Chua's piecewise linear model 5D (Ch 17, page 242)

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

<<VectorFieldPlots`;

System

Piecewise linear function

F[x_ ,a_,b_]=Piecewise[{{b x +(b-a),x<-1},{ a x,-1≤x≤1},{b x +(a-b),x>1}}];

Plot[x+F[x,-1.246,-0.6724],{x,-2,2}]

Parameters

{(Subscript[x, min]),(Subscript[x, max])}={-4,4};
{(Subscript[y, min]),(Subscript[y, max])}={-2,2};
{(Subscript[z, min]),(Subscript[z, max])}={-4,4};

f[x_,y_,z_,u_,w_]:=α(y - x - F[x,a,b])

g[x_,y_,z_,u_,w_]:=x -y + z

h[x_,y_,z_,u_,w_]:=β(u - y)

i[x_,y_,z_,u_,w_]:=-γ(z + w)

j[x_,y_,z_,u_,w_]:=(Subscript[γ, 2])(u + (Subscript[γ, 1])w)

Vector field

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

Fixed Points (left side)

f[x_,y_,z_,u_,w_]:=α (-x+y-(-a+b+b x))

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

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

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

fp1=Sort[fp[[1]]]

{(Subscript[u, 1]),(Subscript[w, 1]),(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={Extract[fp1,{1}][[2]],Extract[fp1,{2}][[2]],Extract[fp1,{3}][[2]],Extract[fp1,{4}][[2]],Extract[fp1,{5}][[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])}}]];

{(Subscript[x, 1])->-(((-a+b) (-1+(Subscript[γ, 1])))/(-1-b+b (Subscript[γ, 1]))),(Subscript[y, 1])->-((-a (Subscript[γ, 1])+b (Subscript[γ, 1]))/(-1-b+b (Subscript[γ, 1]))),(Subscript[z, 1])->-((-a+b)/(-1-b+b (Subscript[γ, 1])))}/.{α-> 9.934,
β->14.47,γ->406.5,(Subscript[γ, 1])->-0.0152,(Subscript[γ, 2])->41000,a->-1.246,b->-0.6724}//N

{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={-1.834770939221283`,-0.027470959688892335`,1.807};
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 (left side)

a11=Simplify[D[f[x,y,z,u,w],x]]

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

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

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

a15=Simplify[D[f[x,y,z,u,w],w]]

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

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

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

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

a25=Simplify[D[g[x,y,z,u,w],w]]

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

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

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

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

a35=Simplify[D[h[x,y,z,u,w],w]]

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

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

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

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

a45=Simplify[D[i[x,y,z,u,w],w]]

a51=Simplify[D[j[x,y,z,u,w],x]]

a52=Simplify[D[j[x,y,z,u,w],y]]

a53=Simplify[D[j[x,y,z,u,w],z]]

a54=Simplify[D[j[x,y,z,u,w],u]]

a55=Simplify[D[j[x,y,z,u,w],w]]

J={{a11,a12,a13,a14,a15},{a21,a22,a23,a24,a25},{a31,a32,a33,a34,a35},{a41,a42,a43,a44,a45},{a51,a52,a53,a54,a55}}

MatrixForm[J]

Fixed Points Stability - Eigenvalues (left side)

Eigenvalues[J]/.{x-> (Subscript[x, 1]),y->(Subscript[y, 1]),z->(Subscript[z, 1]),u->(Subscript[u, 1]),w->(Subscript[w, 1]),α-> 9.934,
β->14.47,γ->406.5,(Subscript[γ, 1])->-0.0152,(Subscript[γ, 2])->41000,a->-1.246,b->-0.6724}//N

Phase Portrait

Parameters

α=9.934;
β=14.47;
γ=406.5;
(Subscript[γ, 1])=-0.0152;
(Subscript[γ, 2])=41000;

f[x_,y_,z_,u_,w_]:=α(y - x - F[x,-1.246,-0.6724])

g[x_,y_,z_,u_,w_]:=x -y + z

h[x_,y_,z_,u_,w_]:=β(u - y)

i[x_,y_,z_,u_,w_]:=-γ(z + w)

j[x_,y_,z_,u_,w_]:=(Subscript[γ, 2])(u + (Subscript[γ, 1])w)

Phase portrait

solution=NDSolve[{x'[t]==f[x[t],y[t],z[t],u[t],w[t]],x[0]==.1,

y'[t]==g[x[t],y[t],z[t],u[t],w[t]],y[0]==.1,

z'[t]==h[x[t],y[t],z[t],u[t],w[t]],z[0]==.1,

u'[t]==i[x[t],y[t],z[t],u[t],w[t]],u[0]==.1,

w'[t]==j[x[t],y[t],z[t],u[t],w[t]],w[0]==.1},{x,y,z,u,w},

{t,0,130},Method->Automatic];phaseplot=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.solution],{t,0,130},PlotPoints->2000,PlotRange->All,PlotStyle->{RGBColor[1,0,0]},ImageSize->{400,400},BoxRatios->{1,1,1},AxesLabel->{"X","y","Z"},DisplayFunction->Identity]

