\(tri = {{1., 0.}, {1.5, 1.5}, {2.5, 0.}}; 
end = 15\)

pts = 
      Union[Table[
          Point[tri[[1]] + k (tri[[2]] - tri[[1]])], {k, 0, 
            1, 1/150}], 
        Table[Point[tri[[2]] + k (tri[[3]] - tri[[2]])], {k, 
            0, 1, 1/150}], 
        Table[Point[tri[[3]] + k (tri[[1]] - tri[[3]])], {k, 
            0, 1, 1/150}]];

objs = 
      Table[{AbsolutePointSize[6], Hue[a/Length[pts]], pts[[a]]}, {a, 1, 
          Length[pts]}];

show1 = Show[Graphics[objs], AspectRatio -> Automatic];

\(\(\(solgen = 
      Table[NDSolve[{\(x'\)[t] == y[t], 
            \(y'\)[t] == \(-Sin[x[t]]\) - .1\ y[t], 
            x[0] == pts[\([i, 1, 1]\)], y[0] == pts[\([i, 1, 2]\)]}, {x[t], 
            y[t]}, {t, 0, end}], 
	{i, 1, Length[pts]}]; \)
\)\)

pts2 = 
      Table[Point[{solgen[[i, 1, 1, 2]], solgen[[i, 1, 2, 2]]} /. 
            t -> T], {i, 1, Length[pts]}];

objs2 = 
      Table[{AbsolutePointSize[4], Hue[a/Length[pts]], pts2[[a]]}, {a, 1, 
          Length[pts2]}];

Do[
      Show[Graphics[objs2 /. {T -> s}], AspectRatio -> Automatic, 
        PlotRange -> {{-2.7, 2.7}, {-2.7, 2.7}}, Axes -> True], {s, 
        0, end, .5}];