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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Stability properties of a class of attractors
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by Jorge Lewowicz PDF
Trans. Amer. Math. Soc. 185 (1973), 183-198 Request permission

Abstract:

Let A be an attractor of an analytical dynamical system defined in ${R^n} \times R$. The class of attractors considered in this paper consists of those A which remain stable as invariant subsets of the complex extension of the flow to ${C^n} \times R$. If A is a critical point or a closed orbit, these are the elementary or generic attractors. It is shown that such an A is always a submanifold of ${R^n}$ and that there exists a Lie group acting on A and containing the given flow as a one parameter dense subgroup; as a consequence, some necessary and sufficient conditions for an analytical dynamical system to have an attracting generic periodic motion are given. It is also shown that for any flow ${C^1}$-close to the given one, there is a unique retraction of a neighbourhood of A onto a submanifold of ${R^n}$ homeomorphic to A that commutes with the flow.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 185 (1973), 183-198
  • MSC: Primary 58F10; Secondary 34D30, 58F15
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0343315-6
  • MathSciNet review: 0343315