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plotting - How to change the axis on this ListPlot?


My bifurcation diagram almost works, but I can't figure out how to replace the point number on the horizontal axis with the x coordinate from my calculation. I have tried a whole bunch of things that fail in various ways. The code is below. Help would be greatly appreciated.



bifurcation


z[0, c_] := c;
z[n_, c_] := z[n - 1, c]^2 + c;
zSeries[n_, len_, c_] := Module[{ser},
   ser = {z[n, c]};
   Do[AppendTo[ser, ser[[i]]^2 + c], {i, 1, len - 1}];
   ser];
ListPlot[Transpose[Table[zSeries[999, 100, x], {x, -2, .25, .0001}]], PlotStyle -> PointSize[Tiny]]

If I change the zSeries module to give each point its own x value, it gets VERY slow.



zSeries[n_, len_, c_] := Module[{ser},
   ser = {{c, z[n, c]}};
   Do[AppendTo[ser, {c, ser[[i]]^2 + c}], {i, 1, len - 1}];
   ser];

Answer



To illustrate my comment, is that what you had in mind?


ListPlot[Transpose[Table[zSeries[999, 100, x], {x, -2, .25, .0001}]], 
    PlotStyle -> PointSize[Tiny], DataRange -> {-2, .25}]

enter image description here



Addendum: Timing


The OP in his answer provides revised code,


z[0, c_] := c;
z[n_, c_] := z[n - 1, c]^2 + c;
ListPlot[Catenate[Table[{x, #} & /@ NestList[#^2 + x &, z[999, x], 100], 
    {x, -2, .25, .0001}]], PlotStyle -> PointSize[Tiny]]

which is much faster than the earlier code, requiring 64 sec (AbsoluteTiming) on my PC to generate the plot. Still, it seemed to me that further improvements in time could be achieved. Table requires 37 sec to generate the array, leaving 27 sec for ListPlot itself. Because as many as 100 duplicate points are plotted for x > -1.4, applying DeleteDuplicates offers an obvious savings. (It, like Catenate, takes negligible time.) Using FixedPointList instead of NestList also saves a bit of time for x > -.76, but only a few seconds, because the reduced list generation is largely offset by the test for a fixed point. Together,


ListPlot[Catenate[Table[{x, #} & /@ DeleteDuplicates[
   FixedPointList[#^2 + x &, z[999, x], 100]], {x, -2, .25, .0001}]], 

   PlotStyle -> PointSize[Tiny], PlotRange -> All]

requires only about 41 sec. The other opportunity for savings involves replacing z[999, x], which takes some 31 sec in all, by the equivalent Nest[#^2 + x &, x, 999],


ListPlot[Catenate[Table[{x, #} & /@ DeleteDuplicates[
    FixedPointList[#^2 + x &, Evaluate[Nest[#^2 + x &, x, 999]], 100]], 
    {x, -2, .25, .0001}]], PlotStyle -> PointSize[Tiny], PlotRange -> All]

which reduces total time to 11 sec, a significant improvement. (Using FixedPoint instead of Nest offers no further advantage.)


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