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Unstable manifold, Conley index and fixed points of flows

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Barge, Héctor and Rodríguez Sanjurjo, José Manuel (2014) Unstable manifold, Conley index and fixed points of flows. Journal of mathematical analysis and applications, 420 (1). pp. 835-841. ISSN 0022-247X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X14005630


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Abstract

We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.


Item Type:Article
Uncontrolled Keywords:Unstable manifold; Parallelizable structure; Attractor; Conley index; Non-saddle set; Morse decomposition
Subjects:Sciences > Mathematics
ID Code:26685
Deposited On:11 Sep 2014 08:56
Last Modified:12 Dec 2018 15:12

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