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[Texts]## 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.

- 37A: Ergodic theory [See also 28DXX]
- 37B: Topological dynamics [See also 54H20]
- 37C: Smooth dynamical systems: general theory [See also 34CXX, 34DXX]
- 37D: Dynamical systems with hyperbolic behavior
- 37E: Low-dimensional dynamical systems
- 37F: Complex dynamical systems [See also 30D05, 32HXX]
- 37G: Local and local bifurcation theory [See also 34CXX]
- 37H: Random dynamical systems
- 37J: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53DXX, 70FXX, 70HXX]
- 37K: Infinite-dimensional Hamiltonian systems [See also 35AXX, 35QXX]
- 37L: Infinite-dimensional dissipative dynamical systems [See also 35BXX, 35QXX]
- 37M: Approximation methods and numerical treatment of dynamical systems [See also 65PXX]
- 37N: Applications

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:

- Babu Joseph, K.: "A chaos primer", Recent developments in theoretical physics (Kottayam, 1986), 305--322; World Sci. Publishing, Singapore, 1987. MR89m:58134
- Conrad, M.: "What is the use of chaos?" Chaos, 3--14, Nonlinear Sci. Theory Appl.; Manchester Univ. Press, Manchester, 1986. CMP 848 803
- "Applications of fractals and chaos. The shape of things", edited by A. J. Crilly, R. A. Earnshaw and H. Jones. Springer-Verlag, Berlin, 1993. 317 pp. ISBN 3-540-56492-6 MR95h:58092
- Mandelbrot, B. B.: "Fractal geometry: what is it, and what does it do?" Fractals in the natural sciences. Proc. Roy. Soc. London Ser. A 423 (1989), no. 1864, 3--16. MR91d:58160

Newsgroups sci.nonlinear, comp.theory.dynamic-sys, comp.theory.cell-automata

There are two mailing lists nonlin_net@complex.nbi.dk (nonlinear systems) and qchaos_net@complex.nbi.dk (quantization/chaos). Here is the information page.

DsTool for dynamical systems

- Nonlinear sites (Good place to start; the following have not yet been examined too closely).
- Symbolic Dynamics on the World Wide Web
- Entropy on the World Wide Web
- Cellular Automata
- Cellular automata
- SIAM's Dynamical Systems page and Nonlinear Science page.
- Dynamical systems (preprint server, etc.)
- UTK archives page on dynamical systems.
- Nonlinear Dynamics Group

- Strings not containing two identical consecutive substrings
- Garden of Eden Patterns (non-successors) in Conway's Game of Life
- Boundedness of terms of a linear recurrence
- Length of chains in a dynamical system on Z (using operations add-one, subtract-one, and double)
- Floyd's algorithm for finding cycles under iterates of maps
- Citation for dynamical systems (in re: Julia set of x^2+c)
- Sarkovskii's Theorem on lengths of cycles under iterates of continuous real functions (period 3 implies chaos)
- Discrete dynamical systems in R^3 showing symmetries in attracting sets
- Example of limit cycles for iterations of a map f: R \mapsto R
- Overview of dynamical systems (p(x)=kx(1-x), Feigenbaum)
- What is Feigenbaum's constant 4.6692...?
- Behaviour (periodic points) of logistic map h(x)=rx(1-x) for large coefficient r
- The Borel-Cantelli lemma: iterates of measure-preserving maps escape sets of finite measure
- Entire functions with no 2-cycles (f(f(x))=x) nor fixed points are translations
- Stability of an elementary autonomous system of ordinary differential equations
- Attractor to the Lorentz chaotic ODE
- Smooth dynamical systems with symmetry over domain
- Pointer to a FAQ of the sci.nonlinear newsgroup (including dynamical systems)

Last modified 2000/01/24 by Dave Rusin. Mail: