Chaos

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Such a motion of a dynamical system, in which arbitrarily small variation of the initial conditions causes exponentially large changes of the future trajectory is called chaotic. In the opposite case the dynamics is called stable or regular.

Can one predict the motion of a single planet for, say, billion of years from now? Is the solar system stable?

This very question attracted attention of several generations of researchers. Initial results of Laplace and Lagrange strongly suggested positive answer. Already at the end of XIX century the issue of stability of the Solar System was considered as one of the crucial challenges for natural sciences and the king of Sweden, Oskar II, founded special prize for solving the problem. The prize was awarded in 1887 to the French mathematician Henri Poincaré, who had obtained important, but not entirely decisive results though. He demonstrated that the frequently used perturbation techniques may

not lead to the correct solution, since the series taking into account terms of higher and higher orders may not converge.

Mathematical theory of stability of motion has been initiated by Aleksander M. Lapunov (1857-1918), who analysed how fast the distance between two neighbouring trajectories increases in time. If the system in question is chaotic, such a distance grows exponentially with time, and the coefficient in the exponent, called Lapunov exponent, is positive. Although such systems were known to mathematicians since the beginnings of the nineteenth century, they were considered rather a mathematical curiosity and researchers were not aware of their implications for physics, astronomy and science in general. The situation has changed during the last thirty years, when the significance of the pioneering work of Edward Lorenz was accepted. In his 1963 paper [2], published in a meteorological journal, Lorenz analysed numerically a certain nonlinear dynamical system and demonstrated that a minor variation of the initial conditions changed dramatically the behaviour of the system. This property, nowadays called the “Butterfly effect”: (flap of wings of a butterfly in Australia may cause a tornado in Florida) turned out to be typical for a majority of dynamical systems used to model various phenomena in physics, chemistry or biology