Example 6.10: Multiplier of Homogeneous Dynamical Systems (Ch 6, page 113)

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

<<VectorFieldPlots`;

System

Parameters

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

g[x_,y_,z_]:=y(-2x -y -z)

h[x_,y_,z_]:=z(x - 2y+ 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[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^c)(((3 x+2 y-4 z))^d)

Flow Curvature Invariance

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)+Tr[J]ϕ]

Collect[((1+a-2 b+c+d) x+(-3+a-b-2 c-d) y+(2+a-b+c+d) z),{x,y,z}]

Solve[{(1+a-2 b+c+d)==0,(-3+a-b-2 c-d) ==0,(2+a-b+c+d) ==0,a+b+c+d==-4},{a,b,c,d}]

Then, choosing arbitrarily d = 2 leads to

{a->-(4/3)-(d/3),b->-1,c->-(5/3)-((2 d)/3)}/.{d->2}

ϕ=(x^a) (y^b) (z^c)(((3 x+2 y-4 z))^d)/.{a->-2,b->-1,c->-3,d->2}

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)+Tr[J]ϕ]