Topology and dynamics of unstable attractors

Trans. Amer. Math. Soc.

2010

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Abstract. Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow ϕ in a space M . Stable attractors are very well known, but otherwise (without the stability assumption) the situation can be extremely wild. In this paper we consider the class of attractors with no external explosions, where a mild form of instability is allowed.After obtaining a simple description of the trajectories in A(K) − K we study how K sits in A(K) by performing an analysis of the Poincaré polynomial of the pair (A(K), K). In case M is a surface we obtain a nice geometric characterization of attractors with no external explosions, as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence. Finally, we explore the strong relations which exist between the shape (in the sense of Borsuk) of K and the shape (in the intuitive sense) of the whole phase space M , much in the spirit of the Morse-Conley theory.

Section: Manifolds Which Contain Attractors With No External Explosionsmentioning

confidence: 95%

Section: Attractors With No External Explosions Definition and Basicmentioning

confidence: 99%

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Unstable attractors in manifolds

Trans. Amer. Math. Soc.

2010

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“…It was shown in [18] and [23] that there exists manifolds M which cannot contain unstable attractors without external explosions, or otherwise stated every isolated connected attractor in M is either stable or has external explosions. The next corollary goes in the same line but it is applicable to arbitrary phase spaces.…”

Section: Attractors Revisitedmentioning

confidence: 98%

“…Proposition 4.1 means that the simplest unstable attractors are those whose explosion points lie all in K. These receive a special name, first introduced by Athanassopoulos in [1]: an attractor with no external explosions is an attractor K such that J + (p) ⊆ K for all p ∈ A(K) − K. They have been studied in [1], [18] and [23] when M is a manifold, and we will see that many of the results contained in those papers still hold for more general phase spaces. The reasons why this generality is of interest are presented at the end of the section.…”

Section: Proposition 41 If An Attractor K Is Unstable There Are Expmentioning

confidence: 98%

An approach to the shape Conley index without index pairs

2010

Rev Mat Complut

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Robbin and Salamon proved that the shape of the homotopy Conley index of an isolated invariant set K coincides with the shape of the one-point compactification of the unstable region W u (K) of K endowed with a certain topology which they called the intrinsic topology. In this paper an equivalent, simplified definition of the latter is given in elementary terms, without resorting to index pairs whatsoever. We then show how our approach allows for a development of the shape index and its basic properties in an index pair-free fashion and simplifies some classical constructions, such as that of the Morse equations of a Morse decomposition. As a final application, some embrionary work about the geometry of W u (K)/K is presented together with an argument about how this may contribute to our understanding of the relations between how an attractor sits in its basin of attraction and the complexity of the dynamics in the latter.

Dynamical systems and shapes

Rev. R. Acad. Cien. Serie A. Mat.

2008

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This survey is an introduction to some of the methods, techniques and concepts from algebraic topology and related areas (homotopy theory, shape theory) which can be fruitfully applied to study problems concerning continuous dynamical systems. To this end two instances which exemplify the interaction between topology and dynamics are considered, namely, Conley's index theory and the study of some properties of certain attractors. Sistemas dinámicos y formasResumen. Este artículo panorámico constituye una introducción a algunos de los métodos, técnicas y conceptos que, desde la topología algebraica y otrasáreas afines (teoría de homotopía, teoría de la forma), permiten abordar problemas que se plantean en el marco de los sistemas dinámicos continuos. Para ello se presentan dos situaciones que ejemplifican esta interacción entre topología y dinámica, como son la construcción delíndice de Conley y el estudio de algunas propiedades de ciertos atractores. Generally speaking, the task of applied sciences is to observe natural phenomena and try to elaborate a theory which explains them. Such a theory is frequently formalized (at least in quantitative sciences) in a mathematical language and can be used to produce simple models of the phenomena, usually in the form of a dynamical system 1 . Then a mathematical analysis of the latter can be performed and may provide explanations for the basic features of the observed behaviour.The study of dynamical systems involves many areas of mathematics, most notably analysis and topology. More specifically, algebraic topology entered the picture through the pioneering work of Poincaré, later continued by Morse, Smale and Conley, which showed that there exists a strong interaction between a dynamical system and the shape (in an informal, intuitive sense) of the phase space it lives in.Their methods can be considered landmarks in the study of dynamical systems through their phase portraits, which are objects of a geometrical nature. This approach gave rise to a whole new branch where tools like homotopy theory, homology and cohomology theories, and later on shape theory, played a prominent role in the investigation of dynamical systems.The aim of this survey is to present a (necessarily partial and strongly biased) illustration of two specific instances which exemplify how the tools mentioned above are brought into the picture of dynamical systems. We construct Conley's index in its shape theoretical version, present the subsequent Morse equations (this is Section 2) and explore some results about attractors (Section 3). The exposition is very unbalanced in the sense that a great space is taken up by Section 2, but this is just a natural consequence of the fact that Presentado por José María Montesinos. Recibido: 26 de junio de 2007. Aceptado: 5 de diciembre de 2007.it condenses nearly one hundred years of beautiful and deeply influential mathematics and provides enough background for an interested reader to explore more advanced topics. Unfortunately we will not be able even to menti...