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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THE EARTH, A SUPER ORGANISM

JOIN MY COURSE: "Become a programmer in a day with python" A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess. It says, "Earth is a self-regulating organism like a human being. The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment". Hence the earth is still suitable for the life to thrive. Imagine, in a particular place, there are lot of flowers. Some flowers are white and some are darkly coloured. We know, white reflects light and heat while dark absorbs the same. White flowers can thrive in hot climate. But dark flowers requires cold climate. The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist. This is the essence of "Gaia" theory. On our earth, the oxygen constitute 20% of the atmosphere. The oxygen level is always mai

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

DISORDER IS THE "ORDER OF THE DAY"

Imagine a balloon full of air. The air molecules are moving randomly inside the balloon. Let us pierce the balloon with a pin. The air rushes out. Why should not the air molecules stay inside the balloon safely and ignore the little hole? That is not the way the world works. The molecules always "want to occupy as many states as possible". Hence the air goes out in the open to occupy more volume. The things always goes into disorder (entropy) and the disorder increases with time. The above statement is what we call "second law of thermodynamics". Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze. Freezing means bringing things to order and arrangement. It violates the second law. Hence it will never happen. Hence heat must flow from high temperature to low temperature and not the other way. The air molecules in y

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