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On Sensitive Dependence on Initial Conditions and Existence of Physical Measure for 3-Flows

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Modeling, Dynamics, Optimization and Bioeconomics I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 73))

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Abstract

After reviewing known results on sensitiveness and also on robustness of attractors together with observations on their proofs, we show that for attractors of three-dimensional flows, robust chaotic behavior (meaning sensitiveness to initial conditions for the past as well for the future for all nearby flows) is equivalent to the existence of certain hyperbolic structures. These structures, in turn, are associated to the existence of physical measures. In short in low dimensions, robust chaotic behavior for smooth flows ensures the existence of a physical measure.

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Acknowledgements

Author was partially supported by CNPq, FAPERJ and PRONEX-Dynamical Systems (Brazil).

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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, Brazil

    Vítor Araújo

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  1. Vítor Araújo
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Correspondence to Vítor Araújo .

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Editors and Affiliations

  1. University of Porto, Department of Mathematics, Porto, Portugal

    Alberto Adrego Pinto

  2. Univ. of California, Dept. of Agricultural and Resource Economics, Berkeley, California, USA

    David Zilberman

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Araújo, V. (2014). On Sensitive Dependence on Initial Conditions and Existence of Physical Measure for 3-Flows. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics I. Springer Proceedings in Mathematics & Statistics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-04849-9_6

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