STABILITY, CHAOS AND BUTTERFLY EFFECT

Consider this equation
p(n+1)=p(n)*r*(1-p(n))

This equation is taken from population modeling. Here p represents population and r the rate of growth. We need not go into the origin of the equation. Here, the important point; p(n) is calculated from the equation and the p(n) is fed back into the equation to get the next p(n). The output to the input feed back calculation is repeated many times(iterated). In computer language, the equation can also be written as
p = p*r*(1-p)

Let initial p be 0.25 and let r be fixed at 1.5. Let us calculate p repeatedly in a spreadsheet and draw a graph between p and number of iterations. The screen shot is shown below. The p initially increases and then stabilizes at 0.333.

Now let us change only r to 2.5. As you can see in the next figure, the curve is slightly disturbed before settling down at 0.6.

When r = 2.9, the curve oscillates for a long time(more iterations) before stabilizing.

When r is above 3, the graph never settles down at one point but oscillates between two or more points.

If r goes above 3.57, we are not able to see any ordered pattern in the graph. But only chaos rules the graph. The simple equation behaves unpredictably. It has moved from good ordered results to chaotic regime.

Butterfly effect: Now let us draw the graph with r=3.9 and initial p = 0.25. We get some chaotic pattern. Again let us draw the graph for the same r but with a slight change in initial p as 0.2501. Since the change is negligible(0.0001), we expect to see more or less same pattern of the graph. But the two graphs are initially similar; as the iteration progress, the patterns are widely different.
Small changes in input causing drastic changes in output is called the 'butterfly effect'. The two graphs are given below for comparison.

Practically, a small mis-measurement in weather may throw off entire weather forecasting. So the saying goes, " the flap of a butterfly wings in Brazil may set of a Tornado in Texas".
chaos theory is used in cryptography to send secret message

Science update: Singapore based researchers have successfully transfered lemonade's taste throw the Internet.

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