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[Texts]## 58: Global analysis, analysis on manifolds |

Global analysis, or analysis on manifolds, studies the global nature of differential equations on manifolds, that is, dynamical systems. In addition to local tools from ordinary differential equation theory, global techniques include the use of topological spaces of mappings. In this heading also we find general papers on manifold theory, including infinite-dimensional manifolds and manifolds with singularities (hence catastrophe theory), as well as optimization problems (thus overlapping the Calculus of Variations.

(The real introduction to this area will have to summarize the Atiyah-Singer Index Theorem! In the meantime see this summary of some lectures by Singer on the theorem.)

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.'(taken from SIAM Review 37(1), pp. 129, 1995). One sometimes hears similar expressions of regret that other topics in this area --- catastrophe theory, dynamical systems, fractal geometry --- have been championed by persons not familiar with the content of the material.

See also 32-XX, 32CXX, 32FXX, 46-XX, 47HXX, 53CXX; for geometric integration theory, See 49FXX, 49Q15

- 58A: General theory of differentiable manifolds
- 58B: Infinite-dimensional manifolds
- 58C: Calculus on manifolds; nonlinear operators, see also 47HXX
- 58D: Spaces and manifolds of mappings (including nonlinear versions of 46EXX)
- 58E: Variational problems in infinite-dimensional spaces
- 58F: (*) Ordinary differential equations on manifolds; dynamical systems, see also 28D10, 34CXX, 34CXX, 54H20
- 58G: (*) Partial differential equations on manifolds; differential operators, see also 35-XX
- 58H: Pseudogroups, differentiable groupoids and general structures on manifolds
- 58J: Partial differential equations on manifolds [See also 35-XX] [new in 2000]
- 58K: Theory of singularities and catastrophe theory [See also 37-XX] [new in 2000]
- 58Z05: Applications to physics

This is among the larger areas within the Math Reviews database. But over half the papers are in subfield 58F (Dynamical systems), one of the largest 3-digit subfields in the database (and containing two(!) of the largest 5-digit areas -- 58F07, Completely integrable systems, and 58F13 Strange attractors; chaos). Apart from 58F, this area would be average in size.

(*) These two sections are to be removed in the year-2000 version of the MSC. A new section 37: Dynamical systems and ergodic theory is being created.

Browse all (old) classifications for this area at the AMS.

Some descriptions of the traditional areas of global analysis:

- Smale, S.: "What is global analysis?", Amer. Math. Monthly 76 1969 4--9. MR38#5248
- Morse, Marston: "What is analysis in the large?", Amer. Math. Monthly 49, (1942). 358--364. MR3,292a
- 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).
- Ambrosetti, Antonio; Prodi, Giovanni: "A primer of nonlinear analysis", Cambridge Studies in Advanced Mathematics, 34. Cambridge University Press, Cambridge, 1993. 171 pp. ISBN 0-521-37390-5 MR94f:58016
- Ewald, Günter: "Probleme der geometrischen Analysis", (German: Problems of geometric analysis), Bibliographisches Institut, Mannheim, 1982. 156 pp. ISBN 3-411-01633-7 MR84g:58001
- Alexander, J. C.: "A primer on connectivity", Fixed point theory (Sherbrooke, Que., 1980), pp. 455--483, Lecture Notes in Math., 886; Springer, Berlin-New York, 1981. MR83e:58013

Some electronic Survey articles in Dynamical systems

Some descriptions of Catastrophe Theory, starting with its creator:

- Thom, René: "What is catastrophe theory about?" Synergetics (Proc. Internat. Workshop, Garmisch-Partenkirchen, 1977), pp. 26--32. Springer, Berlin, 1977. MR58#18536
- Stewart, Ian: "Consumer guide to catastrophe theory", Southeast Asian Bull. Math. 2 (1978), no. 1, 13--16. MR81a:58017
- Poston, Tim; Stewart, Ian: "Catastrophe theory and its applications", Surveys and Reference Works in Mathematics, No. 2. Fearon-Pitman Publishers, Inc., Belmont, Calif., 1978. 491 pp. ISBN 0-273-01029-8 MR58#18535

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

"Reviews in Global Analysis 1980-1986", AMS

Chaos bibliography database [Link dead or restricted; May 1999]

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.

E-text: "Invariance Theory, the heat equation, and the Atiyah-Singer index theorem", by Gilkey.

DsTool for dynamical systems

- Nonlinear sites (Good place to start; the following have not yet been examined too closely).
- UK Nonlinear Dynamics Groups
- Sci.nonlinear FAQ
- Nonlinear Dynamics and Topological Time Series Analysis Archive
- Nonlinear Dynamics Group
- 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.
- Complexity Online (Complex systems)
- Dynamical systems (preprint server, etc.)
- Here are the AMS and Goettingen resource pages for area 58.
- UTK archives page on dynamical systems.
- Research in Applied Nonlinear Mathematics

- Example of limit cycles for iterations of a map f: R \mapsto R
- Overview of dynamical systems (p(x)=kx(1-x), Feigenbaum)
- Pointer to a FAQ of the sci.nonlinear newsgroup (including dynamical systems)
- Citation for dynamical systems (in re: Julia set of x^2+c)
- Can you hear the shape of a drum? (no) Citations, URLs, and a summary.
- The Poincaré Eternal Return theorem.

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org