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    <h1 align="center">
      The logistic map
    </h1>
    <p>
      The logistic equation was first proposed by <b>Robert May</b> as a 
      simple model of population dynamics. This equation can be written as a 
      one-dimensional difference equation that transforms the population in 
      one generation, <i>x<sub>n</sub></i>, into a succeeding generation, <i>x<sub>n+1</sub></i>.
    </p>
    <p align="center">
      <i>x<sub>n+1 </sub>= 4 r x<sub>n</sub> (1-x<sub>n</sub>)</i>
    </p>
    <p>
      Because the population is scaled so that the maximum value is one, the 
      domain of <i>x</i> falls on the interval<i> [0; 1]</i>.
    </p>
    <p>
      The behavior of the logistic equation depends on the value of the growth 
      parameter, <i>r</i>. If the growth parameter is less than a critical 
      value <i>r&lt;0.75</i>..., then <i>x</i> approaches a stable fixed 
      value. Above this value for <i>r</i>, the behavior of <i>x</i> begins to 
      change. First the population begins to oscillate between two values. If <i>r</i> 
      increases further, then <i>x</i> oscillates between four values, then 
      eight values. This doubling ends when <i>r &gt; 0.8924864..</i>. after 
      which almost any <i>x</i> value is possible.
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    <h1 align="center">
      Generalized logistic map
    </h1>
    <p align="center">
      <i>x<sub>n+1 </sub>= 4rx<sub>n</sub>(1-x<sub>n</sub>) = 4 r ( x<sub>n</sub> 
      -x<sub>n</sub><sup>2</sup>) </i>
    </p>
    <p align="center">
      
    </p>
    <p align="left" style="margin-top: 0">
      The logistic series rule sets the next generation proportional to the 
      existing one in its first term. This alone would lead to exponential 
      growth for <i>r &gt; 1/4</i>, and to exponential decline for <i>r &lt; 1/4</i>). 
      The second term introduces a diminution that depends on the square of 
      the existing population (note that in the above formulation <i>x<sub>n</sub> 
      &lt; 1</i> , hence <i>x<sub>n</sub><sup>2</sup> &lt; x<sub>n</sub></i> ).
    </p>
    <p>
      The issue for a given growth rate <i>r</i> is:&#160; will the population 
      approach a stable equilibrium (limit) value of linear growth and 
      quadratic extinction &#8722; assuming an unlimited number of generations under 
      identical conditions. If so, how does the equilibrium value depend on 
      the growth rate<i> r </i>?
    </p>
    <p>
      Growth exists when <i>x<sub>n+1 </sub>&gt; x<sub>n</sub></i>, hence <i>r 
      &gt; 1/(4(1-x<sub>n</sub>))</i>. As <i>0 &lt; x &lt; 1</i> , for <i>r &lt; 
      0.25,</i> all populations will iterate to zero, independent of the 
      starting value. If for <i>r &gt; 0.25</i> an equilibrium value exists, a 
      starting population greater than the limit should shrink to it, smaller 
      ones should expand to it.
    </p>
    <p>
      In the simulation <i>r</i> is increased in steps of 0.001 in the range <i>0 
      &lt; r &lt; 1 </i>. The calculation for each step starts with a random 
      value <i>0 &lt; x<sub>1</sub>&lt; 1 </i>. In a calculation loop 2000 
      members of the series are calculated. The first ones differ largely in 
      dependence on the random initial value. Therefore the first 1000 
      iterations are suppressed in the chart. For each step in <i>r</i> 1000 
      points on the ordinate could represent the iterations 1000 to 2000.
    </p>
    <p>
      In the range <i>0.25 &lt; r &lt; 0.75</i> the iterations are so close 
      together that they appear as one point only, resulting in a <i>&quot;limit 
      curve&quot;</i> in dependence on <i>r</i>.
    </p>
    <p>
      Then the curve splits in two (<i>bifurcation</i>), which means that the 
      iteration now has two accumulation points for a certain <i>r. </i>The 
      bifurcation repeats itself, until no accumulation points are visible any 
      longer.
    </p>
    <p>
      Quite surprisingly between the &quot;filled&quot; bands there are some quasi 
      &quot;empty&quot; bands with only a few accumulation points.
    </p>
    <p>
      The determining term is the product <i>4r</i>; factors different from <i>4</i> 
      just scale the abscissa differently.
    </p>
    <p>
      It is not decisive for the bifurcation that the limiting term is exactly <i>(1-x<sub>n</sub>)</i>. 
      The crucial point is the <u>nonlinearity </u>of the conjunction <i>x<sub>n</sub> 
      -x<sub>n</sub><sup>2</sup>.</i>
    </p>
    <p>
      To demonstrate this, a generalized series rule is used in this 
      simulation, using a term <i>(1-x<sub>n</sub><sup>k</sup>), with k &gt; 0 </i>:
    </p>
    <p align="center">
      <i>x<sub>n+1 </sub>= 4rx<sub>n</sub>(1-x<sub>n</sub><sup>k</sup>)</i>
    </p>
    <p>
      When opening the simulation <i>k =&#160; 1</i>; <b>Start </b>produces 
      the common logistic map<b>.</b>
    </p>
    <p>
      After <b>Stop</b> <i>k</i> can be changed in the range <i>0.1 &lt; k &lt; 2 </i>by 
      a <b>slider</b><i>. </i>The abscissa scaling is adjusted automatically.
    </p>
    <p>
      The left chart displays the total range, the right one that of 
      bifurcation with higher resolution. One can differentiate the 
      calculation steps and the bifurcation structure in more detail if the 
      window is expanded to full screen size.
    </p>
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  </head>
  <body>
    <p>
      <font size="+1"><b>Author</b> von LogisticMap.xlm : <a href="http://fem.um.es/index.html">Francisco 
      Esquembre </a>and Wolfgang Christian. </font>
    </p>
    <p>
      Text and original idea from the <a href="http://www.opensourcephysics.org">Open 
      Source Physics project</a> manual
    </p>
    <p>
      <font size="+1"><b>Date : </b></font>July 2003
    </p>
    <p>
      Generalized by Dieter Roess in August 08
    </p>
    <p style="margin-top: 0">
      This simulation is part of
    </p>
    <p style="margin-top: 0">
      &#8220;<a href="http://www.degruyter.de/cont/fb/ma/detail.cfm?id=IS-9783110250053-1">Learning 
      and Teaching Mathematics using Simulations </a>
    </p>
    <p style="margin-top: 0">
      &#8211; Plus 2000 Examples from Physics&#8221;
    </p>
    <p style="margin-top: 0">
      ISBN 978-3-11-025005-3, Walter de Gruyter GmbH &amp; Co. KG
    </p>
  </body>
</html>

