Comparison of regular and chaotic motions. (Emergence of the new paradigm, its successes and difficulties)

Perhaps one of those events in the mathematics of the second half of the 20th century that raised interest beyond its framework was the discovery of a new class of motions of dynamical systems—motions of a ‘(quasi-)random, chaotic’ nature—and the understanding (even if incomplete) of how the behaviour of systems that are formally totally deterministic may acquire such a character. From a theoretical viewpoint this is connected with the understanding that in certain systems there are ‘large’ sets of unstable trajectories possessing so-called ‘hyperbolicity’ (so that, in the author’s opinion, the events of the 1960s when this was finally realized could be called a ‘hyperbolic revolution’).
Naturally, all this was first understood in the cases where the ‘hyperbolicity’ was, so to speak, as strong as at all possible. However, there are situations in which it is reasonable to suspect (and sometimes even prove) the presence of a weaker hyperbolicity, which also implies (sometimes presumably, and at other times rigorously) a similar ‘chaotization’. The theoretical analysis of such situations is more complex and seems less satisfactory, but some work in this direction has been done.
Since all this is of interest not only in pure mathematics but also in fields of a (semi-)applied character, problems of this type are described in the literature from very different viewpoints—different to such an extent that this is reminiscent of an old fable from India about blind men being introduced to an elephant. (Which is totally natural for ‘chaotic’ topics.) This side of the matter has also found a place in the report (from the viewpoint of a pure mathematician, of course).