0, so that for a sufficiently weak perturbation the phase space volume not filled with invariant curves can be made arbitrarily small. The invariant tori in the zones exduded by the KAM-condition are in most cases destroyed. In the centers of the destroyed zone we have a torus with rational frequency ratio and, hence, periodic motion. In a simplified picture, one can imagine that, for such a periodic motion, the perturbation is also felt periodically, so that initially small changes induced in the trajectories may in time blow up, giving rise to large scale deviations.

At critical bifurcation points the Lyapunov exponent is zero. For an interpretation of the Lyapunov exponent, it is instructive to note its relationship to the loss of information during the process of iteration. e. one has to ask IdN 'yes' or 'no' questions on average. 65) regarding the position of the iterated point. 66) where the factor In 2 converts binary ('bits') to natural ('nats') units of information. It is also of interest to express the Lyapunov exponent in terms of the asymptotic density of points covered by the orbit Xo, XI, X2, ...