Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T04:10:32.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Quenched stochastic stability for eventually expanding-on-average random interval map cocycles

Published online by Cambridge University Press:  25 January 2018

GARY FROYLAND ,
CECILIA GONZÁLEZ-TOKMAN  and
RUA MURRAY
GARY FROYLAND
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia email [email protected]
CECILIA GONZÁLEZ-TOKMAN
Affiliation:
School of Mathematics and Physics, University of Queensland, St Lucia QLD 4072, Australia email [email protected]
RUA MURRAY
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles. Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froyland et al were that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froyland et al, requiring only that the cocycle be eventually expanding on average, and importantly, allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 10 , October 2019 , pp. 2769 - 2792
Copyright
© Cambridge University Press, 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alves, J. F. and Araújo, V.. Random perturbations of nonuniformly expanding maps. Astérisque 286(xvii) (2003), 25–62. Geometric methods in dynamics. I.Google Scholar
Bahsoun, W.. Rigorous numerical approximation of escape rates. Nonlinearity 19(11) (2006), 2529–2542.Google Scholar
Baladi, V.. Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Commun. Math. Phys. 186(3) (1997), 671–700.Google Scholar
Baladi, V. and Viana, M.. Strong stochastic stability and rate of mixing for unimodal maps. Ann. Sci. Éc. Norm. Supér. (4) 29(4) (1996), 483–517.Google Scholar
Benedicks, M. and Young, L.-S.. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12(1) (1992), 13–37.Google Scholar
Blank, M. and Keller, G.. Stochastic stability versus localization in one-dimensional chaotic dynamical systems. Nonlinearity 10(1) (1997), 81–107.Google Scholar
Bollt, E. M. and Santitissadeekorn, N.. Applied and Computational Measurable Dynamics (Mathematical Modeling and Computation, 18) . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.Google Scholar
Bose, C., Froyland, G., González-Tokman, C. and Murray, R.. Ulam’s method for Lasota–Yorke maps with holes. SIAM J. Appl. Dyn. Syst. 13(2) (2014), 1010–1032.Google Scholar
Bose, C. and Murray, R.. Exact rate of approximation in Ulam’s method. Discrete Contin. Dyn. Syst. 7 (2001), 219–235.Google Scholar
Budišić, M. and Mezić, I.. Geometry of the ergodic quotient reveals coherent structures in flows. Physica D 241(15) (2012), 1255–1269.Google Scholar
Buzzi, J.. Exponential decay of correlations for random Lasota–Yorke maps. Commun. Math. Phys. 208(1) (1999), 25–54.Google Scholar
Buzzi, J.. Absolutely continuous S.R.B. measures for random Lasota–Yorke maps. Trans. Amer. Math. Soc. 352(7) (2000), 3289–3303.Google Scholar
Dellnitz, M., Froyland, G., Horenkamp, C. and Padberg, K.. On the approximation of transport phenomena—a dynamical systems approach. GAMM-Mitt. 32(1) (2009), 47–60.Google Scholar
Dellnitz, M., Froyland, G. and Junge, O.. The algorithms behind GAIO—set oriented numerical methods for dynamical systems. Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin, 2001, pp. 145–174.Google Scholar
Dellnitz, M. and Junge, O.. On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36(2) (1999), 491–515.Google Scholar
Dellnitz, M., Junge, O., Koon, W. S., Lekien, F., Lo, M. W., Marsden, J. E., Padberg, K., Preis, R., Ross, S. D. and Thiere, B.. Transport in dynamical astronomy and multibody problems. Internat. J. Bifur. Chaos 15(03) (2005), 699–727.Google Scholar
Deuflhard, P. and Schütte, C.. Molecular conformation dynamics and computational drug design. Applied Mathemetics Entering the 21st Century. Proceedings ICIAM. SIAM, Philadelphia, PA, 2004, pp. 91–119.Google Scholar
Ding, J. and Zhou, A.. Finite approximations of Frobenius–Perron operators. A solution of Ulam’s conjecture to multi-dimensional transformations. Physica D 92(1–2) (1996), 61–68.Google Scholar
Froyland, G.. Finite approximation of Sinai–Bowen–Ruelle measures for Anosov systems in two dimensions. Random Comput. Dyn. 3(4) (1995), 251–264.Google Scholar
