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(* This notebook contains code for viewing the attractor and generating
   time series data for the Lorenz equations. Note that evaluating this 
   notebook requires a computer with a fair bit of memory. A 68040 25MHz 
   Mac with 16M running the student version of Mathemtica 2.2 and a 
   utility that uses the disk to simulate twice the actual RAM was able 
   to correctly evaluate the notebook in a few minutes of real time. *)
:[font = input; preserveAspect]
(* Here is the version of the Lorenz equations given in Wolfram. *) 
:[font = input; preserveAspect]
equationsLorenz1 = Module[{sigmaLorenz = 3, 
			   rhoLorenz = 26.5, 
			   betaLorenz = 1},
			{x'[t] == sigmaLorenz (y[t] - x[t]),
			 y'[t] == rhoLorenz x[t] - y[t] - x[t] z[t],
			 z'[t] == - betaLorenz z[t] + x[t] y[t],
			 x[0] == 0, z[0] == 0, y[0] == 1}] ;
:[font = input; preserveAspect]
(* Here is the version of the Lorenz equations given in Sauer. *) 
:[font = input; preserveAspect]
equationsLorenz2 = Module[{sigmaLorenz = 10, 
                           rhoLorenz = 28, 
			   betaLorenz = 8/3},
			{x'[t] == sigmaLorenz (y[t] - x[t]),
			 y'[t] == rhoLorenz x[t] - y[t] - x[t] z[t],
			 z'[t] == - betaLorenz z[t] + x[t] y[t],
			 x[0] == 0, z[0] == 0, y[0] == 1}] ;
:[font = input; preserveAspect]
(* Set the range for sampling the system. *)
:[font = input; preserveAspect]
samplingRange = {0.0,20.0} ;
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solutionLorenz1 = NDSolve[equationsLorenz1, {x,y,z},
      {t,samplingRange[[1]],samplingRange[[2]]}, MaxSteps -> 3000] ;
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Plot[Evaluate[x[t] /. solutionLorenz1],
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MathPictureEnd

:[font = input; preserveAspect]
(* Set the sampling rate. *) 
:[font = input; preserveAspect]
samplingRate = 100 ;
:[font = input; preserveAspect]
dataLorenz1 = Flatten[Table[Evaluate[x[t] /. solutionLorenz1],
			    {t,samplingRange[[1]],
			       samplingRange[[2]],
			       1/samplingRate}]] ;
:[font = input; preserveAspect]
(* If you want to save the data to a file, execute something like the
   following: dataLorenz1 >> LorenzData.txt    *)
:[font = input; preserveAspect]
(* Here are some routines for looking at phase plots of delay coordinate
   embeddings of time series taken from solutions to the Lorenz equations. *)
:[font = input; preserveAspect]
(* Routine for fixed-width delay vectors with prepended sample times. *)
:[font = input; preserveAspect]
ComputeAugmentedDelayVectors[series_,width_] := 
  Table[Prepend[Table[series[[i+j]],
		      {j,0,width-1}],
		N[(i+width-1) 1/samplingRate]],
        {i,1,Length[series]-width+1}] ;
:[font = input; preserveAspect]
(* Compute augmented width-3 delay vectors from the time series data. *)
:[font = input; preserveAspect]
ViewDelayCoordinateEmbedding[series_,width_] := 
  Module[{v = ComputeAugmentedDelayVectors[series,width], 
	  n = Length[series]-width+1, x, y, z},
  ParametricPlot3D[
    Evaluate[{x[t],y[t],z[t]} /. 	
      MapThread[(#1 -> Interpolation[#2])&,
		{{x,y,z},Table[Table[{v[[i]][[1]], v[[i]][[j]]},
				     {i,1,n}],
			       {j,2,width+1}]}]], 
		   {t,First[v][[1]],Last[v][[1]]}]] ;
:[font = input; preserveAspect; startGroup]
ViewDelayCoordinateEmbedding[dataLorenz1,3] ;
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 268; pictureHeight = 287; endGroup]
%!
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% End of Graphics
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