Skip to main content

Synchronization by noise

Abstract

We provide sufficient conditions for synchronization by noise, i.e. under these conditions we prove that weak random attractors for random dynamical systems consist of single random points. In the case of SDE with additive noise, these conditions are also essentially necessary. In addition, we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point. As a result, synchronization by noise is proven for a large class of SDE with additive noise. In particular, we prove that the random attractor for an SDE with drift given by a (multidimensional) double-well potential and additive noise consists of a single random point. All examples treated in Tearne (Probab Theory Relat Fields 141(1–2):1–18, 2008) are also included.

This is a preview of subscription content, access via your institution.

Notes

  1. For notation and background on RDS see Sect.  1.2 below.

  2. For example, every metric space (Ed) that is both locally compact and \(\sigma \)-compact allows an equivalent metric \(d'\) such that \((E,d')\) is Heine-Borel [49].

  3. Note that we have \(\varphi _t(\omega ,\cdot )\in C^{1,\delta }_{loc}\) and not only \(\varphi _t(\omega ,\cdot )\in C^{1,\beta }_{loc}\) for every \(\beta <\delta \) due to the additive noise in (3.6). This may be seen by considering the transformation \(\tilde{\varphi }= \varphi - \sigma W\) and studying the regularity of the corresponding pathwise ODE.

  4. In fact, [48, Theorem, p. 243] assumes b to be smooth. However, it is an easy exercise to see that only \(b \in C^\delta \) for some \(\delta >0\) is required for the proof (cf. also [29, Theorem 10.4.1] for an according regularity result for linear, non-degenerate second order PDE with Hölder coefficients).

  5. Note that this property is implied by (3.25).

  6. The existence of (weak) random attractors under a weak coercivity condition has been obtained in [21].

References

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

    Book  Google Scholar 

  2. Arnold, L., Chueshov, I.: Order-preserving random dynamical systems: equilibria, attractors, applications. Dynam. Stab. Syst. 13(3), 265–280 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, L., Crauel, H., Wihstutz, V.: Stabilization of linear systems by noise. SIAM J. Control Optim. 21(3), 451–461 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, L., Kliemann, W.: On unique ergodicity for degenerate diffusions. Stochastics 21(1), 41–61 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baxendale, P.H.: Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. In: Spatial stochastic processes, vol. 19 of Progr. Probab., pp. 189–218. Birkhäuser Boston, Boston (1991)

  6. Caraballo, T., Chueshov, I.D., Kloeden, P.E.: Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal. 38(5), 1489–1507 (2006/2007)

  7. Caraballo, T., Crauel, H., Langa, J.A., Robinson, J.C.: The effect of noise on the Chafee-Infante equation: a nonlinear case study. Proc. Am. Math. Soc. 135(2), 373–382 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caraballo, T., Robinson, J.C.: Stabilisation of linear PDEs by Stratonovich noise. Systems Control Lett. 53(1), 41–50 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carverhill, A.: Flows of stochastic dynamical systems: ergodic theory. Stochastics 14(4), 273–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chekroun, M.D., Simonnet, E., Ghil, M.: Stochastic climate dynamics: random attractors and time-dependent invariant measures. Phys. D 240(21), 1685–1700 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, X., Duan, J., Scheutzow, M.: Evolution systems of measures for stochastic flows. Dyn. Syst. 26(3), 323–334 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chueshov, I.: Monotone random systems theory andapplications, vol. 1779 of Lecture Notes in Mathematics. Springer, Berlin (2002)

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Chueshov, I., Schmalfuß, B.: Master-slave synchronization and invariant manifolds for coupled stochastic systems. J. Math. Phys. 51(10), 102702, 23 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Crauel, H.: Markov measures for random dynamical systems. Stoch. Stoch. Rep. 37(3), 153–173 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crauel, H.: Random point attractors versus random set attractors. J. Lond. Math. Soc. 63(2), 413–427 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Crauel, H.: Random probability measures on Polish spaces, vol. 11 of Stochastics Monographs. Taylor & Francis, London (2002)

    MATH  Google Scholar 

  18. Crauel, H., Dimitroff, G., Scheutzow, M.K.R.: Criteria for strong and weak random attractors. J. Dyn. Differ. Equations 21(2), 233–247 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Crucifix, M.: Why could ice ages be unpredictable? Clim. Past 9, 2253–2267 (2013)

    Article  Google Scholar 

  21. Dimitroff, G., Scheutzow, M.K.R.: Attractors and expansion for Brownian flows. Electron. J. Probab. 16(42), 1193–1213 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Flandoli, F., Gess, B., Scheutzow, M.: Synchronization by noise for order-preserving random dynamical systems (2015). arXiv:1503.08737 [to appear in: Ann. Probab. (2015). doi:10.1214/16-AOP1088]