Froyland, G.. Ulam’s method for random interval maps. Nonlinearity 12(4) (1999), 1029–1052.Google Scholar
Froyland, G.. Extracting dynamical behavior via Markov models. Nonlinear Dynamics and Statistics. Birkhauser Boston, Boston, 2001, pp. 281–321.Google Scholar
Froyland, G., González-Tokman, C. and Quas, A.. Stability and approximation of random invariant densities for Lasota–Yorke map cocycles. Nonlinearity 27(4) (2014), 647–660.Google Scholar
Froyland, G., Horenkamp, C., Rossi, V., Santitissadeekorn, N. and Gupta, A. S.. Three-dimensional characterization and tracking of an Agulhas Ring. Ocean Modelling 52 (2012), 69–75.Google Scholar
Froyland, G. and Padberg, K.. Almost-invariant sets and invariant manifolds—connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D 238(16) (2009), 1507–1523.Google Scholar
Froyland, G., Padberg, K., England, M. H. and Treguier, A. M.. Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98(22) (2007), 224503.Google Scholar
Froyland, G. and Padberg-Gehle, K.. Almost-invariant and finite-time coherent sets: directionality, duration, and diffusion. Ergodic Theory, Open Dynamics, and Coherent Structures. Springer, New York, 2014, pp. 171–216.Google Scholar
Froyland, G., Santitissadeekorn, N. and Monahan, A.. Transport in time-dependent dynamical systems: Finite-time coherent sets. Chaos 20(4) (2010), 043116.Google Scholar
Hsu, C. S.. Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Springer, New York, 1987.Google Scholar
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M., Daviller, G., Östh, J., Krajnović, S. and Niven, R. K.. Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754 (2014), 365–414, 009.Google Scholar
Katok, A. and Kifer, Y.. Random perturbations of transformations of an interval. J. Anal. Math. 47 (1986), 193–237.Google Scholar
Keller, G.. Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94(4) (1982), 313–333.Google Scholar
Kifer, Y.. Random Perturbations of Dynamical Systems (Progress in Probability and Statistics, 16) . Birkhäuser Boston, Inc., Boston, MA, 1988.Google Scholar
Kruijt, P. G. M., Galaktionov, O. S., Anderson, P. D., Peters, G. W. M. and Meijer, H. E. H.. Analyzing mixing in periodic flows by distribution matrices: mapping method. AIChE J. 47(5) (2001), 1005–1015.Google Scholar
Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
Li, T. Y.. Finite approximation for the Frobenius–Perron operator. A solution to Ulam’s conjecture. J. Approx. Theory 17(2) (1976), 177–186.Google Scholar
Maximenko, N., Hafner, J. and Niiler, P.. Pathways of marine debris derived from trajectories of Lagrangian drifters. Marine Pollution Bull. 65(1) (2012), 51–62.Google Scholar
Morita, T.. Deterministic version lemmas in ergodic theory of random dynamical systems. Hiroshima Math. J. 18(1) (1988), 15–29.Google Scholar
Murray, R.. Existence, mixing and approximation of invariant densities for expanding maps on R r . Nonlinear Anal. 45(1) (2001), 37–72.Google Scholar
Murray, R.. Ulam’s method for some non-uniformly expanding maps. Discrete Contin. Dyn. Syst. 26(3) (2010), 1007–1018.Google Scholar
Ohno, T.. Asymptotic behaviors of dynamical systems with random parameters. Publ. Res. Inst. Math. Sci. 19(1) (1983), 83–98.Google Scholar
Padberg, K.. Numerical analysis of transport in dynamical systems. PhD Thesis, University of Paderborn, 2005.Google Scholar
Ulam, S. M.. A Collection of Mathematical Problems (Interscience Tracts in Pure and Applied Mathematics, 8) . Interscience Publishers, New York–London, 1960.Google Scholar
Van Sebille, E., England, M. H. and Froyland, G.. Origin, dynamics and evolution of ocean garbage patches from observed surface drifters. Environmental Res. Lett. 7(4) (2012), 044040.Google Scholar
Williams, M. O., Rowley, C. W. and Kevrekidis, I. G.. A kernel-based method for data-driven Koopman spectral analysis. J. Comput. Dyn. 2(2) (2015), 247–265.Google Scholar
Young, L.-S.. Stochastic stability of hyperbolic attractors. Ergod. Th. & Dynam. Sys. 6(2) (1986), 311–319.Google Scholar