Abstract
Given a dynamical system (Ω,ℱ, ℙ,θ) and a random diffeomorphism ϕ(ω): ℝd → ℝd with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h(ω) to make the random diffeomorphism\(\tilde \varphi \)(ω)=h(θω)−1○ϕ(ω)○ h(ω) “as simple as possible,” preferably linear. The linearization Dϕ(ω, 0)=:A(ω) generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance ofθ turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptofε-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anosov, D. V., and Arnold, V. I. (eds.) (1988).Dynamical System I, Springer Verlag, New York.
Arnold, L. (1966). Über die Konvergenz einer zufälligen Potenzreihe.J. Reine Angewandte Math.222, 79–112.
Arnold, L., and Crauel, H. (1991). Random dynamical systems. In Arnold, L., Crauel, H., and Eckmann, J.-P. (eds.),Lyapunov Exponents, Vol. 1486, Lecture Notes in Mathematics, Springer Verlag, Berlin, pp. 1–22.
Arnold, L., and San Martin, L. (1989). A multiplicative ergodic theorem for rotation numbers.J. Dynam. Diff. Eg.1, 95–119.
Arnold, L., and Xu, K.-D. (1991a). Normal forms for random differential equations (submitted for publication).
Arnold, L., and Xu, K.-D. (1991b). Simultaneous normal form and center manifold reduction for random differential equations (submitted for publication).
Arnold, V. I. (1983).Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren 250, Springer Verlag, Berlin.
Bougerol, Ph. (1980). Comparaison des exposants de Lyapounov des processus markoviens multiplicatifs.Ann. Inst. Henri Poincaré 24, 439–489.
Bougerol, Ph., and Lacroix, J. (1985).Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, Boston, Basel.
Boxler, P. (1985). A stochastic version of center manifold theory.Probab. Th. Rel. Fields 83, 509–545.
Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G. (1982).Ergodic Theory, Springer Verlag, Berlin.
Coullet, P. H., Elphick, C., and Tirapegui, E. (1985). Normal form of a Hopf bifurcation with noise.Phys. Lett.111A, 277–282.
Crauel, H. (1990). Extremal exponents of random dynamical systems do not vanish.J. Dynam. Diff. Eq.2, 245–291.
Deimling, K. (1985).Nonlinear Functional Analysis, Springer Verlag, Berlin.
Doob, J. L. (1953).Stochastic Processes, Wiley, New York.
Elphick, C., and Tirapegui, E. (1987). Normal forms with noise. In Tirapegui, E., and Villamoel, D. (eds.),Instabilities and Nonequilibrium Structures, Reidel, Dordrecht, pp. 299–310.
Elphick, C., M. E. Tirapegui, Brachet, M. E., Coullet, P. H., and Iooss, G. (1987). A simple global characterization for normal forms of singular vector fields.Physica 29, 95–127.
Goldsheid, I. Ya., and Margulis, G. A. (1989). Lyapunov indices of products of random matrices.Uspekhi Mat. Nauk 44(6), 13–60. (Russ. Math. Surv.44(6), 11–71, 1989.)
Graham, A. (1981).Kronecker Products and Matrix Calculus with Applications, Ellis Horwood, Chichester.
Guckenheimer, J., and Holmes, Ph. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Berlin, Springer Verlag.
Irwin, M. C. (1980).Smooth Dynamical Systems, Academic Press, London.
Nicolis, C., and Nicolis, G. (1986). Normal form analysis of stochastically forced dynamical systems.Dynam. Stabil. Syst.1, 249–253.
Oseledec, V. I. (1968). A multiplicative ergodic theorem—Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc.19, 197–231.
Poincaré, H. (1892).Les Méthodes Nouvelles de la Mécanique Celeste I–III, Gauthier-Villars, Paris.
Ruelle, D. (1979). Ergodic theory of differentiable dynamical systems.Publ. Math. IHES 50, 27–58.
Sri Namachchivaya, N., and Leng, G. (1990). Equivalence of stochastic averaging and stochastic normal forms.J. Appl. Mech. (ASME) 57(4), 1011–1017.
Sri Namachchivaya, N., and Lin, Y. K. (1991). Method of stochastic normal forms.Int. J. Nonlin. Mech. 26.
Tirapegui, E. (1988). Normal forms for deterministic and stochastic systems. In Bamón, R., Labarca, R., and Palis, J., Jr. (eds.),Dynamical Systems, Valparaiso, 1986, Vol. 1331, Lecture Notes in Mathematics, Springer Verlag, Berlin.
Vanderbauwhede, A. (1989). Center manifolds, normal forms and elementary bifurcations. In Kirchgraber, U., and Walter, H. O. (eds.),Dynamics Reported, Vol. 2, Teubner and Wiley, New York, pp. 89–169.
Wiggins, S. (1990).Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer Verlag, Berlin.
Xu, K.-D. (1990).Normalformen für zufällige dynamische Systeme, Ph.D. thesis, Universität Bremen, Bremen.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arnold, L., Kedai, X. Normal forms for random diffeomorphisms. J Dyn Diff Equat 4, 445–483 (1992). https://doi.org/10.1007/BF01053806
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01053806