Abstract
We consider families of random dynamical systems induced by parametrized one-dimensional stochastic differential equations. We give necessary and sufficient conditions on the invariant measures of the associated Markov semigroups which ensure a stochastic bifurcation. This leads to sufficient conditions on drift and diffusion coefficients for a stochastic pitchfork and transcritical bifurcation of the family of random dynamical systems.
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.
Similar content being viewed by others
5 References
L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg New York, 1998.
L. Arnold and P. Boxler. Stochastic bifurcation: Instructive examples in dimension one. In: Pinsky and Wihstutz, editors, Diffusion processes and related problems in analysis, Vol. II: Stochastic flows. Progress in Probability, Vol. 27, pages 241–256, Boston Basel Stuttgart, 1992 Birkhäuser.
L. Arnold and B. Schmalfuß. Fixed Point and Attractors for Random Dynamical Systems. In A. Naess and S. Krenk, editors, IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, pages 19–28. Kluwer, Dordrecht, 1996.
L. Arnold and M. Scheutzow. Perfect cocycles through stochastic differential equations. Probab. Theory Relat. Fields, 101:65–88, 1995.
P. Baxendale. Asymptotic behaviour of stochastic flows of diffeomorphisms. In K. Itô and T. Hida, editors, Stochastic processes and their applications, volume 1203 of Lecture Notes in Mathematics, pages 1–19. Springer, Berlin Heidelberg New York, 1986.
H. Crauel. Markov measures for random dynamical systems. Stochastics and Stochastics Reports, 37:153–173, 1991.
H. Crauel. Non-Markovian invariant measures are hyperbolic. Stochastics Processes and their Applications, 45:13–28, 1993.
H. Crauel and F. Flandoli. Additive noise destroys a pitchfork bifurcation. Journal of Dynamics and Differential Equations, 10:259–274, 1998.
W. Hackenbroch and A. Thalmaier. Stochastische Analysis. Teubner, Stuttgart, 1994.
W. Horsthemke and D. K. Lefever. Noise-induced transitions. Springer, Berlin Heidelberg New York, 1984.
I. Karatzas and S. Shreve. Brownian motion and stochastic calculus. Springer, Berlin Heidelberg New York, 1988.
F. Ledrappier. Positivity of the exponent for stationary sequences of matrices. In: L. Arnold and V. Wihstutz, editors, Lyapunov Exponents, Proceedings Bremen 1984, Lecture Notes in Mathematics 1097, pages 56–73. Springer, Berlin Heidelberg New York 1986.
Y. Le Jan. Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. H. Poincaré Probab. Statist., 23:111–120, 1987.
Xu Kedai. Bifurcations of random differential equations in dimension one. Random & Computational Dynamics, 1:277–305, 1993.
Rights and permissions
Copyright information
© 1999 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Crauel, H., Imkeller, P., Steinkamp, M. (1999). Bifurcations of One-Dimensional Stochastic Differential Equations. In: Stochastic Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-22655-9_2
Download citation
DOI: https://doi.org/10.1007/0-387-22655-9_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98512-1
Online ISBN: 978-0-387-22655-2
eBook Packages: Springer Book Archive