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    Arbieto, A. Morales, C. A. and Santiago, B. 2015. Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows. Mathematische Annalen, Vol. 361, Issue. 1-2, p. 67.


    López, Andrés Mauricio 2015. Sectional-Anosov Flows in Higher Dimensions. Revista Colombiana de Matemáticas, Vol. 49, Issue. 1, p. 39.


    2013. Lozi Mappings.


    Arbieto, A. Morales, C. and Senos, L. 2012. On the sensitivity of sectional-Anosov flows. Mathematische Zeitschrift, Vol. 270, Issue. 1-2, p. 545.


    Golmakani, A. and Homburg, A.J. 2011. Lorenz attractors in unfoldings of homoclinic-flip bifurcations. Dynamical Systems, Vol. 26, Issue. 1, p. 61.


    Morales, C. A. 2011. An improved sectional-Anosov closing lemma. Mathematische Zeitschrift, Vol. 268, Issue. 1-2, p. 317.


    BAUTISTA, S. and MORALES, C. 2010. A sectional-Anosov connecting lemma. Ergodic Theory and Dynamical Systems, Vol. 30, Issue. 02, p. 339.


    Morales, C. A. 2010. Sectional-Anosov flows. Monatshefte für Mathematik, Vol. 159, Issue. 3, p. 253.


    Morales, C. A. 2007. Singular-Hyperbolic Attractors with Handlebody Basins. Journal of Dynamical and Control Systems, Vol. 13, Issue. 1, p. 15.


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The explosion of singular-hyperbolic attractors

  • C. A. MORALES (a1)
  • DOI: http://dx.doi.org/10.1017/S014338570300052X
  • Published online: 01 March 2004
Abstract

A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if $\Lambda$ is a singular-hyperbolic attractor of a three-dimensional vector field X, then there is a neighborhood U of $\Lambda$ in M such that every attractor in U of a Cr vector field Cr close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of $\Lambda$ (in M) and $\mathcal U$ of X (in the space of Cr vector fields) such that if n denotes the number of singularities of X in $\Lambda$, then $\#\{A\subset U:A$ is an attractor of $Y\in\mathcal U\}\leq n$. Every three-dimensional vector field Cr close to one exhibiting a singular-hyperbolic attractor has a singularity non-isolated in the non-wandering set. A singularity of a three-dimensional Cr vector field Y is stably non-isolated in the non-wandering set if it is the unique singularity of a singular-hyperbolic attractor of Y. These results generalize well-known properties of the geometric Lorenz attractor.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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