Example 6.13: First Integral of Affine and Projective models (Ch 6, page 123)

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

<<VectorFieldPlots`;

Projective Model

According to Darboux Theorem 5, 5 invariant manifolds are required to have a first integral.

Parameters

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

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

h[x_,y_,z_]:=z(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]]];

{(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]}]

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]]

Osculating PLane

X={x,y,z}

ϕ=FullSimplify[X  . Cross[V,γ]]

ϕ=(x^a) (y^b) (z^c)(((x -y))^d)(((y - z))^e)

Flow Curvature Invariance

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)]

Collect[(e x+c (x+y)+d z+b (x+z)+a (y+z)),{x,y,z}]

Solve[{(b+c+e) ==0,(a+c)==0,(a+b+d)==0,a+b+c+d+e==0},{a,b,c,d,e}]

ϕ/.{a->e,b->0,c->-e,d->-e}

ϕ=(x^e) (((x-y))^(-e)) (((y-z))^e) (z^(-e))

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[(V . η)]

In order to find a first integral of the affine model let's pose z = 1 in ϕ

ϕ=(x^e) (((x-y))^(-e)) (((y-z))^e) (z^(-e))/.{z->1}

Affine Model

Parameters

f[x_,y_]:=x(1- x)

g[x_,y_]:=y(1-y)

Vector field

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

Normal vector

η={D[ϕ,x],D[ϕ,y]}

Invariance

Ψ=FullSimplify[(V . η)]

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

Flow Curvature Manifold

ϕ=FullSimplify[Det[{V,γ}]]

ϕ=(x^a) (y^b)(((x-y))^d)(((y-1))^e)

Flow Curvature Invariance

Normal vector

η={D[ϕ,x],D[ϕ,y]}

Invariance

Ψ=FullSimplify[(V . η)]

Collect[(a (-1+x)+b (-1+y)+e y+d (-1+x+y)),{x,y}]

Solve[{-a-b-d==0,(a+d) ==0,(b+d+e) ==0},{a,b,d,e}]

ϕ=(x^a)  (y^b) (((x -y))^d)(((y - z))^e)/.{a->e,b->0,d->-e}

In order to find a first integral in the projective model let's replace x by x/z and y by y/z in ϕ

ϕ=FullSimplify[(x^e) (((x-y))^(-e)) (((y-z))^e)/.{x->(x/z),y->(y/z),λ->-e}]

ϕ=(((y-z))^e) ((((x/z)))^e) (((x-y))^(-e))

Projective Model

Parameters

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

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

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

Vector field

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

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z]}

Invariance

Ψ=FullSimplify[V . η]