Publication: Stability, attraction and shape: a topological study of flows
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2011
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Juliusz Schauder Center for Nonlinear Studies
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Abstract
This paper is a survey on how topological techniques (mainly from algebraic and geometric topology) can be applied to the study of flows. The first five sections constitute a self-contained introduction to the subject, including the proofs of several results which provide an excellent illustration of the sort of topological techniques that can be applied to understand dynamics. The remaining two sections are more expository in nature and discuss open problems and current lines of research. An abundant bibliography is provided for the interested reader.
Since the paper covers a great variety of topics we will only mention very briefly the content of each section. Section 1 is an introduction, containing the relevant definitions needed to follow the rest of the survey. Section 2 is devoted to asymptotically stable attractors. A number of foundational results are stated, dealing with the shape and cohomology of attractors. In Section 3 the author enlarges the class of invariant sets under study by considering non-saddle sets. The dynamics in the vicinity of these sets can be more complicated than the dynamics near attractors, but they still share many of their topological properties.
Section 4 is devoted to the Lorenz attractor. The author is able to compute its shape (hence its cohomology) and its Conley index for certain interesting values of the parameters. In order to achieve this he resorts to a result of his own which describes how an attractor for a parametrized flow changes as the parameter changes. This topic is pursued in more generality in Section 5, where Hopf bifurcations are explored with topological techniques. A beautiful theorem mentioned in this section (again by the author of the survey) can be stated, somewhat imprecisely, as follows. Suppose φ λ is a parametrized family of flows (λ∈[0,1] ) on an n -manifold. Let a fixed point p be an attractor for φ 0 and a repeller for φ λ , λ>0 . Then for small enough λ there exists an attractor K λ very close to p and with the shape of an (n−1) -sphere (Theorem 5.2).
Section 6 enumerates some open problems, and finally Section 7 outlines several recent developments on the following topics: the intrinsic topology of the unstable manifold of an invariant set, the Lyusternik–Shnirelʹman category of isolated invariant compacta, dynamical systems and hyperspaces, the Conley index and Ważewski theories, the regularity of isolating blocks, continuations and robustness, the boundaries of attractors and their regions of attraction, and dynamical systems and exterior spaces.
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Papers from the Juliusz Schauder Center Winter School held at Nicolaus Copernicus University, Toruń, February 9–13, 2009