Happy Every Day (Hopf Bifurcation)

Define Vector Fields

f[x_,y_]:=a-x+x^2*y;
g[x_,y_]:=b-x^2*y;

Parameters

xub=1;
yub=4;
a=.1;
b=.13;

Nullclines

plot1=ContourPlot[f[x,y]==0,{x,0,xub},{y,0,yub},FrameLabel->{"x","y"},LabelStyle->Directive[Black, Large],ContourStyle->{Blue,Thickness[.015]},PlotRange->{{0,xub},{0,yub}}];
plot2=ContourPlot[g[x,y]==0,{x,0,xub},{y,0,yub},FrameLabel->{"x","y"},LabelStyle->Directive[Black, Large],ContourStyle->{Red,Thickness[.015]},PlotRange->{{0,xub},{0,yub}}];
Show[plot1,plot2]

Phase Portrait

plot3=StreamPlot[{f[x,y],g[x,y]},{x,0,xub},{y,0,yub}]

Show[plot1,plot2,plot3]

Fixed Point Analysis

Clear[a,b]

D[f[x,y],x]
D[f[x,y],y]
D[g[x,y],x]
D[g[x,y],y]

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

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

MatrixForm[J[a+b,(b/(((a+b))^2))]]

FullSimplify[Eigenvalues[J[a+b,(b/(((a+b))^2))]]]

Simplify[Det[J[a+b,(b/(((a+b))^2))]]]

Tr[J[a+b,(b/(((a+b))^2))]]