Chua's model (Ch 16, page 227)

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

<<VectorFieldPlots`;

System

Parameters

f[x_,y_,z_]:=y+a (((x-z))^3) (-1+α) (α^3)

g[x_,y_,z_]:=-a (((x-z))^3) (α^4)

h[x_,y_,z_]:=-z+a (((x-z))^3) (α^4)

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])}={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])}}]];

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[α]->0,Dt[a]->0}

Fixed Points Stability - Eigenvalues

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

Flow Curvature Manifold

ϕ=Factor[Γ  . Cross[V,γ]]

Centre Manifold Approximation by Flow Curvature Method (Proposition 10.1)

Coefficient (Subscript[a, 10]) (A10)

A10=Factor[-((D[ϕ,x])/(D[ϕ,z]))/.{z->(Subscript[a, 20]) (x^2) + (Subscript[a, 11]) x y + (Subscript[a, 02]) (y^2)}];

Coefficient (Subscript[a, 01]) (A01)

A01=Factor[-((D[ϕ,y])/(D[ϕ,z]))/.{z->(Subscript[a, 20]) (x^2) + (Subscript[a, 11]) x y + (Subscript[a, 02]) (y^2)}];

Coefficient (Subscript[a, 20])

Factor[Factor[(1/2)D[A10,x]/.{y->0}]/.{x->0}]

Solve[(Subscript[a, 20])==-3 (Subscript[a, 20]),(Subscript[a, 20])]

Coefficient (Subscript[a, 11])

Factor[Factor[D[A10,y]/.{y->0}]/.{x->0}]

Solve[(Subscript[a, 11])==-((3 (2 a (α^4) (Subscript[a, 11])+a (α^3) (Subscript[a, 20])-a (α^4) (Subscript[a, 20])-34 (Subscript[a, 20]^3)))/(a (α^4))),(Subscript[a, 11])]/.{(Subscript[a, 20])->0}

Coefficient (Subscript[a, 02])

Factor[Factor[(1/2)D[A01,y]/.{x->0}]/.{y->0}]

Solve[(Subscript[a, 02])==-((3 (Subscript[a, 2]))/4),(Subscript[a, 02])]

Centre Manifold Approximation by Flow Curvature Method (Identification, cf. remark page 166)

Taking into account the preliminary results, i.e., that the centre manifold is homogeneous polynomial of degree three, let' s plug inside the flow curvature manifold the centre manifold approximation with its unknown coefficients. An identification order by order leads to :

K=Expand[ϕ/.{z->((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3))}];

Coefficient[K,x^5y^2]

Coefficient[K,x^4y^3]

Coefficient[K,x^3y^4]

Coefficient[K,x^2y^5]

Solve[{-3 (a^2) (α^8)+3 a (α^4) (Subscript[a, 30])==0,3 (a^2) (α^8)+3 a (α^4) (Subscript[a, 21])+6 a (α^4) (Subscript[a, 30])==0,3 a (α^4) (Subscript[a, 12])+6 a (α^4) (Subscript[a, 21])==0,3 a (α^4) (Subscript[a, 3])+6 a (α^4) (Subscript[a, 12])==0},{(Subscript[a, 30]) ,(Subscript[a, 21]) ,(Subscript[a, 12]) ,(Subscript[a, 03])}]

There is a slight difference between the term (Subscript[a, 03]) and that found by Algaba et al., 1999. This due on the on the hand to the nonuniqueness of the centre manifold and on the other hand to the fact the the (2^nd) flow curvature manifold represents an aproximation of order three. So, with a centre manfiold of degree three the limit of the method is reached and another approach such as Euler's theorem is required.

Centre Manifold Approximation by Euler's theorem (cf. remark page 174)

Taking into account the preliminary results, i.e., that the centre manifold is homogeneous polynomial of degree three and by using Euler's theorem leads fo the following system:

x(Overscript[z, •]) + ((Overscript[x, •])y - x (Overscript[y, •]))((∂h)/(∂y))=3h(x,y)     Eq. (1)

y(Overscript[z, •]) - ((Overscript[x, •])y - x (Overscript[y, •]))((∂h)/(∂x))=3h(x,y)     Eq. (2)

Equation (1)

D[((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3)),y]

K1=Expand[x h[x,y,z] - (x g[x,y,z]- y f[x,y,z])(3 (y^2) (Subscript[a, 3])+2 x y (Subscript[a, 12])+(x^2) (Subscript[a, 21]))-3((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3))f[x,y,z]/.{z-> ((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3))}]

Coefficient[K1,{x^4y^0}]/.{y->0}

Coefficient[K1,{x^3y^1}]

Coefficient[K1,{x^2y^2}]

Coefficient[K1,{x^1y^3}]

Solve[{a (α^4)-(Subscript[a, 30])==0,-(Subscript[a, 21])-3 (Subscript[a, 30])==0,-(Subscript[a, 12])-2 (Subscript[a, 21])==0,-(Subscript[a, 3])-(Subscript[a, 12])==0},{(Subscript[a, 30]) ,(Subscript[a, 21]) ,(Subscript[a, 12]) ,(Subscript[a, 03])}]

Then, there is no difference between the term (Subscript[a, 03]) and that found by Algaba et al., 1999.

Equation (2)

D[((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3)),x]

K2=Expand[y h[x,y,z] + (x g[x,y,z]- y f[x,y,z])((y^2) (Subscript[a, 12])+2 x y (Subscript[a, 21])+3 (x^2) (Subscript[a, 30]))-3((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3))g[x,y,z]/.{z->((Subscript[a, 30]) (x^3) +(Subscript[a, 21]) (x^2) y + (Subscript[a, 12]) x (y^2) + (Subscript[a, 03]) (y^3))}]

Coefficient[K2,{x^4y^0}]/.{y->0}

Coefficient[K2,{x^3y^1}]

Coefficient[K2,{x^2y^2}]

Coefficient[K2,{x^1y^3}]

Coefficient[K2,{x^0y^4}]/.{x->0}

Solve[{a (α^4)-(Subscript[a, 30])==0,-(Subscript[a, 21])-3 (Subscript[a, 30])==0,-(Subscript[a, 12])-2 (Subscript[a, 21])==0,-(Subscript[a, 3])-(Subscript[a, 12])==0},{(Subscript[a, 30]) ,(Subscript[a, 21]) ,(Subscript[a, 12]) ,(Subscript[a, 03])}]

Then, there is no difference between the term (Subscript[a, 03]) and that found by Algaba et al., 1999.