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Infinitesimal Lyapunov functions for singular flows

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Abstract

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative \(DX\) of the vector field \(X\) together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.

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Acknowledgments

This is part of the second author PhD Thesis [39] at Instituto de Matemática da Universidade Federal do Rio de Janeiro (UFRJ), whose research facilities were most useful. This work have been improved during her postdoctoral research at IMPA with financial support of INCTMat-CAPES.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N, Ondina, 40170-110 , Salvador, BA, Brazil

    Vitor Araujo

  2. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Jardim Botânico, Rio de Janeiro, 22460-320, Brazil

    Luciana Salgado

Authors
  1. Vitor Araujo
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  2. Luciana Salgado
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Correspondence to Vitor Araujo.

Additional information

V.A. was partially supported by CAPES, CNPq, PRONEX-Dyn.Syst. and FAPERJ (Brazil). L.S. was supported by a CNPq doctoral scholarship and is presently supported by a INCTMat-CAPES post-doctoral scholarship.

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Araujo, V., Salgado, L. Infinitesimal Lyapunov functions for singular flows. Math. Z. 275, 863–897 (2013). https://doi.org/10.1007/s00209-013-1163-8

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