Example 6.7: General Integral (Ch 6, page 106)

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

<<VectorFieldPlots`;

System

Parameters

A=1;

B=1;

μ=λ;

ν=λ;

f[x_,y_,z_]:=x(λ + C y + z)

g[x_,y_,z_]:=y(μ + x +A z)

h[x_,y_,z_]:=z(ν+ B x + y)

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]]];fp2=Sort[fp[[2]]];fp3=Sort[fp[[3]]];fp4=Sort[fp[[4]]];fp5=Sort[fp[[5]]];

{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={Extract[fp1,{1}][[2]],Extract[fp1,{2}][[2]],Extract[fp1,{3}][[2]]};
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])}}]];

{(Subscript[x, 2]),(Subscript[y, 2]),(Subscript[z, 2])}={Extract[fp2,{1}][[2]],Extract[fp2,{2}][[2]],Extract[fp2,{3}][[2]]};
fixedpoint2=Show[Graphics3D[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 2]),(Subscript[y, 2]),(Subscript[z, 2])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])},{(Subscript[z, min]),(Subscript[z, max])}}]];

{(Subscript[x, 3]),(Subscript[y, 3]),(Subscript[z, 3])}={Extract[fp3,{1}][[2]],Extract[fp3,{2}][[2]],Extract[fp3,{3}][[2]]};
fixedpoint3=Show[Graphics3D[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 3]),(Subscript[y, 3]),(Subscript[z, 3])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])},{(Subscript[z, min]),(Subscript[z, max])}}]];

{(Subscript[x, 4]),(Subscript[y, 4]),(Subscript[z, 4])}={Extract[fp4,{1}][[2]],Extract[fp4,{2}][[2]],Extract[fp4,{3}][[2]]};
fixedpoint4=Show[Graphics3D[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 4]),(Subscript[y, 4]),(Subscript[z, 4])}]},PlotRange->{{(Subscript[x, min]),(Subscript[x, max])},{(Subscript[y, min]),(Subscript[y, max])},{(Subscript[z, min]),(Subscript[z, max])}}]];

{(Subscript[x, 5]),(Subscript[y, 5]),(Subscript[z, 5])}={Extract[fp5,{1}][[2]],Extract[fp5,{2}][[2]],Extract[fp5,{3}][[2]]};
fixedpoint5=Show[Graphics3D[{PointSize[0.025],Hue[0.33],Point[{(Subscript[x, 5]),(Subscript[y, 5]),(Subscript[z, 5])}]},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=FullSimplify[(J . γ +Dt[J] . V)/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[λ]->0,Dt[μ]->0,Dt[ν]->0,Dt[A]->0,Dt[B]->0,Dt[C]->0}]

Fixed Points Stability - Eigenvalues

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

Flow Curvature Manifold

ϕ=FullSimplify[Γ  . Cross[V,γ]]

ϕ=(x^a) (y^b) (z^p)(((x-C y))^q) (((y-z))^r)

Then, according to Eq. (6.15) we may have :

(∑_(i=1)^5)(Subscript[α, i])(Subscript[h, i])=a + b+ p + q + r =0

Choosing arbitrarily: q -> 1, r -> - C, b -> 0, a -> - 1, p -> C leads to

ϕ=FullSimplify[(x^a) (y^b) (z^p)(((x-C y))^q) (((y-z))^r)/.{q->1,r->-C,b->0,a->-1,p->C}]

Flow Curvature Invariance

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)]