B = -2; A = 1;
s = ParametricNDSolveValue[{Sqrt[-1]*x'[t] ==
B*x[t] - R*x[t]*Abs[x[t]]^2 - A*y[t], x[0] == 1,
Sqrt[-1]*y'[t] == B*y[t] - R*y[t]*Abs[y[t]]^2 - A*x[t],
y[0] == 0}, {x, y}, {t, 0, 100}, {R}];
coll = {};
Table[pp =
ParametricPlot[Re@Through[s[a][t]], {t, 0, 100},
PlotRange -> {{-2, 2}, {-2, 2}}, PerformanceGoal -> "Quality",
MeshFunctions -> (#2 &), Mesh -> {{0.}},
MeshStyle -> {Red, PointSize[0.01]}];
pts = pp[[1, 1]];
AppendTo[
coll, {a, #[[1]]} & /@
pts[[First@Cases[pp[[1]], Point[x__] :> x, -1]]]];, {a, 0.0, 5,
0.01}];
lp = ListPlot[Join @@ coll, Frame -> True, PlotStyle -> Red];
Manipulate[
Column[{ParametricPlot[Re@Through[s[par][t]], {t, 0, 100},
PlotRange -> {{-2, 2}, {-2, 2}}, PerformanceGoal -> "Quality",
MeshFunctions -> (#2 &), Mesh -> {{0.}},
MeshStyle -> {Red, PointSize[0.01]}, Frame -> True,
FrameLabel -> {"x[t]", "y[t]"}],
Show[lp, Graphics[{Gray, Line[{{par, 0}, {par, 4}}]}]]}], {par,
0.0, 5, 0.01}]

coll = {};
Table[pp =
ParametricPlot[Re@Through[s[a][b][t]], {t, 0, 20},
PlotRange -> {{-2, 2}, {-2, 2}}, PerformanceGoal -> "Quality",
MeshFunctions -> (#2 &), Mesh -> {{0.}},
MeshStyle -> {Red, PointSize[0.01]}];
pts = pp[[1, 1]];
AppendTo[
coll, {a, b, #[[1]]} & /@
pts[[First@Cases[pp[[1]], Point[x__] :> x, -1]]]];, {a, 0.0, 5,
0.01}, {b, 0.0, 5, 0.01}];
lp = ListPlot[Join @@ coll, Frame -> True, PlotStyle -> Red];
Manipulate[
Column[{ParametricPlot[Re@Through[s[par1][par2][t]], {t, 0, 20},
PlotRange -> {{-2, 2}, {-2, 2}}, PerformanceGoal -> "Quality",
MeshFunctions -> (#2 &), Mesh -> {{0.}},
MeshStyle -> {Red, PointSize[0.01]}, Frame -> True,
FrameLabel -> {"x[t]", "y[t]"}],
Show[lp,
Graphics[{Gray,
  Line[{{par1, 0}, {par1, 4}, {par2, 0}, {par2, 4}}]}]]}], {par1,
0.0, 5, 0.01}, {par2, 0.0, 5, 0.01}]