Chapter 8, Section 8.8.3, Figure 8.13

Start by defining the Hamiltonian function h3.  This returns the value of the Hamiltonian at given values of x, y and δb.

h3[x_,y_,db_]:=
   (1/2)*db*(x^2+y^2) + (1/4)*(x^2+y^2)^2 + 2*x*(x^2-3y^2)

Plot curves of constant Hamiltonian for \(TraditionalForm\`δ\&_\)=15.  See Fig.8.13a.

ContourPlot[h3[x,y,15],{x,-4,4},{y,-4,4},
   ContourShading->False, 
   Contours->{1,5,10,20,40},PlotPoints->40]

Plot curves of constant Hamiltonian for \(TraditionalForm\`δ\&_\)=9.  See Fig.8.13b.

ContourPlot[h3[x,y,9],{x,-6,6},{y,-6,6},
   ContourShading->False, 
   Contours->{1,4,6.75,7.5,10,15,25,40},
   PlotPoints->200]

Plot curves of constant Hamiltonian for \(TraditionalForm\`δ\&_\)=5.  See Fig.8.13c.

ContourPlot[h3[x,y,5],{x,-8,8},{y,-8,8},
   ContourShading->False, 
   Contours->{-25,-10,0.75,10,25,50},PlotPoints->200]

Plot curves of constant Hamiltonian for \(TraditionalForm\`δ\&_\)=0.  See Fig.8.13d.

ContourPlot[h3[x,y,0],{x,-9,9},{y,-9,9},
   ContourShading->False, 
   Contours->{-100,-50,-10,0,20,50,100},
   PlotPoints->150]

Plot curves of constant Hamiltonian for \(TraditionalForm\`δ\&_\)=–5.  See Fig.8.13e.

ContourPlot[h3[x,y,-5],{x,-10,10},{y,-10,10},
   ContourShading->False, 
   Contours->{-200,-100,-30,-10,-0.483593,50,100},
   PlotPoints->150]

Plot curves of constant Hamiltonian for \(TraditionalForm\`δ\&_\)=–10.  See Fig.8.13e.

ContourPlot[h3[x,y,-10],{x,-12,12},{y,-12,12},
   ContourShading->False, 
   Contours->{-300,-200,-50,-10,-3.36184,100,300,500},
   PlotPoints->150]