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Global attractors and bifurcations. (English) Zbl 0849.58045

Broer, H. W. (ed.) et al., Nonlinear dynamical systems and chaos. Proceedings of the dynamical systems conference, held at the University of Groningen, Netherlands in Dec. 1995, in honour of Johann Bernoulli. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 19, 299-324 (1996).
The author presents some recent developments in the study of attractors of smooth dynamical systems, specially attractors whose basin has a global character. A key point in his approach is to explore the relations between this study and that of main bifurcation mechanisms. The whole paper consists of 5 sections.
Section 1 is “Introduction”. Section 2 analyses the basin of Hénon-like attractors and proves that it contains a neighbourhood of the attractor, at least for a large set of parameters. A more quantitative result, of ergodic flavour, recently established by M. Benedicks and the author is announced: almost every point in the basin of attraction is generic with respect to the SBR measure of the attractor.
Section 3 corresponds to joint work with V. Baladi concerning the ergodic properties of certain nonuniformly hyperbolic unimodal maps of the interval. The main result asserts that those properties, including the fact that such maps are exponentially mixing, are robust under random perturbations of the map (stochastic stability).
Section 4 was written with S. Luzzatto and contains a discussion of an extended geometric model for the behaviour of Lorenz equations. The main statement is that the attractor persists after the appearance of the folds, but only for a positive measure set of parameter values.
Section 5 introduces the joint work with M. J. Pacifico and A. Rovella. They consider smooth flows in 3-dimensional manifolds exhibiting homoclinic connections associated to equilibrium points of saddle-focus type. They prove that a new type of global attractors, with spiraling geometry, occurs (and is even a persistent phenomenon) in such families.
For the entire collection see [Zbl 0830.00032].
Reviewer: Wang Duo (Beijing)

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37G99 Local and nonlocal bifurcation theory for dynamical systems
37A99 Ergodic theory
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