Limit Cycle Example

Definition Vector Fields

f[x_,y_]:=y;
g[x_,y_]:=-x+(1-x^2-y^2)*y;

Phase Portrait

plot1=StreamPlot[{f[x,y],g[x,y]},{x,-3,3},{y,-3,3},FrameLabel->{"x","y"},FrameTicksStyle->Directive[Black,20]];
plot2=ContourPlot[f[x,y]==0,{x,-3,3},{y,-3,3},ContourStyle->{Orange,Thick}];
plot3=ContourPlot[g[x,y]==0,{x,-3,3},{y,-3,3},ContourStyle->{Purple,Thick}];

Show[plot1,plot2,plot3]

Local Analysis

({{D[f[x,y],x], D[f[x,y],y]}, {D[g[x,y],x], D[g[x,y],y]}})

J[x_,y_]:={{0,1},{-1-2 x y,1-(x^2)-3 (y^2)}}

Solve[f[x,y]==0&&g[x,y]==0,{x,y}]

J[0,0]

N[Eigenvalues[J[0,0]]]