]]>
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<PageComment><![CDATA[]]></PageComment>
<Variable>
<Name><![CDATA[r]]></Name>
<Value><![CDATA[0.1]]></Value>
<Type><![CDATA[double]]></Type>
<Dimension><![CDATA[]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[dr]]></Name>
<Value><![CDATA[0.01]]></Value>
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<Comment><![CDATA[]]></Comment>
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<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
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<Value><![CDATA[0]]></Value>
<Type><![CDATA[int]]></Type>
<Dimension><![CDATA[[points]]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[orbitmemory]]></Name>
<Value><![CDATA[0]]></Value>
<Type><![CDATA[int]]></Type>
<Dimension><![CDATA[[points*1100]]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[rmemory]]></Name>
<Value><![CDATA[0]]></Value>
<Type><![CDATA[int]]></Type>
<Dimension><![CDATA[[points*1100]]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[k]]></Name>
<Value><![CDATA[1]]></Value>
<Type><![CDATA[double]]></Type>
<Dimension><![CDATA[]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[rmin]]></Name>
<Value><![CDATA[]]></Value>
<Type><![CDATA[double]]></Type>
<Dimension><![CDATA[]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[rmax]]></Name>
<Value><![CDATA[]]></Value>
<Type><![CDATA[double]]></Type>
<Dimension><![CDATA[]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
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<Name><![CDATA[count]]></Name>
<Value><![CDATA[0]]></Value>
<Type><![CDATA[int]]></Type>
<Dimension><![CDATA[]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[]]></Name>
<Value><![CDATA[]]></Value>
<Type><![CDATA[double]]></Type>
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<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
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<Type>VARIABLE_EDITOR</Type>
<Name>Font</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<PageComment><![CDATA[]]></PageComment>
<Variable>
<Name><![CDATA[font]]></Name>
<Value><![CDATA["normal normal 1.5vw"]]></Value>
<Type><![CDATA[String]]></Type>
<Dimension><![CDATA[]]></Dimension>
<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
<Variable>
<Name><![CDATA[]]></Name>
<Value><![CDATA[]]></Value>
<Type><![CDATA[double]]></Type>
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<Domain><![CDATA[public]]></Domain>
<Comment><![CDATA[]]></Comment>
</Variable>
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<Osejs.Model.Initialization.Page>
<Type>CODE_EDITOR</Type>
<Name>Init Page</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[
//scales are calculated by an approximating formula
rmin = 0.7887*Math.pow(k,-0.7912);
rmax = 1.0447*Math.pow(k,-0.8022);
r = 0.0;

