Let's look at the Lorenz attractor!

As usual, start by loading a special package.

<<"VectorFieldPlots`"

Now we'll look at the vector field for the Lorenz attractor.

lorfield = VectorFieldPlot3D[{-3 x+3 y,-x z+30 x-y,x y-z},{x,-15,15},{y,-25,25},{z,-5,65}]

How's about we generate the data for the trajectory shown in the book?

Clear[t];
Clear[tinitial];
Clear[tfinal];
Clear[numberofsteps];
Clear[x];
Clear[y];
Clear[z];
Clear[deltat];
Clear[datapoints];
datapoints = {};
x=0;
y=1;
z=0;
tinitial=0;
tfinal = 100;
t = tinitial;
numberofsteps = 10000;
deltat = (tfinal- tinitial)/numberofsteps;
For[k=1,k≤numberofsteps,
xprime = -3x+3y; 
yprime =-x z + 30 x - y; 
zprime = x y - z; 
deltax = xprime*deltat; 
deltay = yprime*deltat; 
deltaz = zprime*deltat; 
t=N[t+deltat]; 
x = N[x+deltax]; 
y = N[y + deltay]; 
z = N[z+deltaz];
datapoints =Append[datapoints, {x,y,z}]; 
 k++]

And, we'll look at that trajectory together with the vector field.  Feel free to live-rotate this puppy.

Clear[x]
Clear[y]
Clear[z]
Show[Graphics3D[Line[datapoints]],VectorFieldPlot3D[{-3 x+3 y,-x z+30 x-y,x y-z},{x,-15,15},{y,-25,25},{z,-5,65}],Axes->True,AspectRatio->1]

Here we have the trajectory projected onto the x-y plane...

Clear[t]
ListPlot[Table[Drop[datapoints[[t]],{3}],{t,1,Length[datapoints]}],PlotStyle->PointSize[0.001`],Joined->True]

...and the trajectory projected onto the x-z plane...

ListPlot[Table[Drop[datapoints[[t]],{2}],{t,1,Length[datapoints]}],PlotStyle->PointSize[0.001`],Joined->True]

...and the trajectory projected onto the y-z plane.

ListPlot[Table[Drop[datapoints[[t]],{1}],{t,1,Length[datapoints]}],PlotStyle->PointSize[0.001`],Joined->True]

Now we'll make a 3-D animation of the trajectory so we can see how it develops.  Go ahead and pause the animation and live-rotate the paused image (it's fun).

Animate[Graphics3D[Line[Take[datapoints,k]]],{k,1,Length[datapoints],1}]

Oooh!  We can also do it with the vector field displayed as well!  Again, pause and live-rotate...

Clear[x]
Clear[y]
Clear[z]
Animate[Show[{Graphics3D[Line[Take[datapoints,k]]],lorfield}],{k,1,Length[datapoints],1}]