When we observe evolution in time of various phenomena in the macroscopic world that surrounds us, we often use the terms chaos , or chaotic , meaning that the changes in time are without pattern and out of control, and hence are unpredictable. The most frustrating phenomena are those, which concern long-term weather forecasting. We can never be sure about the change of weather patterns. The temperature, barometric pressure, wind direction, amount of precipitation and other important factors come as a surprise contradicting predictions made a few days ago. Sometimes we are caught in a storm, sometimes in a heat-wave.

The world stock market prices are also an example of a system that fluctuates in time in a random-like, irregular way, and the long-term prognosis does not often come true.

The two examples mentioned belong to the category of huge and complicated dynamical systems, with a huge number of variables. The unpredictability of the evolution in time of these interesting events is intuitively natural.

Simultaneously, it also seems natural that evolution of physical processes in simple systems, the systems governed by simple mathematical rules, should be predictable far into the future.

Suppose we consider a small heavy ball, which can move along a definite track, so that the position of the ball is determined by a single coordinate. Due to Newton s Second Law, the motion of the ball is governed by the second order differential equation. The well known physical system the pendulum belongs to this class of oscillators.

We were told...