Skip to main content
SpringerLink
Log in

Poisson stable motions of monotone nonautonomous dynamical systems

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Basit B, Giinzler H. Spectral criteria for solutions of evolution equations and comments on reduced spectra. ArX-iv:1006.2169, 2010

    Google Scholar 

  2. Bebutov V M. On the shift dynamical systems on the space of continuous functions (in Russian). Bull Inst Math Moscow Univ, 1940, 2: 1–65

    Google Scholar 

  3. Bronshtein I U, Cherny V F. About extensions of dynamical systems with the uniformly asymptotically stable points. Differ Equ, 1974, 10: 1225–1230

    Google Scholar 

  4. Bronsteyn I U. Extensions of Minimal Transformation Group (in Russian). The Hague: Martinus Nijhoff Publishers, 1979

    Book  Google Scholar 

  5. Caraballo T, Cheban D. Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition. I. J Differential Equations, 2009, 246: 108–128

    Article  MathSciNet  MATH  Google Scholar 

  6. Caraballo T, Cheban D. Almost periodic motions in semi-group dynamical systems and Bohr/Levitan almost periodic solutions of linear difference equations without Favard’s separation condition. J Difference Equ Appl, 2013, 19: 872–897

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheban D. On the comparability of the points of dynamical systems with regard to the character of recurrence property in the limit. In: Mathematical Sciences. Kishinev: Shtiintsa, 1977, 66–71

    Google Scholar 

  8. Cheban D. Global pullback attractors of ℂ-analytic nonautonomous dynamical systems. Stoch Dyn, 2001, 1: 511–535

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheban D. Levitan almost periodic and almost automorphic solutions of V-monotone differential equations. J Dynam Differential Equations, 2008, 20: 669–697

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheban D. Asymptotically Almost Periodic Solutions of Differential Equations. New York: Hindawi Publ Corp, 2009

    Book  MATH  Google Scholar 

  11. Cheban D. Global Attractors of Set-Valued Dynamical and Control Systems. New York: Nova Sci Publ, 2010

    MATH  Google Scholar 

  12. Cheban D. Global Attractors of Nonautonomous Dynamical and Control Systems, 2nd ed. Interdisciplinary Mathematical Sciences, vol. 18. River Edge: World Scientific, 2015

    Google Scholar 

  13. Cheban D. Levitan/Bohr almost periodic and almost automorphic solutions of the scalar differential equations. In: Dynamical Systems. Abingdon: Taylor & Francis, 2016, 18–19

    Google Scholar 

  14. Cheban D, Shcherbakov B A. Poisson asymptotic stability of motions of dynamical systems and their comparability with regard to the recurrence property in the limit. Differ Equ, 1977, 13: 898–906

    Google Scholar 

  15. Chueshov I. Order-preserving skew-product flows and nonautonomous parabolic systems. Acta Appl Math, 2001, 65: 185–205

    Article  MathSciNet  MATH  Google Scholar 

  16. Ellis R. Lectures on Topological Dynamics. New York: W A Benjamin Inc, 1969

    MATH  Google Scholar 

  17. Fink A M. Almost Periodic Differential Equations. Lecture Notes in Mathematics, vol. 377. Berlin-New York: Springer-Verlag, 1974

    Google Scholar 

  18. Hale J K. Theory of Functional Differential Equations, 2nd ed. Applied Mathematical Sciences, vol. 3. New York-Heidelberg: Springer-Verlag, 1977

    Google Scholar 

  19. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Berlin-New York: Springer-Verlag, 1981

    Book  MATH  Google Scholar 

  20. Hirsch M, Smith H. Monotone dynamical systems. In: Handbook of Differential Equations: Ordinary Differential Equations, vol. 2. Amsterdam: Elsevier, 2006, 239–357

    Book  Google Scholar 

  21. Husemoller D. Fibre Bundles, 3rd ed. Graduate Texts in Mathematics, vol. 20. New York: Springer-Verlag, 1994

    Google Scholar 

  22. Jiang J, Zhao X. Convergence in monotone and uniformly stable skew-product semiflows with applications. J Reine Angew Math, 2005, 589: 21–55