]]></Code>
</Content>
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<Type>CODE_EDITOR</Type>
<Name>Init Page 2</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[
var y = Math.random();
for (var i=0; i<points; i++) {
  y = 4*r*y*(1.0-Math.pow(y,k)); //Ignore the first iterations
}

orbit[0] = y;
for (var i=1; i<points; i++) {
  orbit[i] = 4*r*orbit[i-1]*(1.0-Math.pow((orbit[i-1]),k));
}

]]></Code>
</Content>
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<Type>CODE_EDITOR</Type>
<Name>axes</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[
_view.plottingPanel.getAxisX().setPosition([0,0.045]);
//_view.plottingPanel.getTitleX().setPosition([0.95,0.5]);
_view.plottingPanel.getAxisY().setPosition([0.045,0]);
//_view.plottingPanel.getTitleY().setPosition([0.5,0.95]);
]]></Code>
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<Name>MathJax</Name>
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<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
]]></Code>
</Content>
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<Name>Evol Page</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[
r += dr*rmax;
if (r >= rmax+0.1) {
  _pause();
}
count=count+1;


]]></Code>
</Content>
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<Type>CODE_EDITOR</Type>
<Name>Cons Page</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[

var y = Math.random();
for (var i=0; i<points; i++) {
  y = 4*r*y*(1.0-Math.pow(y,k)); //Ignore the first iterations
}

orbit[0] = y;
orbitmemory[0] =y; // 
for (var i=1; i<points; i++) {
  orbit[i] = 4*r*orbit[i-1]*(1.0-Math.pow((orbit[i-1]),k));
  orbitmemory[i+count*points] = orbit[i];
rmemory[i+count*points]= r;
}

]]></Code>
</Content>
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<Type>CODE_EDITOR</Type>
<Name>axes</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<Comment><![CDATA[]]></Comment>
<Code><![CDATA[
if (_model.isPlaying()===true) { //NEED this for conflict with initialize page values

