Example 9.2: Local Invariance (Ch 9, page 150)

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

<<VectorFieldPlots`;

System

Parameters

ϵ=0.1;

{(Subscript[x, min]),(Subscript[x, max])}={-1.5,1.5};
{(Subscript[y, min]),(Subscript[y, max])}={-1.5,1.5};

f[x_,y_]:=(1/ϵ)(y - (x^2))

g[x_,y_]:=-x

Vector field

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

Fixed Points & Nullclines

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]]];

{(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])}}]];

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

Show[nullclines,fixedpoint1]

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])}

Phase Portrait

solution=NDSolve[{x'[t]==f[x[t],y[t]],x[0]==.3,

y'[t]==g[x[t],y[t]],y[0]==0.9},{x,y},

{t,0,100},Method->Automatic];phaseplot=ParametricPlot[Evaluate[{x[t],y[t]}/.solution],{t,90,100},PlotPoints->1000,PlotRange->All,AxesLabel->{"X","Y"},PlotStyle->{{Red,Thickness[0.005]}},AspectRatio->1,Frame->True]

Show[phaseplot,fixedpoint1,nullclines]

plotX=Plot[Evaluate[x[t]/.solution],{t,80,100},PlotRange->{(Subscript[x, min]),(Subscript[x, max])},PlotStyle->{RGBColor[0,0,1]}]

plotY=Plot[Evaluate[y[t]/.solution],{t,80,100},PlotRange->{(Subscript[y, min]),(Subscript[y, max])},PlotStyle->{RGBColor[0,1,0]}]

Show[plotX,plotY]

Flow Curvature Manifold

ϕ=FullSimplify[Det[{V,γ}]]

flowmanifold=ContourPlot[{ϕ==0},{x,(Subscript[x, min]),(Subscript[x, max])},{y,(Subscript[y, min]),(Subscript[y, max])},ContourStyle->{{Blue,Thickness[0.005]}},AspectRatio->1,Frame->True]

Flow Curvature Invariance

Normal Vector

η={D[ϕ,x],D[ϕ,y]}

Invariance

Ψ=FullSimplify[V . η]

invariantpart=ContourPlot[{Ψ==0},{x,(Subscript[x, min]),(Subscript[x, max])},{y,(Subscript[y, min]),(Subscript[y, max])},ContourStyle->{{Magenta,Thickness[0.005],Dashing[{.025}]}},Frame->True,AspectRatio->1]

Show[phaseplot,fixedpoint1,flowmanifold,invariantpart,Frame->True,ImageSize->{600,600} ,DisplayFunction->$DisplayFunction]

Flow

tableplot={};
For[t=10,t≤15,t=t+ 0.5,
x=Evaluate[{x[t],y[t]}/.solution][[1]][[1]];
y=Evaluate[{x[t],y[t]}/.solution][[1]][[2]];
vx=f[x,y];
vy=g[x,y];
tmptable={{x,y},{vx,vy}};
tableplot=Append[tableplot,tmptable];];
flow=ListVectorFieldPlot[tableplot ,ColorFunction->(Hue[0.66]&),ScaleFactor->0.1,MaxArrowLength->5,Frame->True,AspectRatio->1]

Show[phaseplot,fixedpoint1,nullclines,flow,flowmanifold,invariantpart, ImageSize->{600,600},DisplayFunction->$DisplayFunction]