Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Collapse of attractors for ODEs under small random perturbations
Download PDF
Download PDF
  • Published: 30 November 2007

Collapse of attractors for ODEs under small random perturbations

  • Oliver M. Tearne1 

Probability Theory and Related Fields volume 141, pages 1–18 (2008)Cite this article

  • 190 Accesses

  • 6 Citations

  • Metrics details

Abstract

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Arnold, L.: Random dynamical systems. Springer Monographs in Mathematics. Springer, Berlin (1998)

  2. Caraballo T., Langa J.A. and Robinson J.C. (1998). Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commun. Partial Differ. Equ. 23(9–10): 1557–1581

    Article  MATH  MathSciNet  Google Scholar 

  3. Chueshov I. and Scheutzow M. (2004). On the structure of attractors and invariant measures for a class of monotone random systems. Dyn. Syst. 19(2): 127–144

    Article  MATH  MathSciNet  Google Scholar 

  4. Crauel H. (1999). Global random attractors are uniquely determined by attracting deterministic compact sets. Ann. Mat. Pura Appl. 176(4): 57–72

    Article  MATH  MathSciNet  Google Scholar 

  5. Crauel H., Debussche A. and Flandoli F. (1997). Random attractors. J. Dyn. Differ. Equ. 9(2): 307–341

    Article  MATH  MathSciNet  Google Scholar 

  6. Crauel H. and Flandoli F. (1994). Attractors for random dynamical systems. Probab. Theory Relat. Fields 100(3): 365–393

    Article  MATH  MathSciNet  Google Scholar 

  7. Crauel H. and Flandoli F. (1998). Additive noise destroys a pitchfork bifurcation. J. Dyn. Differ. Equ. 10(2): 259–274

    Article  MATH  MathSciNet  Google Scholar 

  8. Feller W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55(2): 468–519

    Article  MathSciNet  Google Scholar 

  9. Friedman, A.: Stochastic differential equations and applications, vol. 1. Academic (Harcourt Brace Jovanovich Publishers), New York (1975)

  10. Hale, J.K.: Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence, RI (1988)

  11. Has´minskiĭ R.Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor. Verojatnost. i Primenen. 5: 196–214

    MathSciNet  Google Scholar 

  12. Humphries, A.R., Stuart, A.M.: Deterministic and random dynamical systems: theory and numerics. In: Modern methods in scientific computing and applications (Montréal, QC, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 75, pp. 211–254. Kluwer, Dordrecht (2002)

  13. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, North- Holland Mathematical Library, vol. 24. North-Holland Publishing, Amsterdam (1989)

  14. Martinelli F., Sbano L. and Scoppola E. (1994). Small random perturbation of dynamical systems: recursive multiscale analysis. Stochastics Stochastics Rep. 49(3–4): 253–272

    MATH  MathSciNet  Google Scholar 

  15. McKean, H.P., Jr.: Stochastic integrals. Probability and Mathematical Statistics, vol. 5. Academic, New York (1969)

  16. Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales, vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000)

  17. Schenk-Hoppé K.R. (1998). Random attractors—general properties, existence and applications to stochastic bifurcation theory. Discrete Contin. Dyn. Syst. 4(1): 99–130

    MATH  Google Scholar 

  18. Schmalfuss, B.: Backward cocycles and attractors of stochastic differential equations. In: Koksch, N., Reitmann, V., Riedrich, T. (eds.) International Seminar on Applied Mathematics—Nonlinear Dynamics: Attractor Approximation and Global Behaviour, pp. 185–192. Technische Universität, Dresden (1992)

  19. Tearne O.M. (2005). Boundary approximation of deterministic and random attractors. Nonlinearity 18: 2011–2034

    Article  MATH  MathSciNet  Google Scholar 

  20. Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)

  21. Ventcel´ A.D. and Freĭdlin M.I. (1970). Small random perturbations of dynamical systems. Uspehi Mat. Nauk 25(1 (151)): 3–55

    Google Scholar 

  22. Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Oliver M. Tearne

Authors
  1. Oliver M. Tearne
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver M. Tearne.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tearne, O.M. Collapse of attractors for ODEs under small random perturbations. Probab. Theory Relat. Fields 141, 1–18 (2008). https://doi.org/10.1007/s00440-006-0051-0

Download citation

  • Received: 07 November 2005

  • Revised: 24 May 2006

  • Published: 30 November 2007

  • Issue Date: May 2008

  • DOI: https://doi.org/10.1007/s00440-006-0051-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Global attractor
  • Random attractor
  • Omega limit set

Mathematics Subject Classification (2000)

  • 34D45
  • 34F05
  • 37C70
  • 60H10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature