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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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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