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

  • Shaobo Gan
  • Ming Li
  • Sergey B. Tikhomirov
Article

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.

Keywords

Vector fields Oriented shadowing \(\Omega \)-stability 

Mathematics Subject Classification

37C50 

Notes

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).

References

  1. 1.
    Abdenur, F., Diaz, L.J.: Pseudo-orbit shadowing in the \(C^{1}\) topology. Discret. Contin. Dyn. Syst. 17, 223–245 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arbieto, A., Reis, J.E., Ribeiro, R.: On Various Types of Shadowing for Geometric Lorenz Flows, http://arxiv.org/abs/1306.2061
  3. 3.
    Anosov, D.V.: On a class of invariant sets of smooth dynamical systems. Proc. 5th Int. Conf. Nonlinear Oscil. 2, 39–45 (1970)zbMATHGoogle Scholar
  4. 4.
    Aoki, N.: The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Bras. Mat. 23, 21–65 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975)zbMATHGoogle Scholar
  6. 6.
    Ding, H.: Disturbance of the homoclinic trajectory and applications. Acta Sci. Nat. Univ. Pekin. 1, 53–63 (1986)zbMATHGoogle Scholar
  7. 7.
    Gan, S., Wen, L.: Nonsingular star flows satisfy Axiom A and the nocycle condition. Invent. Math. 164, 279–315 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Guchenheimer, J.: A strange, strange attractor. The Hopf bifurcation theorems and its applications. Applied Mathematical Series. Springer, Berlin (1976)Google Scholar
  9. 9.
    Hayashi, S.: Diffeomorphisms in \({\cal F}^{1}(M)\) satisfy Axiom A. Ergod. Theory Dyn. Syst. 12, 233C253 (1992)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)Google Scholar
  11. 11.
    Kiriki, S., Soma, T.: Parameter-shifted shadowing property for geometric Lorenz attractors. Trans. Am. Math. Soc. 357, 1325–1339 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Komuro, M.: One-parameter flows with the pseudo orbit tracing property. Monat. Math. 98, 219–253 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Komuro, M.: Lorenz attractors do not have the pseudo-orbit tracing property. J. Math. Soc. Japan 37, 489–514 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Lee, K., Sakai, K.: Structural stability of vector fields with shadowing. J. Differ. Equ. 232, 303–313 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Li, C., Wen, L.: \({\cal X}^{*}\) plus Axiom A does not imply no-cycle. J. Differ. Equ. 119, 395C400 (1995)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Li, M., Gan, S., Wen, L.: Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discret. Contin. Dyn. Syst. 13, 239–269 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Liao, S.T.: Obstruction sets (II). Acta Sci. Nat. Univ. Pekin. 2, 1–36 (1981)MathSciNetGoogle Scholar
  18. 18.
    Mañé, R.: An ergodic closing lemma. Ann. Math. 116, 503–540 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Morimoto, A.: The method of pseudo-orbit tracing and stability of dynamical systems. Sem. Note, Tokyo University, vol. 39 (1979)Google Scholar
  20. 20.
    Palmer, K.J.: Shadowing in Dynamical Systems: Theory and Applications. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  21. 21.
    Palmer, K.J., Pilyugin, S.Y., Tikhomirov, S.B.: Lipschitz shadowing and structural stability of flows. J. Differ. Equ. 252, 1723–1747 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Pilyugin, S.Y.: Introduction to Structurally Stable Systems of Differential Equations. Birkhauser-Verlag, Basel (1992)CrossRefzbMATHGoogle Scholar
  23. 23.
    Pilyugin, S.Y.: Shadowing in Dynamical Systems. Lecture Notes in Math, vol. 1706. Springer, Berlin (1999)Google Scholar
  24. 24.
    Pilyugin, S.Y.: Shadowing in structurally stable flows. J. Differ. Equ. 140(2), 238–265 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Pilyugin, S.Y., Rodionova, A.A., Sakai, K.: Orbital and weak shadowing properties. Discret. Contin. Dyn. Syst. 9, 287–308 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Pilyugin, S.Y., Tikhomirov, S.B.: Sets of vector fields with various shadowing properties of pseudotrajectories. Doklady Math. 422, 30–31 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Pilyugin, S.Y., Tikhomirov, S.B.: Vector fields with the oriented shadowing property. J. Differ. Equ. 248, 1345–1375 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Pilyugin, S.Y., Tikhomirov, S.B.: Lipschitz shadowing implies structural stability. Nonlinearity 23, 2509–2515 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Ribeiro, R.: Hyperbolicity and types of shadowing for \(C^1\) generic vector fields. Discret. Contin. Dyn. Syst. 34, 2963–2982 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Robinson, C.: Stability theorems and hyperbolicity in dynamical systems. Rocky Mt. J. Math. 7, 425–437 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Sakai, K.: Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds. Osaka J. Math. 31, 373–386 (1994)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Sawada, K.: Extended \(f\)-orbits are approximated by orbits. Nagoya Math. J. 79, 33–45 (1980)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Shilnikov, L. P., Shilnikov, A. L., Taruve, D. V., Chua, L. O.: Methods of qualitative theory in nonlinear dynamics. World Scientific Series A, vol. 5Google Scholar
  34. 34.
    Thomas, R.F.: Stability properties of one-parameter flows. Proc. Lond. Math. Soc. 54, 479–505 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Tikhomirov, S.B.: Interiors of sets of vector fields with shadowing properties that correspond to some classes of reparametrizations. Vestnik St. Petersburg Univ. Math. 41(4), 360–366 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Tikhomirov, S.: An example of a vector field with the oriented shadowing property. http://arxiv.org/abs/1403.7378
  37. 37.
    Wen, L.: A uniform \(C^1\) connecting lemma. Discret. Contin. Dyn. Syst. 8, 257–265 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Wen, L., Xia, Z.: \(C^1\) connecting lemmas. Trans. Am. Math. Soc. 352, 5213–5230 (2000)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.LMAM and School of Mathematical SciencesPeking UniversityBeijingPeople’s republic of China
  2. 2.School of Mathematical Sciences and LPMCNankai UniversityTianjinPeople’s Republic of China
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  4. 4.Chebyshev LaboratorySaint-Petersburg State UniversitySaint-PetersburgRussia

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