phaseplotXY=ParametricPlot[Evaluate[{y[t],x[t]}/.solution],{t,20,130},PlotPoints->2000,PlotRange->All,PlotStyle->{RGBColor[1,0,0]},ImageSize->{400,400},AxesLabel->{"X","Y"},AspectRatio->1]

plotX=Plot[Evaluate[x[t]/.solution],{t,30,130},PlotRange->{(Subscript[x, min]),(Subscript[x, max])},PlotStyle->{RGBColor[0,0,1]}]

plotY=Plot[Evaluate[y[t]/.solution],{t,30,130},PlotRange->{(Subscript[y, min]),(Subscript[y, max])},PlotStyle->{RGBColor[0,1,0]}]

plotZ=Plot[Evaluate[z[t]/.solution],{t,30,130},PlotRange->{(Subscript[z, min]),(Subscript[z, max])},PlotStyle->{RGBColor[1,0,0]}]

plotU=Plot[Evaluate[u[t]/.solution],{t,30,130},PlotRange->{(Subscript[z, min]),(Subscript[z, max])},PlotStyle->{Cyan,Thickness[0.005]}]

plotW=Plot[Evaluate[w[t]/.solution],{t,30,130},PlotRange->{(Subscript[z, min]),(Subscript[z, max])},PlotStyle->{Pink,Thickness[0.005]}]

Show[plotX,plotY,plotZ,plotU,plotW]

Flow Curvature Manifold (left side)

α=.
β=.
γ=.
(Subscript[γ, 1])=.
(Subscript[γ, 2])=.

f[x_,y_,z_,u_,w_]:=α (-x+y-(-a+b+b x))

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

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

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

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

V5=Dt[V4]/.{Dt[x]->f[x,y,z,u,w],Dt[y]->g[x,y,z,u,w], Dt[z]->h[x,y,z,u,w],Dt[u]->i[x,y,z,u,w],Dt[w]->j[x,y,z,u,w],Dt[α]->0,Dt[β]->0,Dt[γ]->0,Dt[(Subscript[γ, 1])]->0,Dt[(Subscript[γ, 2])]->0,Dt[a]->0,Dt[b]->0}

K={V1,V2,V3,V4,V5};

ϕ=Factor[Det[K]/.{u->0,w->0}];

Osculating plane (left side)

X={x+(((-a+b) (-1+(Subscript[γ, 1])))/(-1-b+b (Subscript[γ, 1]))),y+((-a (Subscript[γ, 1])+b (Subscript[γ, 1]))/(-1-b+b (Subscript[γ, 1]))),z+((-a+b)/(-1-b+b (Subscript[γ, 1]))),u+((-a (Subscript[γ, 1])+b (Subscript[γ, 1]))/(-1-b+b (Subscript[γ, 1]))),w+((a-b)/(-1-b+b (Subscript[γ, 1])))}

Π=Det[{X,V1,V2,V3,V4}/.{u->0,w->0}];

Factor[(ϕ/Π)]

FullSimplify[Det[J]]

This a consequence of Theorem 8.2 and Corollary 8.2 (Eq. 8.8) page 141.

ϕL=Factor[Det[K]/.{u->0,w->0,α-> 9.934,
β->14.47,γ->406.5,(Subscript[γ, 1])->-0.0152,(Subscript[γ, 2])->41000,a->-1.246,b->-0.6724}]

flowmanifoldL=ContourPlot3D[ϕL==0,{x,-4,0},{y,-4,(Subscript[y, max])},{z,-0.75,5},ColorFunction->Function[{x,y,z,f},ColorData["Rainbow"][z]],Mesh->None,PlotPoints->{7,7,7},BoxRatios->{1,1,1},Axes->True,AxesLabel->{"x","y","z"},BoxRatios->{1,1,1}]

Flow Curvature Invariance (left side)

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z],D[ϕ,u],D[ϕ,w]};

Invariance

Ψ=Factor[V . η/.{u->0,w->0}];

Factor[(Ψ/ϕ)/.{x->0,y->0,z->0}]

Factor[Tr[J]]

This a consequence of Theorem 8.2 and Corollary 8.2 (Eq. 8.7) page 141. Thus, the invariance of the osculating plane is stated ipso facto.

Vector field

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

Vector field

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

Fixed Points (right side)

f[x_,y_,z_,u_,w_]:=α (-x+y-(a-b+b x))

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

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

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

fp1=Sort[fp[[1]]]

{(Subscript[u, 1]),(Subscript[w, 1]),(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={Extract[fp1,{1}][[2]],Extract[fp1,{2}][[2]],Extract[fp1,{3}][[2]],Extract[fp1,{4}][[2]],Extract[fp1,{5}][[2]]};
fixedpoint2=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])}}]];

