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Here we treat the logistic map which yields chaotic orbit. It's written as xn+1=axn(1-xn). (where 2 < a < 4) This map receives a real number between 0 and 1, then returns a real number in [0,1] again. The various sequences are generated depending on the parameter a and the initial value x0. |
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The most simple example is the case where the sequence xn converge to a fixed point xe independent on the initial value x0. Such situations are realized when the parameter a is between 2 and 3. You can confirm this fact in this page.
When parameter a exceeds 3, what happens to the sequence xn ? As you see with the help of this page again, the sequence converges to periodic orbit of period 2. |
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If you make the parameter a larger, the period of the periodic orbit will be doubled, that is, 4, 8, 16.... This is called period doubling cascade, and beyond this cascade, the stable periodic orbit disapears and Chaos appears. This transition of the orbit structure with the change of parameter is called bifurcation phenomena. The above applet visualizes this cascade from a fixed point to chaos. The horizontal axis denotes parameter a, and vertical one is variable x. When you drag the area with your mouse, the region is expanded. |