  23. Gess, B.: Random attractors for singular stochastic evolution equations. J. Differ. Equations 255(3), 524–559 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ghil, M., Chekroun, M.D., Simonnet, E.: Climate dynamics and fluid mechanics: Natural variability and related uncertainties. Phys. D 237(14–17), 2111–2126 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Homburg, A.J.: Synchronization in iterated function systems (2013). arXiv:1303.6054

  26. Jäger, T., Keller, G.: Random minimality and continuity of invariant graphs in random dynamical systems (2013). arXiv:1211.5885 [to appear in Trans. Am. Math. Soc. (2013)]

  27. Keller, G., Jafri, H.H., Ramaswamy, R.: Nature of weak generalized synchronization in chaotically driven maps. Phys. Rev. E 87 (2013)

  28. Khasminskii, R.: Stochastic stability of differentialequations, vol. 66 of Stochastic Modelling and AppliedProbability. In: Milstein, G.N., Nevelson, M.B. (eds.) 2nd edn. Springer, Heidelberg (2012)

  29. Krylov, N.V.: Lectures on elliptic and parabolic equationsin Hölder spaces, vol. 12 of Graduate Studies inMathematics. American Mathematical Society, Providence (1996)

  30. Kuksin, S., Shirikyan, A.: Mathematics of two-dimensional turbulence. Cambridge University Press, Cambridge (2012) (books online)

  31. Kuksin, S.B., Shirikyan, A.: On random attractors for systems of mixing type. Funktsional. Anal. i Prilozhen. 38(1), 34–46 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Le Jan, Y.: Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. H. Poincaré Probab. Statist. 23(1), 111–120 (1987)

    MathSciNet  MATH  Google Scholar 

  33. Lemaire, V., Pagès, G., Panloup, F.: Invariant distribution of duplicated diffusions and application to richardson-romberg extrapolation. arXiv:1302.1651 [to appear in Ann. Instit. Henri Poincaré, Probab. Stat. (2014)]

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Martinelli, F., Scoppola, E.: Small random perturbations of dynamical systems: exponential loss of memory of the initial condition. Comm. Math. Phys. 120(1), 25–69 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mohammed, S.-E.A., Scheutzow, M.K.R.: The stable manifold theorem for stochastic differential equations. Ann. Probab. 27(2), 615–652 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Newman, J.: Necessary and sufficient conditions for stable synchronisation in random dynamical systems (2014). arXiv:1408.5599

  38. Ochs, G.: Weak random attractors. Report 449, Institut für Dynamische Systeme, Universität Bremen, pp. 1–18 (1999)

  39. Pikovsky, A., Rosenblum, M., Kurths, J., Hilborn, R.C.: Synchronization: a universal concept in nonlinear science. Am. J. Phys. 70(6) (2002)

  40. Pyragas, K.: Weak and strong synchronization of chaos. Phys. Rev. E 54, R4508–R4511 (1996)

    Article  Google Scholar 

  41. Ruelle, D.: Ergodic theory of differentiable dynamical systems. Inst. Hautes Études Sci. Publ. Math. 50, 27–58 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994 (1995)

    Article  Google Scholar 

  43. Scheutzow, M.K.R.: Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12(1), 41–80 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  44. Scheutzow, M.K.R.: Comparison of various concepts of a random attractor: a case study. Arch. Math. 78(3), 233–240 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sirovich, R., Sacerdote, L., Villa, A.E.P.: Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Math. Biosci. Eng. 11(2), 385–401 (2014)

    MathSciNet  MATH  Google Scholar 

  46. Tearne, O.M.: Collapse of attractors for ODEs under small random perturbations. Probab. Theory Relat. Fields 141(1–2), 1–18 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Teschl, G.: Ordinary differential equations and dynamicalsystems, vol. 140 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2012)

  48. Varadhan, S.R.S.: Lectures on diffusion problems and partialdifferential equations, vol. 64 of Tata Institute of FundamentalResearch Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay (1980) (With notes by P.l. Muthuramalingam and T.R. Nanda)

  49. Williamson, R., Janos, L.: Constructing metrics with the Heine-Borel property. Proc. Am. Math. Soc. 100(3), 567–573 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank the anonymous referees for their careful reading or our manuscript and their many insightful comments and suggestions that helped to improve the presentation of the manuscript and simplify several proofs.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Scheutzow.

Additional information

BG has been partially supported by the Research Project “Random dynamical systems and regularization by noise for stochastic partial differential equations” funded by the German Research Foundation.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Flandoli, F., Gess, B. & Scheutzow, M. Synchronization by noise. Probab. Theory Relat. Fields 168, 511–556 (2017). https://doi.org/10.1007/s00440-016-0716-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-016-0716-2

Keywords

  • Synchronization
  • Random dynamical system
  • Random attractor
  • Lyapunov exponent
  • Stochastic differential equation
  • Statistical equilibrium

Mathematics Subject Classification

  • 37B25
  • 37G35
  • 37H15