{(Subscript[x, 1])->-(((a-b) (-1+(Subscript[γ, 1])))/(-1-b+b (Subscript[γ, 1]))),(Subscript[y, 1])->-((a (Subscript[γ, 1])-b (Subscript[γ, 1]))/(-1-b+b (Subscript[γ, 1]))),(Subscript[z, 1])->-((a-b)/(-1-b+b (Subscript[γ, 1])))}/.{α-> 9.934,
β->14.47,γ->406.5,(Subscript[γ, 1])->-0.0152,(Subscript[γ, 2])->41000,a->-1.246,b->-0.6724}

{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={1.834770939221283`,0.027470959688892335`,-1.8072999795323905`};
fixedpoint2=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 (right side)

a11=Simplify[D[f[x,y,z,u,w],x]]

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

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

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

a15=Simplify[D[f[x,y,z,u,w],w]]

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

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

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

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

a25=Simplify[D[g[x,y,z,u,w],w]]

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

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

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

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

a35=Simplify[D[h[x,y,z,u,w],w]]

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

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

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

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

a45=Simplify[D[i[x,y,z,u,w],w]]

a51=Simplify[D[j[x,y,z,u,w],x]]

a52=Simplify[D[j[x,y,z,u,w],y]]

a53=Simplify[D[j[x,y,z,u,w],z]]

a54=Simplify[D[j[x,y,z,u,w],u]]

a55=Simplify[D[j[x,y,z,u,w],w]]

J={{a11,a12,a13,a14,a15},{a21,a22,a23,a24,a25},{a31,a32,a33,a34,a35},{a41,a42,a43,a44,a45},{a51,a52,a53,a54,a55}}

MatrixForm[J]

Fixed Points Stability - Eigenvalues (right side)

Eigenvalues[J]/.{x-> (Subscript[x, 1]),y->(Subscript[y, 1]),z->(Subscript[z, 1]),u->(Subscript[u, 1]),w->(Subscript[w, 1]),α-> 9.934,
β->14.47,γ->406.5,(Subscript[γ, 1])->-0.0152,(Subscript[γ, 2])->41000,a->-1.246,b->-0.6724}//N

Flow Curvature Manifold (right side)

α=.
β=.
γ=.
(Subscript[γ, 1])=.
(Subscript[γ, 2])=.

f[x_,y_,z_,u_,w_]:=α (-x+y-(a-b+b x))

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

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

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

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

V5=Dt[V4]/.{Dt[x]->f[x,y,z,u,w],Dt[y]->g[x,y,z,u,w], Dt[z]->h[x,y,z,u,w],Dt[u]->i[x,y,z,u,w],Dt[w]->j[x,y,z,u,w],Dt[α]->0,Dt[β]->0,Dt[γ]->0,Dt[(Subscript[γ, 1])]->0,Dt[(Subscript[γ, 2])]->0,Dt[a]->0,Dt[b]->0}

K={V1,V2,V3,V4,V5};

ϕ=Factor[Det[K]/.{u->0,w->0}];

Osculating plane (right side)

X={x+(((a-b) (-1+(Subscript[γ, 1])))/(-1-b+b (Subscript[γ, 1]))),y+((a (Subscript[γ, 1])-b (Subscript[γ, 1]))/(-1-b+b (Subscript[γ, 1]))),z+((a-b)/(-1-b+b (Subscript[γ, 1]))),u+((a (Subscript[γ, 1])-b (Subscript[γ, 1]))/(-1-b+b (Subscript[γ, 1]))),w+((-a+b)/(-1-b+b (Subscript[γ, 1])))}

Π=Det[{X,V1,V2,V3,V4}/.{u->0,w->0}];

Factor[(ϕ/Π)]

FullSimplify[Det[J]]

This a consequence of Theorem 8.2 and Corollary 8.2 (Eq. 8.8) page 141.

ϕR=Factor[Det[K]/.{u->0,w->0,α-> 9.934,
β->14.47,γ->406.5,(Subscript[γ, 1])->-0.0152,(Subscript[γ, 2])->41000,a->-1.246,b->-0.6724}]

flowmanifoldR=ContourPlot3D[ϕR==0,{x,0,4},{y,-2,4},{z,-5,0.75},ColorFunction->Function[{x,y,z,f},ColorData["Rainbow"][z]],Mesh->None,PlotPoints->{7,7,7},BoxRatios->{1,1,1},Axes->True,AxesLabel->{"x","y","z"},BoxRatios->{1,1,1}]

Flow Curvature Invariance (right side)

Normal vector

η={D[ϕ,x],D[ϕ,y],D[ϕ,z],D[ϕ,u],D[ϕ,w]};

Invariance

Ψ=Factor[V . η/.{u->0,w->0}];

Factor[(Ψ/ϕ)/.{x->0,y->0,z->0}]

Factor[Tr[J]]

This a consequence of Theorem 8.2 and Corollary 8.2 (Eq. 8.7) page 141. Thus, the invariance of the osculating plane is stated ipso facto.

{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])}={0,0,0};
fixedpoint0=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])}}]];

Show[phaseplot,flowmanifoldL,flowmanifoldR,fixedpoint1,fixedpoint0,fixedpoint2,BoxRatios->{1,1,1},Boxed->False,Axes->False,ImageSize->{600,600}]