Chaos Project (page1)

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CHAOS IN THE SOLAR SYSTEM

In 1887 King Oscar of Sweden offered a prize of 2500 crowns to anyone who could tell him whether the solar system was stable. The solar system is a very complex entity with a large number of massive bodies and hence its analysis is an extremely formidable task. Among the many who attempted to answer this question was Henri Poincare, the great French mathematician. Even though Poincare looked at a highly simplified model he found that under certain conditions this model behaved in a way he had never encountered before, and in an attempt to understand it Poincare sowed the seeds of the theory of Chaos.

Chaos has come a long way since Poincares time. It now forms an entirely new branch of mathematics and has applications in meteorology, fluid dynamics, astronomy, economics, biology and many other fields.

What Is Chaos?

Simply speaking, chaos means sensitive dependence on initial conditions. Two identical chaotic systems starting very close to each other will evolve in totally different ways.

Consider a billiard table with convex obstacles. A ball moving on this table will hit an obstacle, rebound with the angle of reflection equal to the angle of incidence, hit another and so on. A ball starting from a slightly different point will strike the first obstacle at a slightly different point, the next at a significantly different point and so on. The difference will grow till at some stage the second ball completely misses an obstacle the first one will have hit. Thereafter these paths have no correlation, whatsoever. This means that we cannot predict how the system will evolve because however accurately we determine the initial conditions, the small initial error in measurement will grow till the predicted and actual trajectories are hopelessly different! This is so despite the fact that very simple laws govern the system, i.e. it is deterministic. This is the beauty of chaos: the combination of determinism and unpredictability. Neither all unpredictable systems nor all deterministic systems are chaotic. A non-deterministic system (which must be unpredictable) is called a random system. On the other hand, many deterministic systems are predictable there has to be some non-linearity present for a deterministic system to be unpredictable.

A ball falling under the influence of gravity is a simple example of a predictable system. Note that if two identical balls are dropped from slightly different heights their trajectories even after a long time will remain close together. The solar system is a storehouse of many such predictable systems. Eclipses and transits can be predicted to the second. Astrological charts accurately give the position of planets at any point in time. Other predictable systems include a simple pendulum of a known length and mass, a single planet orbiting a star etc.