Example 6.5: Integrable System (Ch 6, page 102)

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

<<VectorFieldPlots`;

System

Parameters

{(Subscript[x, min]),(Subscript[x, max])}={-2,2};
{(Subscript[y, min]),(Subscript[y, max])}={-2,2};
{(Subscript[z, min]),(Subscript[z, max])}={-2,2};

f[x_,y_,z_]:=x(z-y)

g[x_,y_,z_]:=y(x -z)

h[x_,y_,z_]:=z(y - x)

Vector field

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

Fixed Points

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

fp1=Sort[fp[[1]]];

{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={0,0,0};
fixedpoint1=Show[Graphics3D[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])},{(Subscript[z, min]),(Subscript[z, max])}}]];

Jacobian

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

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

a13=Simplify[D[f[x,y,z],z]]

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

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

a23=Simplify[D[g[x,y,z],z]]

a31=Simplify[D[h[x,y,z],x]]

a32=Simplify[D[h[x,y,z],y]]

a33=Simplify[D[h[x,y,z],z]]

J={{a11,a12,a13},{a21,a22,a23},{a31,a32,a33}}

MatrixForm[J]

Acceleration

γ=J . V

Over-Acceleration (jerk)

Γ=(J . γ +Dt[J] . V)/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[parameters]->0}

Fixed Points Stability - Eigenvalues

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

Phase Portrait

x=.
y=.
t=.

solution=NDSolve[{x'[t]==f[x[t],y[t],z[t]],x[0]==0.5,

y'[t]==g[x[t],y[t],z[t]],y[0]==0.5,

z'[t]==h[x[t],y[t],z[t]],z[0]==2},{x,y,z},

{t,0,60},Method->Automatic];phaseplot=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.solution],{t,0,60},PlotStyle->{{Red,Thickness[0.005]}},PlotPoints->2000,PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])},{(Subscript[z, min]),(Subscript[z, max])}},ImageSize->{400,400},AspectRatio->1,AxesLabel->{"X","Y","Z"},DisplayFunction->Identity];

Show[phaseplot,fixedpoint1,ImageSize->{600,600}]

plotX=Plot[Evaluate[x[t]/.solution],{t,0,50},PlotRange->{(Subscript[x, min]),(Subscript[x, max])},PlotStyle->{RGBColor[0,0,1]}]

plotY=Plot[Evaluate[y[t]/.solution],{t,0,50},PlotRange->{(Subscript[y, min]),(Subscript[y, max])},PlotStyle->{RGBColor[0,1,0]}]

plotZ=Plot[Evaluate[z[t]/.solution],{t,0,50},PlotRange->{(Subscript[z, min]),(Subscript[z, max])},PlotStyle->{RGBColor[1,0,0]}]

Flow Curvature Manifold

ϕ=FullSimplify[Γ  . Cross[V,γ]]

Osculating Plane

X={x,y,z}

ϕ=FullSimplify[X . Cross[V,γ]]

Solve[{(x^3)+(y^3)-3 x y z+(z^3)==0},z]

ϕ=x y z (x+y+z)

flowmanifold=ContourPlot3D[{x y z==0.5},{x,0,(Subscript[x, max])},{y,0,(Subscript[y, max])},{z,0,(Subscript[z, max])},PlotPoints->{20,20,20},BoxRatios->{1,1,1},Axes->True,AxesLabel->{"x","y","z"},Mesh->None,ContourStyle->Directive[Orange,Opacity[0.3],Specularity[White,30]],AspectRatio->1,DisplayFunction->Identity];

plan=ContourPlot3D[x + y+z==3,{x,0,(Subscript[x, max])},{y,0,(Subscript[y, max])},{z,0,(Subscript[z, max])},PlotPoints->{2,2,2},BoxRatios->{1,1,1},Axes->True,AxesLabel->{"x","y","z"},Mesh->None,ContourStyle->Directive[Blue,Opacity[0.2],Specularity[White,30]],AspectRatio->1,DisplayFunction->Identity];

Show[flowmanifold,plan,phaseplot,Boxed->False,Axes->False,BoxRatios->{1,1,1},ImageSize->{600,600},DisplayFunction->$DisplayFunction]

Flow Curvature Invariance

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[V . η]