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Lorenz Attractors and Generalizations: Geometric and Topological Aspects

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Nonlinear Maps and their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 57))

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Abstract

In this paper, the problem of topological distinguishing of Lorenz-like attractors and their generalizations is discussed. New cardinal-valued topological invariants for Lorenz-type attractors and generalizations are constructed. A generalization is considered of Williams’s well-known model of the attractor in the Lorenz system, the inverse limit of semiflows on branched manifolds that are suspensions over a discontinuous expanding map of a closed linear interval. The generalization consists in the consideration of maps with several discontinuity points, rather than one. A cardinal-valued topological invariant L-manuscript is constructed that distinguishes a continuum of nonhomeomorphic generalized models. A topological invariant distinguishing a continuum of nonhomeomorphic geometric Lorenz attractors is obtained as a consequence. An analogous cardinal-valued invariant is constructed for attractors of Lorenz-type maps. The kneading invariant is not a topological invariant distinguishing attractors as sets; there exists an uncountable set of mutually nonconjugating Lorenz-type maps having homeomorphic attractors.

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Authors and Affiliations

  1. Obninsk Institute for Nuclear Power Engineering, 249040, Studgorodok 1, Obninsk, Kaluga Region, Russia

    Natalia Klinshpont

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  1. Natalia Klinshpont
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Correspondence to Natalia Klinshpont .

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Editors and Affiliations

  1. Colégio Luís António Verney, University of Évora CIMA-UE, Évora, Portugal

    Clara Grácio

  2. LAAS-INSA, Toulouse, France

    Daniele Fournier-Prunaret

  3. Tokushima University, Tokushima, Japan

    Tetsushi Ueta

  4. E. E. Eng., Tokushima University, Tokushima, Japan

    Yoshifumi Nishio

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Klinshpont, N. (2014). Lorenz Attractors and Generalizations: Geometric and Topological Aspects. In: Grácio, C., Fournier-Prunaret, D., Ueta, T., Nishio, Y. (eds) Nonlinear Maps and their Applications. Springer Proceedings in Mathematics & Statistics, vol 57. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9161-3_12

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