Chua's cubic model 4D (Ch 17, page 244)

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

<<VectorFieldPlots`;

System

Nonlinear (cubic) function

(Subscript[α, 1])=10.1428;
(Subscript[α, 2])=-0.225;
(Subscript[β, 1])=0.121;
(Subscript[β, 2])=0.0047;
a=-0.525;
b=1.75;
d=1.85;

(Subscript[c, 1])=-((35((((d^2)-1))^2)(a-b))/(16(d^7)));
(Subscript[c, 2])=(((45(d^4)-50(d^2)+21)(a-b))/(16(d^5)))+b;

F[x_ ]=(Subscript[c, 1]) (x^3)+(Subscript[c, 2])x;
Plot[F[x],{x,-2,2}]

Parameters

{(Subscript[x, min]),(Subscript[x, max])}={-6,6};
{(Subscript[y, min]),(Subscript[y, max])}={-6,6};
{(Subscript[z, min]),(Subscript[z, max])}={-6,6};

f[x_,y_,z_,u_ ]:=(Subscript[α, 1])(z-F[x])

g[x_,y_,z_,u_ ]:=-(Subscript[α, 2])y-z-u

h[x_,y_,z_,u_ ]:=(Subscript[β, 1])( y-x- z)

i[x_,y_,z_,u_ ]:=(Subscript[β, 2])y

Vector field

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

Fixed Points

(Subscript[x, 1])=.
(Subscript[y, 1])=.
(Subscript[z, 1])=.
(Subscript[u, 1])=.

fp=Solve[{f[(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1]),(Subscript[u, 1])]==0,g[(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1]),(Subscript[u, 1])]==0,h[(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1]),(Subscript[u, 1])]==0,i[(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1]),(Subscript[u, 1])]==0},{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1]),(Subscript[u, 1])}]//N

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,u],x]]

a12=Simplify[D[f[x,y,z,u],y]]

a13=Simplify[D[f[x,y,z,u],z]]

a14=Simplify[D[f[x,y,z,u],u]]

a21=Simplify[D[g[x,y,z,u],x]]

a22=Simplify[D[g[x,y,z,u],y]]

a23=Simplify[D[g[x,y,z,u],z]]

a24=Simplify[D[g[x,y,z,u],u]]

a31=Simplify[D[h[x,y,z,u],x]]

a32=Simplify[D[h[x,y,z,u],y]]

a33=Simplify[D[h[x,y,z,u],z]]

a34=Simplify[D[h[x,y,z,u],u]]

a41=Simplify[D[i[x,y,z,u],x]]

a42=Simplify[D[i[x,y,z,u],y]]

a43=Simplify[D[i[x,y,z,u],z]]

a44=Simplify[D[i[x,y,z,u],u]]

J={{a11,a12,a13,a14},{a21,a22,a23,a24},{a31,a32,a33,a34},{a41,a42,a43,a44}}

MatrixForm[J]

Fixed Points Stability - Eigenvalues

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

Phase Portrait

Phase portrait

solution=NDSolve[{x'[t]==f[x[t],y[t],z[t],u[t]],x[0]==.5,

y'[t]==g[x[t],y[t],z[t],u[t]],y[0]==.3,

z'[t]==h[x[t],y[t],z[t],u[t]],z[0]==.05,

u'[t]==i[x[t],y[t],z[t],u[t]] ,u[0]==.05},{x,y,z,u},

{t,0,1000},Method->"ExplicitRungeKutta",MaxSteps->Infinity,StartingStepSize->0.0001];phaseplot=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.solution],{t,700,1000},PlotPoints->5000,PlotRange->All,PlotStyle->{RGBColor[1,0,0]},ImageSize->{400,400},BoxRatios->{1,1,1},AxesLabel->{"X","Y","Z"}]

phaseplotXY=ParametricPlot[Evaluate[{y[t],x[t]}/.solution],{t,10,1000},PlotPoints->10000,PlotRange->All,PlotStyle->{RGBColor[1,0,0]},ImageSize->{400,400},AspectRatio->1,AxesLabel->{"X","Y","Z"}]

