TABLE OF CONTENTS In a most surprising manner, modern theory of nonlinear dynamics has disclosed a number of misconceptions in our understanding of the laws of motion. The notion we have about predictability of classical mechanical systems, for instance, is not correct in general, but is founded on a few exceptional cases where the equations of motion can be solved analytically. For most conservative systems, the space of initial conditions is scattered with regions in which the slightest distortion can lead to a completely different trajectory [1, 2, 3]. Moreover, in these regions the systems show forms of behavior that differ qualitatively from the simple motions we know for integrable systems [4, 5].
Dissipative systems, i.e., systems with friction, can also show unpredictability and chaos, provided that their motion is maintained through a sufficient supply of energy or through strong external forcing [6, 7, 8, 9]. Nonlinear systems can also exhibit two or more simultaneous solutions such that varying initial conditions can cause a system to choose one stationary solution rather than another. Moreover, the boundary between the conditions that lead to the different stationary states may be fractal and show structure down to the smallest detail [10, 11]. Close to such boundaries, the uncertainty in the final outcome will diminish more slowly than does the uncertainty in the initial conditions [12]. If the initial conditions are specified with twice the accuracy, our ability to predict the resulting stationary state will only improve by a factor less than two.
