Example 2.10: Unforced Duffing oscillator (Ch 2, page 45)

Clear["Global`*"]
Off[General::spell1]

System

Parameters

δ=1.5;

{(Subscript[x, min]),(Subscript[x, max])}={-2.5,2.5};
{(Subscript[y, min]),(Subscript[y, max])}={-5,5};

f[x_,y_]:=y

g[x_,y_]:=x-(x^3)-δ y +(x^2) y

Vector field

V={f[x,y],g[x,y]}

Fixed Points & Nullclines

(Subscript[x, 1])=.
(Subscript[y, 1])=.

fp=Solve[{f[(Subscript[x, 1]),(Subscript[y, 1])]==0,g[(Subscript[x, 1]),(Subscript[y, 1])]==0},{(Subscript[x, 1]),(Subscript[y, 1])}]

fp1=Sort[fp[[1]]];fp2=Sort[fp[[2]]];fp3=Sort[fp[[3]]];

{(Subscript[x, 1]),(Subscript[y, 1])}={Extract[fp1,{1}][[2]],Extract[fp1,{2}][[2]]};
fixedpoint1=Show[Graphics[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 1]),(Subscript[y, 1])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])}}]];

{(Subscript[x, 2]),(Subscript[y, 2])}={Extract[fp2,{1}][[2]],Extract[fp2,{2}][[2]]};
fixedpoint2=Show[Graphics[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 2]),(Subscript[y, 2])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])}}]];

{(Subscript[x, 3]),(Subscript[y, 3])}={Extract[fp3,{1}][[2]],Extract[fp2,{2}][[2]]};
fixedpoint3=Show[Graphics[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 3]),(Subscript[y, 3])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])}}]];

nullclines=ContourPlot[{f[x,y]==0,g[x,y]==0},{x,(Subscript[x, min]),(Subscript[x, max])},{y,(Subscript[y, min]),(Subscript[y, max])},ContourStyle->{{Green,Thickness[0.005]}},DisplayFunction->Identity];

Show[nullclines,fixedpoint1,fixedpoint2,fixedpoint3]

Jacobian

a11=Simplify[D[f[x,y],x]]

a12=Simplify[D[f[x,y],y]]

a21=Simplify[D[g[x,y],x]]

a22=Simplify[D[g[x,y],y]]

J={{a11,a12},{a21,a22}}

MatrixForm[J]

Acceleration

γ=J . V

Fixed Points Stability - Eigenvalues

Eigenvalues[J]/.{x-> (Subscript[x, 1]),y->(Subscript[y, 1])}

Eigenvalues[J]/.{x-> (Subscript[x, 2]),y->(Subscript[y, 2])}

Eigenvalues[J]/.{x-> (Subscript[x, 3]),y->(Subscript[y, 3])}

Phase Portrait

solution=NDSolve[{x'[t]==f[x[t],y[t]],x[0]==2.1497,

y'[t]==g[x[t],y[t]],y[0]==1.3173},{x,y},

{t,0,7},Method->Automatic];phaseplot1=ParametricPlot[Evaluate[{x[t],y[t]}/.solution],{t,3,7},PlotPoints->1000,PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])}},AxesLabel->{"X","Y"},PlotStyle->{{Red,Thickness[0.005]}},AspectRatio->1,Frame->True]

Show[phaseplot1,phaseplot1,fixedpoint1,fixedpoint2,fixedpoint3]

Poincaré-Bendixson criterion

poincarébendixson=ContourPlot[{x==-(Sqrt[1.5]),x==(Sqrt[1.5])},{x,(Subscript[x, min]),(Subscript[x, max])},{y,(Subscript[y, min]),(Subscript[y, max])},ContourStyle->{{Magenta,Thickness[0.005],Dashing[{.025}]}},AspectRatio->1,Frame->True]

Show[phaseplot1,phaseplot1,poincarébendixson,Frame->True,ImageSize->{600,600} ,DisplayFunction->$DisplayFunction]