37: Dynamical systems and ergodic theory
Dynamical systems is the study of iteration of functions from a space to
itself -- in discrete repetitions or in a continuous flow of time. Thus
in principle this field is closely allied to differential equations on
manifolds, but in practice the focus is on the underlying sets (invariant
sets or limit sets) and on the chaotic behaviour of limiting systems.
This heading includes the topic of Chaos, well-known in the popular press,
but not a particularly large part of mathematics. At best, it provides a
paradigm for the phrasing of situations in the applications of mathematics.
A quote by Philip Holmes (SIAM Review 37(1), pp. 129, 1995) illustrates this situation well:
`In spite of all the hype and my enthusiasm for the area, I do not
believe that chaos theory exists, at least not in the manner of
quantum theory, or the theory of self-adjoint linear operators. Rather
we have a loose collection of tools and techniques, many of them from
the classical theory of differential equations, and a guiding global
geometrical viewpoint that originated with Poincaré over a hundred
years ago and that was further developed by G D Birkhoff, D V Anosov
and S Smale and other mathematicians. I therefore prefer a sober
description of new tools, rather than grand claims that the problems
of life, the universe, and everything will shortly be solved.'
One sometimes hears similar
expressions of regret that other valid topics in nearby area --- catastrophe
theory, dynamical systems, fractal geometry --- have been championed by persons
not familiar with the content of the material.
This heading includes the topic of Chaos, well-known in the popular press, but not a particularly large part of mathematics. At best, it provides a paradigm for the phrasing of situations in the applications of mathematics. A quote by Philip Holmes (SIAM Review 37(1), pp. 129, 1995) illustrates this situation well:
This area was added to the MSC in 2000; there are no papers to map with this classification, although much of what was previously classified in areas 34, 35, 58, and other sections should be shown here. In particular, the former subfield 58F (Dynamical systems) is one of the largest 3-digit subfields in the MR database (and contains two(!) of the largest 5-digit areas -- 58F07, Completely integrable systems, and 58F13 Strange attractors; chaos). Papers in those areas would now mostly be classed in 37D, 37J, 37K, or to portions of areas 34, 35, or 70.
Browse all (old) classifications for this area at the AMS.
A survey article by Sell, George R., "What is a dynamical system? Studies in ordinary differential equations" pp. 32--51. Stud. in Math., Vol. 14, Math. Assoc. of America, Washington, D.C., 1977. MR57#16840
There are some survey papers in the text, "What is integrability?", Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1991. MR91k:58005; see in particular Flaschka, H.; Newell, A. C.; Tabor, M.: "Integrability" (pp. 73--114, MR92i:58074) and Veselov, A. P.: "What is an integrable mapping?" (pp 251--272, MR92c:58119).
Some electronic Survey articles in Dynamical systems
Reviews of Chaos and fractal geometry:
Newsgroups sci.nonlinear, comp.theory.dynamic-sys, comp.theory.cell-automata
There are two mailing lists email@example.com (nonlinear systems) and firstname.lastname@example.org (quantization/chaos). Here is the information page.
DsTool for dynamical systems