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136 lines
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136 lines
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<article id="content" role="main">
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<h1 class="center">Bifurcation</h1>
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<p class="center"><img src="images/path-fork.jpg" alt="path fork" height="122" width="600"><br>
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Bifurcation means splitting into two parts: "bi" (two), and "furca" (fork).</p>
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<p>As some functions evolve they suddenly split into two!</p>
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<p>First we will need a function:</p>
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<p class="center larger"><b>rx(1−x)</b> is a good one.</p>
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<p><b>x</b> is the input value, and <b>r</b> is a value we want to investigate.</p>
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<p>We will calculate the function over and over again, each time using the result as the new x value.</p>
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<p>Let us try <b>r=2</b>, and start with <b>x=0.2</b>:</p>
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<p class="center large">2 × 0.2(1−0.2) = 2 × 0.2 × 0.8 = <b>0.32</b></p>
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<p class="center">Now with the new <b>x</b> value: 2 × 0.32 × 0.68 = <b>0.4352</b><br>
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And once again: 2 × 0.4352 × 0.5648 = <b>0.49160...</b><br>
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And again: 2 × 0.49160... × 0.50840... = <b>0.49986...</b><br>
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And again: 2 × 0.49986... × 0.50014... = <b>0.50000...</b></p>
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<p>We see it is settling to <b>0.5</b></p>
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<p>But is not always so simple. Try some other <b>r</b> values here:</p>
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<div class="script" style="height: 340px;">
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images/bifurc.js?mode=simple
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</div>
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<p>What do you notice?</p>
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<ul>
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<li>How about when r is around <b>3.2</b> ? It jumps between <b>two</b> values.</li>
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<li>And around r = <b>3.5</b> it jumps between <b>four</b> values.</li>
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<li>And what is the story around r = <b>3.7</b> ? It seems like complete <b>chaos</b>.</li>
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<li>But then try around <b>3.84</b>: it briefly settles down, then goes crazy again.</li></ul>
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<p>We need to investigate more!</p>
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<p>So try here. Same idea as above, but a lower plot keeps track of the last few values achieved at each r-value. It needs your help to make the plot:</p>
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<div class="script" style="height: 600px;">
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images/bifurc.js?mode=both
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</div>
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<p>Here is a closer look at it, notice the "forks" (where the name bifurcation comes from), and also notice how it goes from order to chaos and sometimes back again:</p>
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<p class="center"><img src="images/bifurc.gif" alt="Bifurcation Plot" height="253" width="360"></p>
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<p>The change between order and chaos is also seen in nature.</p>
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<p>For example populations of animals can be steady, or show this "one year many, next year few" pattern, or be just very chaotic.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/drips.svg" alt="drips" height="401" width="274"></p>
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<h2>Dripping Taps</h2>
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<p>Get a tap dripping.</p>
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<p>It can be steady (drip, drip, drip), but at a different flow rate you may get a "double drip" (like when <b>r</b> is around <b>3.2</b> above).</p>
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<p>Change the flow again and the drips can seem random.</p>
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<p>Like in these photos:</p>
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<h2>Mandelbrot Set</h2>
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<p>There is a relationship between the bifurcation diagram and the <a href="../numbers/mandelbrot.html">Mandelbrot set</a>. You may like to investigate that!</p>
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<h2>More</h2>
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<p>There are more types of bifurcation in mathematics!</p>
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<p>And there is a whole lot more to learn in this really interesting subject called Dynamical Systems.</p>
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<p><br></p>
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<div class="related">
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<a href="index.html">Calculus Index</a>
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