Abstract
We prove a theorem on the existence of a random attractor for a multivalued random dynamical system dissipative with respect to probability. Abstract results are used for the analysis of the qualitative behavior of solutions of a system of ordinary differential equations with continuous right-hand side perturbed by a stationary random process. In terms of the Lyapunov function, for an unperturbed system, we give sufficient conditions for the existence of a random attractor.
Similar content being viewed by others
REFERENCES
L. Arnold (1998) Random Dynamical Systems Springer Berlin
H. Crauel F. Flandoli (1994) ArticleTitleAttractors for random dynamical systems Probab. Theory Related Fields 100 365–393
H. Crauel (1999) ArticleTitleGlobal random attractors are uniquely determined by attracting deterministic compact sets Ann. Mat. Pura Appl. 126 IssueID4 57–72
K. R. Schenk-Hoppe (1998) ArticleTitleRandom attractors — general properties, existence and applications to stochastic bifurcation theory Discr. Cont. Dynam. Syst. 4 IssueID1 99–130
A. V. Kapustyan V. S. Mel’nik (1998) ArticleTitleAttractors of multivalued semidynamical systems and their approximations Dopov. Nats. Akad. Nauk Ukr. 10 21–25
T. Caraballo J. A. Langa J. Valero (2002) ArticleTitleGlobal attractors for multivalued random dynamical systems Nonlin. Analysis 48 805–829
R. Z. Khas’minskii (1969) Stability of Systems of Differential Equations under Random Perturbations of Their Parameters Nauka Moscow
J.-P. Aubin H. Frankowska (1990) Set-Valued Analysis Birkhäuser Boston
A. F. Fillipov (1985) Differential Equations with Discontinuous Right-Hand Side Nauka Moscow
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 892–900, July, 2004.
Rights and permissions
About this article
Cite this article
Kapustyan, O.V. Random attractors for ambiguously solvable systems dissipative with respect to probability. Ukr Math J 56, 1063–1073 (2004). https://doi.org/10.1007/PL00022177
Received:
Issue Date:
DOI: https://doi.org/10.1007/PL00022177