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<title>Cobweb plot</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Cobweb plot</span></span>
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<p>A <b>cobweb plot</b>, known also as <b>Lémeray Diagram</b> or <b>Verhulst diagram</b> is a visual tool used in <a href="Dynamical_system" title="Dynamical system">dynamical systems</a>, a field of <a href="Mathematics" title="Mathematics">mathematics</a> to investigate the qualitative behaviour of one-dimensional <a href="Iterated_function" title="Iterated function">iterated functions</a>, such as the <a href="Logistic_map" title="Logistic map">logistic map</a>. The technique was introduced in 1822 by <a href="Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Using a cobweb plot, it is possible to infer the long-term status of an <a href="Initial_condition" title="Initial condition">initial condition</a> under <a href="Recurrence_relation" title="Recurrence relation">repeated application</a> of a map.<sup id="cite_ref-stoop_2-0" class="reference"><a href="#cite_note-stoop-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
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<div class="mw-heading mw-heading2"><h2 id="Method">Method</h2></div>
<p>For a given iterated function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }">
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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }</annotation>
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</math></span><img src="./85e6e186aabef9e51814bbce62e625dc67e825f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }" loading="lazy"></span>, the plot consists of a diagonal (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}">
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</math></span><img src="./409a91214d63eabe46ec10ff3cbba689ab687366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}" loading="lazy"></span>) line and a curve representing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}">
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<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>
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</math></span><img src="./2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy"></span>. To plot the behaviour of a value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>, apply the following steps.
</p>
<ol><li>Find the point on the function curve with an x-coordinate of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>. This has the coordinates (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},f(x_{0})}">
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<mi>x</mi>
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<mi>f</mi>
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<mi>x</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle x_{0},f(x_{0})}</annotation>
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</math></span><img src="./197db5bdc05f1ef6a248feb434a7fa1bb35e183c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.89ex; height:2.843ex;" alt="{\displaystyle x_{0},f(x_{0})}" loading="lazy"></span>).</li>
<li>Plot horizontally across from this point to the diagonal line. This has the coordinates (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0}),f(x_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
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<mi>x</mi>
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<mn>0</mn>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle f(x_{0}),f(x_{0})}</annotation>
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</math></span><img src="./9d3b9cf902f6345e6b230aa51849b5a29e45aa18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.978ex; height:2.843ex;" alt="{\displaystyle f(x_{0}),f(x_{0})}" loading="lazy"></span>).</li>
<li>Plot vertically from the point on the diagonal to the function curve. This has the coordinates (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0}),f(f(x_{0}))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle f(x_{0}),f(f(x_{0}))}</annotation>
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</math></span><img src="./14937e261fd3f85c5a4268dbdb4cf9dc19af602d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.065ex; height:2.843ex;" alt="{\displaystyle f(x_{0}),f(f(x_{0}))}" loading="lazy"></span>).</li>
<li>Repeat from step 2 as required.</li></ol>
<div class="mw-heading mw-heading2"><h2 id="Interpretation">Interpretation</h2></div>
<p>On the Lémeray diagram, a stable <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a> corresponds to the segment of the staircase with progressively decreasing stair lengths or to an inward <a href="Spiral" title="Spiral">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="Converge_(mathematics)" class="mw-redirect" title="Converge (mathematics)">converge</a> whereas spirals center at a point where the <a href="Identity_function" title="Identity function">diagonal</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle y=x}</annotation>
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</math></span><img src="./d0abe2e7da593fb7b41d44e87a97fefdd8998b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}" loading="lazy"></span> line crosses the function graph. A period-2 <a href="Orbit_(dynamics)" title="Orbit (dynamics)">orbit</a> is represented by a <a href="Rectangle" title="Rectangle">rectangle</a>, while greater period cycles produce further, more complex closed loops. A <a href="Chaos_theory" title="Chaos theory">chaotic</a> orbit would show a "filled-out" area, indicating an infinite number of non-repeating values.<sup id="cite_ref-stoop_2-1" class="reference"><a href="#cite_note-stoop-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Jones_diagram" title="Jones diagram">Jones diagram</a> – similar plotting technique</li>
<li><a href="Fixed-point_iteration" title="Fixed-point iteration">Fixed-point iteration</a> – iterative algorithm to find fixed points (produces a cobweb plot)</li></ul>
<p><br>
</p>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFRosa2021" class="citation journal cs1">Rosa, Alessandro (2021). <a rel="nofollow" class="external text" href="https://wydawnictwa.ptm.org.pl/index.php/antiquitates-mathematicae/article/viewArticle/7056">"An episodic history of the staircased iteration diagram"</a>. <i>Antiquitates Mathematicae</i>. <b>15</b>: <span class="nowrap">3–</span>90. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.14708%2Fam.v15i1.7056">10.14708/am.v15i1.7056</a>.</cite></span>
</li>
<li id="cite_note-stoop-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-stoop_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-stoop_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFStoopSteeb2006" class="citation book cs1 cs1-prop-foreign-lang-source">Stoop, Ruedi; Steeb, Willi-Hans (2006). <i>Berechenbares Chaos in dynamischen Systemen</i> [<i>Computable Chaos in dynamic systems</i>] (in German). Birkhäuser Basel. p. 8. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-7643-7551-5">10.1007/3-7643-7551-5</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-7551-5</bdi>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Cobweb_plots" class="extiw external" title="c:Category:Cobweb plots"><b><i>Cobweb plot</i></b></a>.</div></div>
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This article is issued from <a class="external text" title="Last edited on 2025-07-29" href="https://en.wikipedia.org/wiki/?title=Cobweb_plot&oldid=1303135916">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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