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Oriented Shadowing Property and \(\Omega \)-Stability for Vector Fields

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Abstract

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.

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Acknowledgments

Shaobo Gan is supported by 973 project 2011CB808002, NSFC 11025101 and 11231001. Ming Li is partially supported by the National Science Foundation of China (Grant No. 11201244), the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP, Grant No. 20120031120024), and the LPMC of Nankai University. Sergey Tikhomirov is partially supported by Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026, JSC “Gazprom neft”, by the Saint-Petersburg State University research grant 6.38.223.2014 and by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD).

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s republic of China

    Shaobo Gan

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Ming Li

  3. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Sergey B. Tikhomirov

  4. Chebyshev Laboratory, Saint-Petersburg State University, 14th line of Vasilievsky island, 29B, Saint-Petersburg, 199178, Russia

    Sergey B. Tikhomirov

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  1. Shaobo Gan
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  2. Ming Li
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  3. Sergey B. Tikhomirov
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Correspondence to Ming Li.

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Gan, S., Li, M. & Tikhomirov, S.B. Oriented Shadowing Property and \(\Omega \)-Stability for Vector Fields. J Dyn Diff Equat 28, 225–237 (2016). https://doi.org/10.1007/s10884-014-9399-5

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  • DOI: https://doi.org/10.1007/s10884-014-9399-5

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