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Bifurcations of stationary measures of random diffeomorphisms

Published online by Cambridge University Press:  01 October 2007

HICHAM ZMARROU  and
ALE JAN HOMBURG
HICHAM ZMARROU
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands (email: [email protected])
ALE JAN HOMBURG
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands (email: [email protected])
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Abstract

Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss the dependence of stationary measures on an auxiliary parameter, thus describing bifurcations of families of random diffeomorphisms. A bifurcation theory is developed under mild regularity assumptions on the diffeomorphisms and the noise distribution (e.g. smooth diffeomorphisms with uniformly distributed additive noise are included). We distinguish bifurcations where the density function of a stationary measure varies discontinuously or where the support of a stationary measure varies discontinuously. We establish that generic random diffeomorphisms are stable. The densities of stable stationary measures are shown to be smooth and to depend smoothly on an auxiliary parameter, except at bifurcation values. The bifurcation theory explains the occurrence of transients and intermittency as the main bifurcation phenomena in random diffeomorphisms. Quantitative descriptions by means of average escape times from sets as functions of the parameter are provided. Further quantitative properties are described through the speed of decay of correlations as a function of the parameter. Random differentiable maps which are not necessarily injective are studied in one dimension; we show that stable one-dimensional random maps occur open and dense and that in one-parameter families bifurcations are typically isolated. We classify codimension-one bifurcations for one-dimensional random maps; we distinguish three possible kinds, the random saddle node, the random homoclinic and the random boundary bifurcation. The theory is illustrated on families of random circle diffeomorphisms and random unimodal maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1651 - 1692
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Abraham, R., Marsden, J. E. and Ratiu, T.. Manifolds, Tensor Analysis, and Applications. Addison-Wesley, Reading, MA, 1983.Google Scholar
[2]Alves, J. F. and Araújo, V.. Random perturbations of nonuniformly expanding maps. Astérisque 286 (2003), 25–62.Google Scholar
[3]Alves, J. F., Araújo, V. and Vásquez, C. H.. Random perturbations of diffeomorphisms with dominated splitting. Preprint, CMUP, 2004.Google Scholar
[4]Araújo, V.. Attractors and time averages for random maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 307–369.CrossRefGoogle Scholar
[5]Arnold, L.. Random Dynamical Systems. Springer, Berlin, 1998.Google Scholar
[6]Baladi, V.. Positive Transfer Operators and Decay of Correlations. World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[7]Baladi, V. and Viana, M.. Strong stochastic stability and rate of mixing for unimodal maps. Ann. Sci. École Norm. Sup. (4) 29 (1996), 483–517.Google Scholar
[8]Baladi, V. and Young, L.-S.. On the spectra of randomly perturbed expanding maps. Comm. Math. Phys. 156 (1993), 355–385.CrossRefGoogle Scholar
[9]Berger, M. S.. Nonlinearity and Functional Analysis. Academic Press, New York, 1977.Google Scholar
[10]Bonatti, C., Diaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. Springer, Berlin, 2005.Google Scholar
[11]Buffoni, B. and Toland, J.. Analytic Theory of Global Bifurcation. Princeton University Press, Princeton, NJ, 2003.Google Scholar
[12]Chernov, N., Markarian, R. and Troubetzkoy, S.. Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 1049–1073.Google Scholar
[13]Chow, S.-N. and Hale, J. K.. Methods of Bifurcation Theory. Springer, Berlin, 1982.Google Scholar
[14]Colonius, F. and Kliemann, W.. The Dynamics of Control. Birkhäuser, Basel, 2000.CrossRefGoogle Scholar
[15]Dellnitz, M. and Junge, O.. On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 (1999), 491–515.CrossRefGoogle Scholar
