Example 9.7: Nondecomposable quadrics (Ch 9, page 158)

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

<<VectorFieldPlots`;

System

Parameters

f[x_,y_,z_]:=k y-ν x - y z

g[x_,y_,z_]:=k x - ν y + x z

h[x_,y_,z_]:=x y-ν z

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=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[k]->0,Dt[ν]->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]]

Since there are no linear invariant manifolds in factor in the flow curvature manifold a nonlinear invariant manifold may sought through extatic determinant.

Extatic Determinant (Lagutinskii)

L1 = {(x^2),(y^2),(z^2)}

L2=FullSimplify[Dt[L1]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[k]->0,Dt[ν]->0}]

L3=FullSimplify[Dt[L2]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[k]->0,Dt[ν]->0}];

K={L1,L2,L3};

MatrixForm[K];

Factor[Det[K]]

It may be easily checked that neither (x^2) (y^2) nor (x y - z ν) are invariant algebraic manifolds.

Flow Manifold Invariance

ϕ=(x^2) (y^2)

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)]

FullSimplify[(Ψ/ϕ)]

ϕ=(x y-z ν)

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)]

FullSimplify[(Ψ/ϕ)]