Time-evolution in low-dimensional phase spaces is a fundamental chapter of
Modern Dynamics. Despite more than a century of brilliant history, the inflow of
new ideas in this area continues. By now one can clearly see only the lower levels
of this giant "tree" planted by Poincare, while the "top" is hardly observable.
The objective of this survey is to give an updated, detailed picture in the
case when dimension of the phase space does not exceed 2; in other words,
continuous flows on 2-dimensional manifolds are considered. Moreover, we adopt
a "geometric" philosophy: the less analysis is used to obtain the phase portrait
of dynamical system, the better.
The necessity of such a volume is due primarily to the absence of a study
trying to cover the entire spectrum of emerging problems. The recent monograph
of Aranson et al. [28] is of introductory nature and is addressed mostly to the
students; for the people working at the edge of research the subject requires a
wider treatment. We hope, however, that this volume inherits the elementary
style of [28] and will be helpful both to the advanced students and researchers.
Secondly, certain old results have been revised recently using statistical,
algebraic and combinatorial methods, a possibility which seemed unrealistic earlier.
We tried to include all of them, paying special tribute to the classical results.
We did not hesitate to make historical comments, sometimes bold, but not
intentionally so. We bring our excuses to people whose names are omitted or
unproportionally cited; we thank those who brought to our attention their
contributions.