Abstract
Suppose (*)Ȧ=A(v t w)x is hyperbolic, i.e. all of its Lyapunov exponents are different from zero. Then Ȧ=A(v t w)x+f(v t w,x)+b(v t w) with f(w,·) locally Lipschitz and f(w,0)=0 has a (unique) stationary solution in a neighborhood of x=0 provided f and b are ‘small’. ‘Smallness’ in being described in terms of a random norm measuring the non-uniformity of the hyperbolicity of (*).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnold, L. and H. Crauel: Iterated function systems and multiplicative ergodic theory. M. Pinsky, V. Wihstutz (eds): Stochastic flows. Birkhäuser 1991.
Boxler, P.: A stochastic version of center manifold theory. Probab. Th. Rel. Fields 83, 509–545 (1989).
Bunke, H.: Gewöhnliche Differentialgleichungen mit zufälligen Parametern. Akademie-Verlag, Berlin 1972.
Coppel, W.A.: Stability and asymptotic behaviour of differential equations. Heath, Boston 1965.
Hale, J.K.: Ordinary differential equations. Wiley, New York 1969.
Hasminskii, R.Z.: Stochastic stability of differential equations. Sijthoff and Noordhoff, Alphen 1980.
Meyer, K.R. and G.R. Sell: Melnikov transforms, Bernoulli bundles, and almost periodic perturbations. Trans. Amer. Math. Soc. 314, 63–105 (1989).
Palmer, K.: Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynamics Reported, Vol. 1. Wiley, New York 1988, 265–306.
Scheurle, J.: Chaotic solutions of systems with almost periodic forcing. J. Applied Math. and Phys. (ZAMP) 37, 12–26 (1986).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Arnold, L., Boxler, P. (1991). Additive noise turns a hyperbolic fixed point into a stationary solution. In: Arnold, L., Crauel, H., Eckmann, JP. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086665
Download citation
DOI: https://doi.org/10.1007/BFb0086665
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54662-7
Online ISBN: 978-3-540-46431-0
eBook Packages: Springer Book Archive