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Multi-sensitivity with respect to a vector for semiflows

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Abstract

The notions of multi-sensitivity with respect to a vector, \({\mathscr {N}}\)-sensitivity and strong multi-sensitivity are introduced and studied on semiflows under the action of the most general possible semigroups. Using the concept of Furstenberg families, some other stronger forms of sensitivity are also studied. Moreover, all these notions are studied on hyperspaces and on the product of semiflows.

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References

  1. Auslander, J., Yorke, J.A.: Interval maps, factors of maps, and chaos. Tohoku Math. J. 32(2), 177–188 (1980)

    Article  MathSciNet  Google Scholar 

  2. Chen, Z., Li, J., Lü, J.: On multi-transitivity with respect to a vector. Sci. China Math. 57(8), 1639–1648 (2014)

    Article  MathSciNet  Google Scholar 

  3. Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA (1989)

  4. Iglesias, J., Portela, A.: Almost open semigroup actions. Semigroup Forum 98(2), 261–270 (2019)

    Article  MathSciNet  Google Scholar 

  5. Jiao, L., Wang, L., Li, F., Liu, H.: On multi-sensitivity with respect to a vector. Modern Phys. Lett. B 32(15), 1850166 (2018)

    Article  MathSciNet  Google Scholar 

  6. Li, R., Zhou, X.: A note on chaos in product maps. Turkish J. Math. 37(4), 665–675 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Monthly 82(10), 985–992 (1975)

    Article  MathSciNet  Google Scholar 

  8. Miller, A.: A note about various types of sensitivity in general semiflows. Appl. Gen. Topol. 19(2), 281–289 (2018)

    Article  MathSciNet  Google Scholar 

  9. Miller, A., Money, C.: Syndetic sensitivity in semiflows. Topol. Appl. 196(part A), 1–7 (2015)

    Article  MathSciNet  Google Scholar 

  10. Miller, A., Money, C.: Chaos-related properties on the product of semiflows. Turkish J. Math. 41(5), 1323–1336 (2017)

    Article  MathSciNet  Google Scholar 

  11. Moothathu, T.K.S.: Stronger forms of sensitivity for dynamical systems. Nonlinearity 20(9), 2115–2126 (2007)

    Article  MathSciNet  Google Scholar 

  12. Sharma, P., Nagar, A.: Inducing sensitivity on hyperspaces. Topol. Appl. 157(13), 2052–2058 (2010)

    Article  MathSciNet  Google Scholar 

  13. Tan, F., Zhang, R.: On F-sensitive pairs. Acta Math. Sci. Ser. B (Engl. Ed.) 31(4), 1425–1435 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Thakur, R., Das, R.: Devaney chaos and stronger forms of sensitivity on the product of semiflows. Semigroup Forum 98(3), 631–644 (2019)

    Article  MathSciNet  Google Scholar 

  15. Thakur, R., Das, R.: Stronger forms of sensitivity on product dynamical system via Furstenberg families. Studia Sci. Math. Hungar. 56(4), 440–453 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Thakur, R., Das, R.: Strongly Ruelle–Takens, strongly Auslander–Yorke and Poincaré chaos on semiflows. Commun. Nonlinear Sci. Numer. Simul. 81, 105018 (2020)

    Article  MathSciNet  Google Scholar 

  17. Wang, T., Yin, J., Yan, Q.: The sufficient conditions for dynamical systems of semigroup actions to have some stronger forms of sensitivities. J. Nonlinear Sci. Appl. 9(3), 989–997 (2016)

    Article  MathSciNet  Google Scholar 

  18. Wu, X., Wang, J., Chen, G.: F-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429(1), 16–26 (2015)

    Article  MathSciNet  Google Scholar 

  19. Zeng, T.: Multi-transitivity and \(\Delta\)-transitivity for semigroup actions. Topol. Appl. 226, 1–15 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Authors are thankful to the referee for his/her valuable comments and suggestions for improvement of the paper. The first author is supported by CSIR-SRF Sr.No. 1121641416 Ref.No: 18/12/2016(ii) EU-V (File No: 09/045(1532)/2017-EMR-I) for carrying out this research work.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Rahul Thakur & Ruchi Das

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  1. Rahul Thakur
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  2. Ruchi Das
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Correspondence to Ruchi Das.

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Communicated by Jimmie D. Lawson.

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Thakur, R., Das, R. Multi-sensitivity with respect to a vector for semiflows. Semigroup Forum 101, 452–464 (2020). https://doi.org/10.1007/s00233-020-10125-2

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  • DOI: https://doi.org/10.1007/s00233-020-10125-2

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