plotX=Plot[Evaluate[x[t]/.solution],{t,200,1000},PlotRange->{(Subscript[x, min]),(Subscript[x, max])},PlotStyle->{RGBColor[0,0,1]}]

plotY=Plot[Evaluate[y[t]/.solution],{t,200,1000},PlotRange->{(Subscript[y, min]),(Subscript[y, max])},PlotStyle->{RGBColor[0,1,0]}]

plotZ=Plot[Evaluate[z[t]/.solution],{t,200,1000},PlotRange->{(Subscript[z, min]),(Subscript[z, max])},PlotStyle->{RGBColor[1,0,0]}]

plotU=Plot[Evaluate[u[t]/.solution],{t,200,1000},PlotRange->{(Subscript[z, min]),(Subscript[z, max])},PlotStyle->{Cyan,Thickness[0.005]}]

Show[plotX,plotY,plotZ,plotU]

Flow Curvature Manifold

V1={f[x,y,z,u],g[x,y,z,u],h[x,y,z,u],i[x,y,z,u]}

V2=Dt[V1]/.{Dt[x]->f[x,y,z,u],Dt[y]->g[x,y,z,u], Dt[z]->h[x,y,z,u],Dt[u]->i[x,y,z,u],Dt[(Subscript[α, 1])]->0,Dt[(Subscript[α, 2])]->0,Dt[(Subscript[β, 1])]->0,Dt[(Subscript[β, 2])]->0,Dt[a]->0,Dt[b]->0,Dt[d]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0 }

V3=Dt[V2]/.{Dt[x]->f[x,y,z,u],Dt[y]->g[x,y,z,u], Dt[z]->h[x,y,z,u],Dt[u]->i[x,y,z,u],Dt[(Subscript[α, 1])]->0,Dt[(Subscript[α, 2])]->0,Dt[(Subscript[β, 1])]->0,Dt[(Subscript[β, 2])]->0,Dt[a]->0,Dt[b]->0,Dt[d]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0 };

V4=Dt[V3]/.{Dt[x]->f[x,y,z,u],Dt[y]->g[x,y,z,u], Dt[z]->h[x,y,z,u],Dt[u]->i[x,y,z,u],Dt[(Subscript[α, 1])]->0,Dt[(Subscript[α, 2])]->0,Dt[(Subscript[β, 1])]->0,Dt[(Subscript[β, 2])]->0,Dt[a]->0,Dt[b]->0,Dt[d]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0 };

K={V1,V2,V3,V4};

ϕ=Factor[Det[K]/.{u->0}]//Chop

Since the functional Jacobian matrix is locally stationary the nonstationary part of the flow curvature manifold may be removed in order to improve the figure.

V1={f[x,y,z,u],g[x,y,z,u],h[x,y,z,u],i[x,y,z,u]}

V2=J . V1;

V3=J . V2;

V4=J . V3;

K={V1,V2,V3,V4};

ϕ=Factor[Det[K]/.{u->0}]//N

flowmanifoldL=ContourPlot3D[ϕ==0,{x,-6,-1},{y,-5,3},{z,-1.5,0.6},ColorFunction->Function[{x,y,z,f},ColorData["Rainbow"][y]],Mesh->None,PlotPoints->{15,15,15},BoxRatios->{1,1,1},Axes->True,AxesLabel->{"x","y","z"},BoxRatios->{1,1,1}]

flowmanifoldR=ContourPlot3D[ϕ==0,{x,1,6},{y,-3.5,6},{z,-.6,2},ColorFunction->Function[{x,y,z,f},ColorData["Rainbow"][y]],Mesh->None,PlotPoints->{15,15,15},BoxRatios->{1,1,1},Axes->True,AxesLabel->{"x","y","z"},BoxRatios->{1,1,1}]

Show[phaseplot,flowmanifoldL,fixedpoint1,flowmanifoldR,Boxed->False,Axes->False,BoxRatios->{1,1,1},ImageSize->{600,600}]