FillColor  =  
        RGBColor[1,  .75,  .75];
<< 
      Graphics`Arrow`
<< 
      Graphics`InequalityGraphics`
$TextStyle = {FontFamily -> "Times", 
            FontSize -> 10, FontWeight -> "Bold", 
            FontSlant -> "Italic"};

  
B =  {{-10, 
            12}, {0, -1}};
G[
        omega_]  :=  {{0, -omega}, {omega, 
          0}}
R[t_, omega_]  :=  
      MatrixExp[G[omega] t]
A[t_]  :=  R[t, -6] . B . R[t, 6]

  
A2  =  
        B  -  G[-6];
v1  =  (1/
              Norm[Eigenvectors[A2][[1]]]) Eigenvectors[
              A2][[1]];
v2  =  (1/
                  Norm[Eigenvectors[A2][[2]]]) Eigenvectors[
                  A2][[2]];

  
f1[t_]  :=  
        MatrixExp[G[-6] t] . MatrixExp[(B  - G[-6]) t]

  
prx1  =  -3;
prx2  =  
        3;
pry1 = -3;
pry2 = 3;
stepsize  =  
        2.9/50;
nframes  =  
        50;
x0  =  -2.44;
y0  =  1.23;

 (* 
        Initialize t1 so that t1 begins just slightly off  of 1 *) 
t1  =  -stepsize +  .001;

\(animation\  = \ 
\ \ Table[
\ \ \ \ t1\  = \ 
              t1\  + \ 
                stepsize; 

\ \ \ \ Graphics[Show[

 (*\ 
                  Shade\ the\ unstable\ regions\ *) 
InequalityPlot[{x1, y1} . A[t1] . {x1, y1} > 0, \ {x1, 
                    prx1, prx2}, {y1, pry1, pry2}, AspectRatio\  -> \ 1, 
                  PlotRange\  -> \ {{prx1, \ prx2}, \ {pry1, \ pry2}}, 
                  AxesLabel\  -> \ {\*""(Subscript[x, 1])(t)"", \*""(Subscript[x, 2])(t)""}, Ticks -> None, 
                  DisplayFunction -> \ Identity, 
                  Fills\  -> \ 
                    FillColor], 

 (*\ 
                  Plot\ the\ solution\ up\ to\ t1\ *) 
\ \ \ \ \ \ ParametricPlot[f1[t] . {x0, y0}, \ {t, \ 0, \ t1}, \ 
                  PlotStyle\  -> \ 
\ \ \ \ \ \ \ \ \ {Thickness[ .0075]}, \ 
                  DisplayFunction -> \ 
                    Identity], 

\ \ \ \  (*Plot\ the\ Eigenvectors\ *) 
Graphics[
                  RGBColor[0, 0, 
                    1]], 
Graphics[{Thickness[ .0075], 
                    Line[{{0, 0}, 4  R[\(-6\), t1] . v1}]}, 
                  DisplayFunction -> \ 
                    Identity], 
Graphics[{Thickness[ .0075], Line[{{0, 0}, 4  R[\(-6\), t1] . v2}]}, 
                  DisplayFunction -> \ 
                    Identity], 
Graphics[
                  Arrow[3  R[\(-6\), t1] . v1, 1.75  R[\(-6\), t1] . v1, 
                    HeadLength -> \  .04], 
                  DisplayFunction -> \ 
                    Identity], 
Graphics[
                  Arrow[{0, 0}, 1.75  R[\(-6\), t1] . v2, \ 
                    HeadLength -> \  .04], 
                  DisplayFunction -> \ 
                    Identity], 

 (*\ 
                  Plot\ the\ current\ posisition\ of\ the\ solution\ *) 
ListPlot[{f1[t1] . {x0, y0}}, \ 
                  PlotStyle\  -> \ {RGBColor[0, 1, 0], \ PointSize[ .02]}, 
                  DisplayFunction\  -> \ 
                    Identity]

]], 

\ \ \ \ {n, \ 1, \ nframes, \ 1}];\)

 (* 
        Define an animation that pauses for two frames at the end *) 
animation2  =  
          Join[animation, {animation[[nframes]], 
              animation[[nframes]]}];

  
\(Export["C:/Files/work/paper/webpage/vinograd_animationX.gif", \ animation2, 
\ \ \ ConversionOptions\  -> \ {"Loop"\  -> \ 
              True, \ "AnimationDisplayTime" ->  .125}, 
        ImageResolution\  -> \ 144]\)

  
Export["vinograd_animationX_tn.gif", 
      animation[[nframes - 5]]]
Null

\(Export["vinograd_animationX.gif", \ 
      animation2, 
\ \ \ ConversionOptions\  -> \ {"Loop"\  -> \ 
            True, \ "AnimationDisplayTime" ->  .125}, 
      ImageResolution\  -> \ 144]\)