's not respected. If i change f1 and f2 with something else it works as intended, like:

mm = 2;
(*Boundary on state space*)
f1 = x1;
f2 = -x1;

... rest of the code ...

Answer 1

I guess complex values confuse StreamPlot/RegionFunction. Try this:

mm = 2;
(*Boundary on state space*)
f1 = Sqrt[-2*x1 - x1^2];
f2 = -Sqrt[2*x1 - x1^2];
x2f = Piecewise[{
  {f1, -2*x1 - x1^2 >= 0 && x1 <= 0},
  {f2,  2*x1 - x1^2 >= 0 && x1 >  0}}];
x2ff = Function[x1, Evaluate@x2f];
plot1 = Plot[x2ff[x1], {x1, -mm, mm}];

(*Dynamical system*)
A = {{0, 1}, {-1, 0}};
b = {{0}, {1}};
eqdyn = Flatten[A . {{x1}, {x2}} + b];

StreamPlot[eqdyn, {x1, -mm, mm}, {x2, -mm, mm}, 
 RegionFunction -> Function[{x1, x2}, x2 < x2ff[x1]]]

Answer 2

This doesn't answer why RegionFunction isn't working, but as a work around you can use an explicit region instead of RegionFunction:

reg = ImplicitRegion[x2 < x2ff[x1], {{x1, -mm, mm}, {x2, -mm, mm}}];
StreamPlot[eqdyn, {x1, x2} \[Element] reg, AspectRatio -> Automatic]

The problem with RegionFunction could be a bug, worth mentioning to Wolfram support.

Answer 3

RegionMember can be helpful.

Using the given:

A = {{0, 1}, {-1, 0}};
b = {{0}, {1}};
eqdyn = Flatten[A . {{x1}, {x2}} + b];

Defining the region and the region member function and plotting:

reg = RegionDifference[
  RegionUnion[Disk[{-1, 0}, 1, {0, Pi}], Rectangle[{-2, -2}, {2, 0}]],
   Disk[{1, 0}, 1, {Pi, 2 Pi}]]
rm = RegionMember[reg]

StreamPlot[eqdyn, {x1, -2, 2}, {x2, -2, 2}, 
 RegionFunction -> Function[{x, y}, rm[{x, y}]]]