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Ważewski method and Conley index. (English) Zbl 1091.37006
Cañada, A. (ed.) et al., Ordinary differential equations. Vol. I. Amsterdam: Elsevier/North Holland (ISBN 0-444-51128-8/hbk). Handbook of Differential Equations, 591-684 (2004).
The author gives a survey on topological methods in dynamical systems based on the Ważewski retract theorem. He provides a historical context as well as the current exposition of the method with applications to ordinary differential equations and inclusions and for retarded functional-differential equations. He proves many interesting properties of isolating blocks concerning their homological properties and connections with the Ljusternik-Schnirelmann theory. A variant of the Ważewski set, namely isolating segments, is used for detecting periodic solutions and chaos in nonautonomous periodic in time ODEs. The paper is finished by a chapter about isolated invariant sets and the Conley index.

MSC:
37B30Index theory, Morse-Conley indices
34A34Nonlinear ODE and systems, general
34A60Differential inclusions
34K99Functional-differential equations
34B15Nonlinear boundary value problems for ODE
55M30Lyusternik-Shnirel’man category of a space