b=8/3;
σ=10;

Lorenz Atrractor

Manipulate[s=NDSolve[{x'[t]==σ*(y[t]-x[t]),y'[t]==r*x[t]-y[t]-x[t]*z[t],z'[t]==x[t]*y[t]-b*z[t],x[0]==x0,y[0]==y0,z[0]==z0},{x,y,z},{t,0,T}];
ParametricPlot3D[{Evaluate[{x[t],y[t],z[t]}/.s],Evaluate[{-1*x[t],-1*y[t],z[t]}/.s]},{t,0,T},PlotRange->{{-30,30},{-30,30},{-10,50}},PlotStyle->{{Thick,Red},{Thick,Blue}},Boxed->False,ViewPoint->{6,3,2}],
{T,0,100},{x0,0,20},{y0,0,20},{z0,0,30},{r,0,30}]

x-dynamics

Manipulate[s=NDSolve[{x'[t]==σ*(y[t]-x[t]),y'[t]==r*x[t]-y[t]-x[t]*z[t],z'[t]==x[t]*y[t]-b*z[t],x[0]==x0,y[0]==y0,z[0]==z0},{x,y,z},{t,0,T}];
Plot[{Evaluate[x[t]/.s],Evaluate[-x[t]]/.s},{t,0,T},PlotRange->{-30,30}],
{T,0,100},{x0,0,20},{y0,0,20},{z0,0,30},{r,0,30}]

Lorenz Section

r=30;
x0=.1;
y0=0;
z0=20;
s=NDSolve[{x'[t]==σ*(y[t]-x[t]),y'[t]==r*x[t]-y[t]-x[t]*z[t],z'[t]==x[t]*y[t]-b*z[t],x[0]==x0,y[0]==y0,z[0]==z0},{x,y,z},{t,0,50}];
ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.s],{t,40,50},PlotRange->{{-30,30},{-30,30},{-10,50}},PlotStyle->Red,Boxed->False,ViewPoint->Right]