Cobweb plot

<h2 id="method">Method</h2>
<p>For a given iterated function 
  
    
      
        f
        :
        
          R
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }
  
, the plot consists of a diagonal (
  
    
      
        x
        =
        y
      
    
    {\displaystyle x=y}
  
) line and a curve representing 
  
    
      
        y
        =
        f
        (
        x
        )
      
    
    {\displaystyle y=f(x)}
  
. To plot the behaviour of a value 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
, apply the following steps.
</p>
<ol><li>Find the point on the function curve with an x-coordinate of 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
. This has the coordinates (
  
    
      
        
          x
          
            0
          
        
        ,
        f
        (
        
          x
          
            0
          
        
        )
      
    
    {\displaystyle x_{0},f(x_{0})}
  
).</li>
<li>Plot horizontally across from this point to the diagonal line. This has the coordinates (
  
    
      
        f
        (
        
          x
          
            0
          
        
        )
        ,
        f
        (
        
          x
          
            0
          
        
        )
      
    
    {\displaystyle f(x_{0}),f(x_{0})}
  
).</li>
<li>Plot vertically from the point on the diagonal to the function curve. This has the coordinates (
  
    
      
        f
        (
        
          x
          
            0
          
        
        )
        ,
        f
        (
        f
        (
        
          x
          
            0
          
        
        )
        )
      
    
    {\displaystyle f(x_{0}),f(f(x_{0}))}
  
).</li>
<li>Repeat from step 2 as required.</li></ol>
<h2 id="interpretation">Interpretation</h2>
<p>On the Lémeray diagram, a stable <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> corresponds to the segment of the staircase with progressively decreasing stair lengths or to an inward <a href="/facts/Spiral/4R331aba">spiral</a>, while an unstable fixed point is the segment of the staircase with growing stairs or an outward spiral.  It follows from the definition of a fixed point that the staircases <a href="/facts/Converge_(mathematics)/Ktge2RwL">converge</a> whereas spirals center at a point where the <a href="/facts/Identity_function/9AtsP0LC">diagonal</a> 
  
    
      
        y
        =
        x
      
    
    {\displaystyle y=x}
  
 line crosses the function graph.  A period-2 <a href="/facts/Orbit_(dynamics)/WBpc3bUH">orbit</a> is represented by a <a href="/facts/Rectangle/mdLfIGKA">rectangle</a>, while greater period cycles produce further, more complex closed loops. A <a href="/facts/Chaos_theory/LTC17vsF">chaotic</a> orbit would show a "filled-out" area, indicating an infinite number of non-repeating values.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Jones_diagram/DBRwJtQs">Jones diagram</a> – similar plotting technique</li>
<li><a href="/facts/Fixed-point_iteration/05dQGazd">Fixed-point iteration</a> – iterative algorithm to find fixed points (produces a cobweb plot)</li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Cobweb plot.

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lémeray, E.-M. (1897). "Sur la convergence des substitutions uniformes" (PDF). Nouvelles annales de mathématiques, 3e série. 16: 306–319. <a href="http://www.numdam.org/item/NAM_1898_3_17__75_1.pdf" target="_blank">http://www.numdam.org/item/NAM_1898_3_17__75_1.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Stoop, Ruedi; Steeb, Willi-Hans (2006). Berechenbares Chaos in dynamischen Systemen [Computable Chaos in dynamic systems] (in German). Birkhäuser Basel. p. 8. doi:10.1007/3-7643-7551-5. ISBN 978-3-7643-7551-5. <a href="978-3-7643-7551-5" target="_blank">978-3-7643-7551-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Stoop, Ruedi; Steeb, Willi-Hans (2006). Berechenbares Chaos in dynamischen Systemen [Computable Chaos in dynamic systems] (in German). Birkhäuser Basel. p. 8. doi:10.1007/3-7643-7551-5. ISBN 978-3-7643-7551-5. <a href="978-3-7643-7551-5" target="_blank">978-3-7643-7551-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Cobweb plot open-in-new

Cobweb plot