[16]Demers, M. S. and Young, L.-S.. Escape rates and conditionally invariant measures. Nonlinearity 19 (2006), 377–397.Google Scholar
[17]Doob, J. L.. Stochastic Processes. Wiley & Sons, New York, 1953.Google Scholar
[18]Eckmann, J.-P., Thomas, L. and Wittwer, P.. Intermittency in the presence of noise. J. Phys. A: Math. Gen. 14 (1981), 3153–3168.Google Scholar
[19]Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P.. Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 (1995), 501–521.Google Scholar
[20]Fort, M. K.. Points of continuity of semi-continuous functions. Publ. Math. Debrecen 2 (1951), 100–102.Google Scholar
[21]Gayer, T.. On Markov chains and the spectra of the corresponding Frobenius–Perron operators. Stoch. Dyn. 1 (2001), 477–491.Google Scholar
[22]Gayer, T.. Control sets and their boundaries under parameter variation. J. Differential Equations 201 (2004), 177–200.Google Scholar
[23]Grebogi, C., Ott, E., Romeiras, F. and Yorke, J. A.. Critical exponents for crisis-induced intermittency. Phys. Rev. A 36 (1987), 5365–5380.Google Scholar
[24]Hirsch, J. E., Huberman, B. A. and Scalapino, D. J.. Theory of intermittency. Phys. Rev. A 25 (1982), 519–532.CrossRefGoogle Scholar
[25]Hirsch, M. W.. Differential Topology. Springer, Berlin, 1976.CrossRefGoogle Scholar
[26]Homburg, A. J. and Young, T.. Intermittency in families of unimodal maps. Ergod. Th. & Dynam. Sys. 22 (2002), 203–225.CrossRefGoogle Scholar
[27]Homburg, A. J. and Young, T.. Intermittency and Jakobson’s theorem near saddle-node bifurcations. Discrete Contin. Dyn. Syst. 17 (2007), 21–58.CrossRefGoogle Scholar
[28]Homburg, A. J. and Young, T.. Hard bifurcations in dynamical systems with bounded random perturbations. Regul. Chaotic Dyn. 11 (2006), 247–258.Google Scholar
[29]Kato, T.. Perturbation Theory for Linear Operators. Springer, Berlin, 1966.Google Scholar
[30]Kifer, Y.. Ergodic Theory of Random Transformations. Birkhäuser, Basel, 1986.CrossRefGoogle Scholar
[31]Kifer, Y.. Random Perturbations of Dynamical Systems. Birkhäuser, Basel, 1988.CrossRefGoogle Scholar
[32]Kreyszig, E.. Introductory Functional Analysis with Applications. Wiley, New York, 1978.Google Scholar
[33]Lasota, A. and Mackey, M. C.. Chaos, Fractals and Noise. Springer, Berlin, 1994.Google Scholar
[34]Lasserre, J. B. and Pearce, C. E. M.. On the existence of a quasistationary measure for a Markov chain. Ann. Probab. 29 (2001), 437–446.Google Scholar
[35]MacKay, R. S.. An extension of Zeeman’s notion of structural stability to noninvertible maps. Phys. D 52 (1991), 246–253.Google Scholar
[36]de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993.CrossRefGoogle Scholar
[37]Milnor, J. W.. Dynamics: introductory lectures. http://www.math.sunysb.edu/∼jack/DYNOTES/.Google Scholar
[38]Pianigiani, G.. Conditionally invariant measures and exponential decay. J. Math. Anal. Appl. 82 (1981), 75–88.Google Scholar
[39]Pianigiani, G. and Yorke, J. A.. Expanding maps on sets which are almost invariant. Decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351–366.Google Scholar
[40]Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980), 189–197.CrossRefGoogle Scholar
[41]Quas, A. N.. On representations of Markov chains by random smooth maps. Bull. London Math. Soc. 23 (1991), 487–492.CrossRefGoogle Scholar
[42]Ruelle, D.. An extension of the theory of Fredholm determinants. Publ. Math. Inst. Hautes Études Sci. 72 (1990), 175–193.CrossRefGoogle Scholar
[43]Ruffino, P. R. C.. A sampling theorem for rotation numbers of linear processes in R2. Random Oper. Stochastic Equations 8 (2000), 175–188.Google Scholar
[44]Schaefer, H. H.. Banach Lattices and Positive Operators. Springer, Berlin, 1974.Google Scholar
[45]Viana, M.. Stochastic Dynamics of Deterministic Systems. Col. Bras. de Matemática, 1997.Google Scholar
[46]Yosida, K.. Functional Analysis. Springer, Berlin, 1980.Google Scholar
[47]Young, L.-S.. Stochastic stability of hyperbolic attractors. Ergod. Th. & Dynam. Sys. 6 (1986), 311–319.CrossRefGoogle Scholar
[48]Zeeman, E. C.. Stability of dynamical systems. Nonlinearity 1 (1988), 115–155.Google Scholar