Example 2.  Solve the Lorenz I. V. P.  
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Use Mathematica's NDSolve procedure.  

Solution 2.

Caution. The syntax must be followed carefully.  The symbol " is the boolean equal, you can use two ordinary equal signs next to each other if you prefer ".  

Mathematica's numerical D.E. solver.  It is more robust than the Runge-Kutta method, we can solve the D.E.'s over the larger interval  .  

Caution.  If you do not use the option  "" then Mathematica will give you the error message:

NDSolve::"mxst": "Maximum number of 1000 steps reached at the point t == 11.146875180780313`."

After you solve the problem successfully, you should make copies of the commands and delete the MaxSteps option so you can see "numerical analysis in action."  (Or the lack of it, because no one wants to report a bad answer.)

We need to get the interpolating functions in three coordinates. The statement ReplaceAll is uses replacement rules to obtain a new form which in this case is needed to draw a graph.  

Now we can get a nice graph.  Unfortunately, ParametricPlot3D does not support colors.

Caution.  If you do not use the option  "" then Mathematica not use enough points and you will not see the "true curve."

After you solve the problem successfully, you should make copies of the commands and delete the PlotPoints option so you can see the "mess."  It takes a lot of computation to make complicated figures that are accurate.  

Caveat. If you want to take these nice pictures home on a disk then you can.  However the numerical computations are tremendous and the size on the disk is about 180K.  So you cannot draw very many 3D pictures and store them on a floppy disk.  We usually suggest that you delete the output by going to the Kernel menu, and use the Delete All Output submenu.  All the necessary Mathematica commands can then be saved and you can re-execute (every step) later.  To re-execute everything in a Mathematica notebook use the Kernel menu, Evaluation submenu, Evaluate Notebook submenu.

Remark.  If you want to use NDSolve  and graph the projections. Then the following commands are needed.

We can plot the projection on the xy-plane.

We can plot the coordinate function  x = x[t].

(c) John H. Mathews 2004