_view.plottingPanel.getAxisX().setAbsoluteY(_view.plottingPanel.getGrid().getFixedTickY());
_view.plottingPanel.getAxisY().setAbsoluteX(_view.plottingPanel.getGrid().getFixedTickX());
}
]]></Code>
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<Osejs.Model.Elements.Element.Classname>org.colos.ejss.model_elements.numerics.MathJaxJavascriptElement</Osejs.Model.Elements.Element.Classname>
<Osejs.Model.Elements.Element.Name>mathJax</Osejs.Model.Elements.Element.Name>
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<Osejs.View.Creation>
</Osejs.View.Creation>
</Osejs.View>
<Osejs.HtmlView>
<Osejs.HtmlView.Page>
<Type>HTML_VIEW_EDITOR</Type>
<Name>Simulation View</Name>
<Active>true</Active>
<Internal>false</Internal>
<Content>
<SizeOption>0</SizeOption>
<X>0</X>
<Y>0</Y>
<Width>800</Width>
<Height>600</Height>
<KeepHidden>true</KeepHidden>
<RootProperties>
</RootProperties>
<Tree>
<HtmlView.Element>
<Expanded>true</Expanded>
<Type>Elements.Panel</Type>
<Name><![CDATA[panel2]]></Name>
</HtmlView.Element>
<HtmlView.Element>
<Type>Elements.Label</Type>
<Name><![CDATA[kText]]></Name>
<Parent><![CDATA[panel2]]></Parent>
<Property name="Text"><![CDATA["k = "+k.toFixed(2)]]></Property>
<Property name="Font"><![CDATA[font]]></Property>
</HtmlView.Element>
<HtmlView.Element>
<Type>Elements.Slider</Type>
<Name><![CDATA[slider]]></Name>
<Parent><![CDATA[panel2]]></Parent>
<Property name="Minimum"><![CDATA[0.1]]></Property>
<Property name="Maximum"><![CDATA[2]]></Property>
<Property name="Format"><![CDATA["k = 0.0"]]></Property>
<Property name="Value"><![CDATA[k]]></Property>
<Property name="OnChange"><![CDATA[_initialize();
_view._update();
]]></Property>
<Property name="Width"><![CDATA["50vw"]]></Property>
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<Type>Elements.TwoStateButton</Type>
<Name><![CDATA[playPauseButton]]></Name>
<Parent><![CDATA[panel2]]></Parent>
<Property name="OffClick"><![CDATA[%_pause%]]></Property>
<Property name="TextOn"><![CDATA["▶"]]></Property>
<Property name="State"><![CDATA[_isPaused]]></Property>
<Property name="TextOff"><![CDATA["❚❚"]]></Property>
<Property name="Tooltip"><![CDATA["Play/Pause the simulation"]]></Property>
<Property name="OnClick"><![CDATA[%_play%]]></Property>
<Property name="Font"><![CDATA[font]]></Property>
</HtmlView.Element>
<HtmlView.Element>
<Type>Elements.Button</Type>
<Name><![CDATA[resetButton]]></Name>
<Parent><![CDATA[panel2]]></Parent>
<Property name="Tooltip"><![CDATA["Reset the simulation"]]></Property>
<Property name="Text"><![CDATA["↻"]]></Property>
<Property name="OnClick"><![CDATA[%_reset%]]></Property>
<Property name="Font"><![CDATA[font]]></Property>
</HtmlView.Element>
<HtmlView.Element>
<Type>Elements.Label</Type>
<Name><![CDATA[logisticmaplatex]]></Name>
<Parent><![CDATA[panel2]]></Parent>
<Property name="Text"><![CDATA["$$x_{n+1} = 4 r x_{n}( 1-x_{n})$$"]]></Property>
</HtmlView.Element>
<HtmlView.Element>
<Expanded>true</Expanded>
<Type>Elements.Panel</Type>
<Name><![CDATA[panel]]></Name>
</HtmlView.Element>
<HtmlView.Element>
<Expanded>true</Expanded>
<Type>Elements.PlottingPanel</Type>
<Name><![CDATA[plottingPanel]]></Name>
<Parent><![CDATA[panel]]></Parent>
<Property name="Gutters"><![CDATA[[0,0,0,0]]]></Property>
<Property name="YScalePrecision"><![CDATA[1]]></Property>
<Property name="XFixedTick"><![CDATA[0]]></Property>
<Property name="MaximumY"><![CDATA[1.3]]></Property>
<Property name="MaximumX"><![CDATA[1.3]]></Property>
<Property name="YFixedTick"><![CDATA[0]]></Property>
<Property name="TRMessage"><![CDATA["Logistic Map"]]></Property>
<Property name="YAutoTicks"><![CDATA[false]]></Property>
<Property name="MinimumX"><![CDATA[0]]></Property>
<Property name="MinimumY"><![CDATA[0]]></Property>
<Property name="XTickStep"><![CDATA[0.1]]></Property>
<Property name="TitleY"><![CDATA["Population x"]]></Property>
<Property name="YTickStep"><![CDATA[0.1]]></Property>
<Property name="AutoScaleY"><![CDATA[false]]></Property>
<Property name="TitleX"><![CDATA["Parameter r"]]></Property>
<Property name="AutoScaleX"><![CDATA[false]]></Property>
<Property name="MarginX"><![CDATA[10]]></Property>
<Property name="MarginY"><![CDATA[10]]></Property>
<Property name="XAutoTicks"><![CDATA[false]]></Property>
<Property name="Height"><![CDATA["90vh"]]></Property>
<Property name="XScalePrecision"><![CDATA[1]]></Property>
<Property name="Width"><![CDATA["100%"]]></Property>
</HtmlView.Element>
<HtmlView.Element>
<Type>Elements.ShapeSet2D</Type>
<Name><![CDATA[shapeSetinstant]]></Name>
<Parent><![CDATA[plottingPanel]]></Parent>
<Property name="NumberOfElements"><![CDATA[points]]></Property>
<Property name="FillColor"><![CDATA["Green"]]></Property>
<Property name="SizeX"><![CDATA[0.01]]></Property>
<Property name="ShapeType"><![CDATA["ELLIPSE"]]></Property>
<Property name="X"><![CDATA[r]]></Property>
<Property name="LineColor"><![CDATA["Green"]]></Property>
<Property name="Y"><![CDATA[orbit]]></Property>
<Property name="Visibility"><![CDATA[false]]></Property>
<Property name="SizeY"><![CDATA[0.01]]></Property>
</HtmlView.Element>
<HtmlView.Element>
<Type>Elements.ShapeSet2D</Type>
<Name><![CDATA[shapeSetmemory]]></Name>
<Parent><![CDATA[plottingPanel]]></Parent>
<Property name="NumberOfElements"><![CDATA[points*500]]></Property>
<Property name="FillColor"><![CDATA["Green"]]></Property>
<Property name="SizeX"><![CDATA[0.01]]></Property>
<Property name="ShapeType"><![CDATA["ELLIPSE"]]></Property>
<Property name="X"><![CDATA[rmemory]]></Property>
<Property name="LineColor"><![CDATA["Green"]]></Property>
<Property name="Y"><![CDATA[orbitmemory]]></Property>
<Property name="SizeY"><![CDATA[0.01]]></Property>
</HtmlView.Element>
</Tree>
</Content>
</Osejs.HtmlView.Page>
</Osejs.HtmlView>
</Osejs>
