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.
References
Arnold, L.: Random dynamical systems. Springer Monographs in Mathematics. Springer, Berlin (1998)
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
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
Crauel H. (1999). Global random attractors are uniquely determined by attracting deterministic compact sets. Ann. Mat. Pura Appl. 176(4): 57–72
Crauel H., Debussche A. and Flandoli F. (1997). Random attractors. J. Dyn. Differ. Equ. 9(2): 307–341
Crauel H. and Flandoli F. (1994). Attractors for random dynamical systems. Probab. Theory Relat. Fields 100(3): 365–393
Crauel H. and Flandoli F. (1998). Additive noise destroys a pitchfork bifurcation. J. Dyn. Differ. Equ. 10(2): 259–274
Feller W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55(2): 468–519
Friedman, A.: Stochastic differential equations and applications, vol. 1. Academic (Harcourt Brace Jovanovich Publishers), New York (1975)
Hale, J.K.: Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence, RI (1988)
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
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)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, North- Holland Mathematical Library, vol. 24. North-Holland Publishing, Amsterdam (1989)
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
McKean, H.P., Jr.: Stochastic integrals. Probability and Mathematical Statistics, vol. 5. Academic, New York (1969)
Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales, vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000)
Schenk-Hoppé K.R. (1998). Random attractors—general properties, existence and applications to stochastic bifurcation theory. Discrete Contin. Dyn. Syst. 4(1): 99–130
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)
Tearne O.M. (2005). Boundary approximation of deterministic and random attractors. Nonlinearity 18: 2011–2034
Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)
Ventcel´ A.D. and Freĭdlin M.I. (1970). Small random perturbations of dynamical systems. Uspehi Mat. Nauk 25(1 (151)): 3–55
Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-006-0051-0
Keywords
- Global attractor
- Random attractor
- Omega limit set
Mathematics Subject Classification (2000)
- 34D45
- 34F05
- 37C70
- 60H10