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Something is wrong with my cobweb plot code but I couldn't figure out where do the mistakes come from.

So basically we have the iterative map $x_n == x_{n-1}^{2} - 3 \mu$. I would like to draw a cobweb diagram that shows in the range $-\frac{1}{12}<\mu<\frac{1}{4}$ there exists a fixed point $\frac{1}{2}(1-\sqrt{1+12\mu})$.

Below are the code and the diagram:

enter image description here

enter image description here

enter image description here

My questions are:
1. Why the diagonal line in the cobweb is not appearing and why is it replaced by a horizontal blue line?
2. The label on top is wrong and I couldn't fix it.
3. How can I fix the cobweb code so that I can show, say let $-\frac{1}{12}<\mu=0.2<\frac{1}{4}$ and initial value 2, there is a convergence towards a fixed point $\frac{1}{2}(1-\sqrt{1+12\mu})$.

Many thanks. I really appreciate any helps.

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"Why the diagonal line in the cobweb is not appearing and why is it replaced by a horizontal blue line?" - if you'll look at the scale of your axes, you should see why the line $y=x$ ought to look flat... –  J. M. May 19 '13 at 12:46
    
@J.M. Then how can I change the scale to make it look correct? I have some troubles changing the options, it will make the plot even worse. –  user71346 May 19 '13 at 12:51
    
I'd say you have an even more fundamental problem: your iteration is diverging quite wildly, as your starting point is rather far away from the domain of convergence of your iteration. Try a smaller starting value. –  J. M. May 19 '13 at 13:05
    
@J.M. I chose x0 = 2 because it is close to the fixed point which is approx 1.4. I tried x0 ranging from -2 to 2 but no big changes. –  user71346 May 19 '13 at 13:38
1  
When $\mu=0.2$, I calculate the fixed point $\frac{1}{2}(1-\sqrt{1+12\mu})$ to be around $-0.422$ rather than 1.4. The latter is a different fixed point. And as already said, you need to start close to the fixed point. –  murray May 19 '13 at 14:40
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closed as too localized by whuber, Oleksandr R., Ajasja, Artes, rm -rf May 19 '13 at 16:33

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1 Answer

As Murray noted, you must have made an error calculating the attraction point:

1/2 (1 - Sqrt[1 + 12  μ]) /.  μ -> 0.2

-0.4219544457

Trying a point close to this with your code (Please, please, never provide code again as a bitmap. Typing this is no fun.)

cobweb::usage = 
  "cobweb[f,x0,nmax,ndrop] produces a cobweb plot for the recursive
  function f[x] with initial value x0, suppressing the first ndrop
  iterates. Options may also be passed to Plot. ";
cobweb[f_, x0_, range_, nmax_, ndrop_: 0, opts___Rule] :=
 Module[{plot1, plot2, p1ot3},
  plot1 = 
    Plot[{x, f[x]}, {x, range[[1]], range[[2]]}, opts, Frame -> True, 
       FrameLabel -> {"\!\(\*SubscriptBox[\(x\), \(i\)]\)", 
      "\!\(\*SubscriptBox[\(x\), \(i + 1\)]\)"}, 
       PlotLabel -> StringJoin["Cobweb Plot for: ", ToString[f[x], TraditionalForm]]]; 
  plot2 = Graphics[
            Map[Line[{{#, #}, {#, f[#]}, {f[#], f[#]}}] &, 
            Drop[NestList[f, x0, nmax], ndrop]]];
  p1ot3 = ListPlot[{{x0, x0}}, PlotStyle -> PointSize[0.03]];
  If[ndrop == 0, Return[Show[{plot1, plot2, p1ot3}]], 
   Return[Show[{plot1, plot2}]]]]

mapCobweb[μ_, x0_, range_, nmax_, ndrop_: 0, opts___Rule] := 
 cobweb[#^2 - 3 μ &, x0, range, nmax, ndrop, opts]

mapCobweb[0.2, -0.6, {-1, 1}, 40, 0, ImageSize -> 800]

enter image description here

share|improve this answer
    
OP might want to contemplate why the attractive fixed point is the one much nearer to the vertex of the parabola... –  J. M. May 19 '13 at 15:27
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