    Article  MathSciNet  MATH  Google Scholar 

  23. Kwietniak D, Li J, Oprocha P, et al. Multi-recurrence and van der Waerden systems. Sci China Math, 2017, 60: 59–82

    Article  MathSciNet  MATH  Google Scholar 

  24. Levitan B M. Almost Periodic Functions (in Russian). Moscow: Gosudarstv Izdat Tehn-Teor Lit, 1953

    Google Scholar 

  25. Levitan B M, Zhikov V V. Almost Periodic Functions and Differential Equations (in Russian). Moscow: Moscow State University Press, 1978

    MATH  Google Scholar 

  26. Novo S, Obaya R, Sanz A M. Stability and extensibility results for abstract skew-product semiflows. J Differential Equations, 2007, 235: 623–646

    Article  MathSciNet  MATH  Google Scholar 

  27. Opial Z. Sur les solutions presque-périodiques des équations differéntielles du premier et du second ordre (in French). Ann Polon Math, 1959, 7: 51–61

    Article  MathSciNet  MATH  Google Scholar 

  28. Sacker R, Sell G. Skew-product flows, finite extensions of minimal transformation groups and almost periodic differential equations. Bull Amer Math Soc, 1973, 79: 802–805

    Article  MathSciNet  MATH  Google Scholar 

  29. Sacker R, Sell G. Lifting properties in skew-product flows with applications to differential equations. Mem Amer Math Soc, 1977, 11: 0448325

    MathSciNet  MATH  Google Scholar 

  30. Sell G. Lectures on Topological Dynamics and Differential Equations. Van Nostrand Reinhold Mathematical Studies, 33. London: Van Nostrand Reinhold, 1971

    Google Scholar 

  31. Shcherbakov B A. Classification of Poisson-stable motions. Pseudo-recurrent motions (in Russian). Dokl Akad Nauk SSSR, 1962, 146: 322–324

    MATH  Google Scholar 

  32. Shcherbakov B A. On classes of Poisson stability of motion. Pseudorecurrent motions (in Russian). Bul Akad Štiince RSS Moldoven, 1963, 1963: 58–72

    MathSciNet  Google Scholar 

  33. Shcherbakov B A. Topologic Dynamics and Poisson Stability of Solutions of Differential Equations (in Russian). Chişinău: Ştiinţa, 1972

    Google Scholar 

  34. Shcherbakov B A. The compatible recurrence of the bounded solutions of first order differential equations (in Russian). Differencial’nye Uravnenija, 1974, 10: 270–275

    MathSciNet  Google Scholar 

  35. Shcherbakov B A. The comparability of the motions of dynamical systems with regard to the nature of their recurrence (in Russian). Differencial’nye Uravnenija, 1975, 11: 1246–1255

    MathSciNet  Google Scholar 

  36. Shcherbakov B A. Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations (in Russian). Chişinău: Ştiinţta, 1985

    Google Scholar 

  37. Shen W, Yi Y. Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem Amer Math Soc, 1998, 136: 1445493

    MathSciNet  MATH  Google Scholar 

  38. Sibirsky K S. Introduction to Topological Dynamics. Leyden: Noordhoff International Publishing, 1975

    MATH  Google Scholar 

  39. Smith H. Monotone semiflows generated by functional differential equations. J Differential Equations, 1987, 66: 420–442

    Article  MathSciNet  MATH  Google Scholar 

  40. Smith H. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. Providence: Amer Math Soc, 1995

    Google Scholar 

  41. Xing J, Yang X, Li Y. Affine-periodic solutions by averaging methods. Sci China Math, 2018, 61: 439–452

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhikov V V. On problem of existence of almost periodic solutions of differential and operator equations (in Russian). Akad Nauk Azerbaĭdžan SSR Dokl, 1970, 26: 3–7

    Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11271151 and 11522104), and the Startup and Xinghai Jieqing Funds from Dalian University of Technology. The authors sincerely thank the anonymous referees for their careful reading and helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    David Cheban & Zhenxin Liu

  2. Department of Mathematics, Faculty of Mathematics and Informatics, State University of Moldova, Chişinău, MD-2009, Moldova

    David Cheban

Authors
  1. David Cheban
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenxin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenxin Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheban, D., Liu, Z. Poisson stable motions of monotone nonautonomous dynamical systems. Sci. China Math. 62, 1391–1418 (2019). https://doi.org/10.1007/s11425-018-9407-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-018-9407-8

Keywords

MSC 2010

Search

Navigation