Feedback Systems

(Author: Jorge Finke)

Clear All;

f[x_,y_]:=-Sin[x]Cos[y];
g[x_,y_]:=-Cos[x]Sin[y];
A=(({{D[f[x,y],x],  D[f[x,y],y]}, {D[g[x,y],x], D[g[x,y],y]}}));
eq={{0,0},{0,Pi},{Pi,0},{Pi,Pi},{(1/2)Pi,(1/2)Pi},{(1/2)Pi,(3/2)Pi},{(3/2)Pi,(1/2)Pi},{(3/2)Pi,(3/2)Pi}};
linearSystems = Table[A /. x->eq[[i,1]]/.y->eq[[i,2]],{i,1,8}];
linearPhasePlane = Table[VectorPlot[Dot[linearSystems[[i]],{x,y}],{x,-2Pi,2Pi},{y,-2Pi,2Pi},FrameLabel->{(Subscript[z, 1]),(Subscript[z, 2])},VectorPoints-> Automatic,ImageSize->400,PlotLabel-> "Linearized vector space around (Subscript[x, eq]) no. "<> ToString[i]],{i,1,8}];
Show[VectorPlot[{f[x,y],g[x,y]},{x,0,2Pi},{y,0,2Pi},FrameLabel->{(Subscript[x, 1]),(Subscript[x, 2])},VectorPoints-> Fine],
Graphics[Table[Tooltip[{PointSize[Large],Red,Point[eq[[i]]]},linearPhasePlane[[i]]],{i,1,8}],ImageSize->600]]