Hostname: page-component-76d6cb85b7-rxvq6 Total loading time: 0 Render date: 2026-07-12T06:45:02.160Z Has data issue: false hasContentIssue false
  • English
  • Français

On–off intermittency and chaotic walks

Part of: Random dynamical systems Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  30 January 2019

ALE JAN HOMBURG Open the ORCID record for ALE JAN HOMBURG [Opens in a new window]  and
VAHATRA RABODONANDRIANANDRAINA
ALE JAN HOMBURG
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Science park 107, 1098 XG Amsterdam, The Netherlands email v.f.rabodonandrianandraina@uva.nl Department of Mathematics, VU University Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands email a.j.homburg@uva.nl
VAHATRA RABODONANDRIANANDRAINA
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Science park 107, 1098 XG Amsterdam, The Netherlands email v.f.rabodonandrianandraina@uva.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

Keywords

intermittencylow-dimensional dynamicsrandom dynamicsrandom walks

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37H20: Bifurcation theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1805 - 1842
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Aaronson, J. and Keane, M.. The visits to zero of some deterministic random walks. Proc. Lond. Math. Soc. 44 (1982), 535–553.CrossRefGoogle Scholar
Ashwin, P., Aston, P. and Nicol, M.. On the unfolding of a blowout bifurcation. Phys. D 111 (1998), 81–95.CrossRefGoogle Scholar
Athreya, K. B. and Dai, J.. Random logistic maps I. J. Theoret. Probab. 13 (2000), 595–608.CrossRefGoogle Scholar
Athreya, K. B. and Schuh, H. J.. Random logistic maps II. The critical case. J. Theoret. Probab. 16 (2003), 813–830.CrossRefGoogle Scholar
Atkinson, G.. Recurrence of co-cycles and random walks. J. Lond. Math. Soc. 13 (1976), 486–488.CrossRefGoogle Scholar
Avila, A., Dolgopyat, D., Duryev, E. and Sarig, O.. The visits to zero of a random walk driven by an irrational rotation. Israel J. Math. 207 (2015), 653–717.CrossRefGoogle Scholar
Bonifant, A. and Milnor, J.. Schwarzian derivatives and cylinder maps. Fields Inst. Commun. 53 (2008), 1–21.Google Scholar
Boyarsky, A. and Góra, P.. Laws of chaos. Invariant Measures and Dynamical Systems in One Dimension. Birkhäuser, Boston, MA, 1997.Google Scholar
Field, M., Melbourne, I. and Török, A.. Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets. Ergod. Th. & Dynam. Sys. 25 (2005), 517–551.CrossRefGoogle Scholar
Fuh, C.-D. and Zhang, C.-H.. Poisson equation, moment inequalities and quick convergence for Markov random walks. Stoch. Process. Appl. 87 (2000), 53–67.CrossRefGoogle Scholar
Gharaei, M. and Homburg, A. J.. Random interval diffeomorphisms. Discrete Contin. Dyn. Syst. Ser. S 10 (2017), 241–272.Google Scholar
Gouëzel, S.. Statistical properties of a skew product with a curve of neutral points. Ergod. Th. & Dynam. Sys. 27 (2007), 123–151.CrossRefGoogle Scholar
Guivarch, Y.. Propriétés ergodiques, en mesure infinie, de certains systemes dynamiques fibrés. Ergod. Th. & Dynam. Sys. 9 (1989), 433–453.CrossRefGoogle Scholar
Heagy, J. F., Platt, N. and Hammel, S. M.. Characterization of on–off intermittency. Phys. Rev. E 49 (1994), 1140–1150.CrossRefGoogle ScholarPubMed
Krámli, A. and Szász, D.. Random walks with internal degrees of freedom. II: First-hitting probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 68 (1984), 53–64.CrossRefGoogle Scholar
Levin, D. A., Peres, Y. and Wilmer, E. L.. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, 2009.Google Scholar
Meyn, S. P. and Tweedie, R. L.. Markov Chains and Stochastic Stability. Springer, London, 1993.CrossRefGoogle Scholar
Moreira, C. G. and Smania, D.. Metric stability for random walks (with applications in renormalization theory). Frontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday. Princeton University Press, Princeton, NJ, 2014.Google Scholar
Niţică, V.. Examples of topologically transitive skew products. Discrete Contin. Dyn. Syst. 6 (2000), 351–360.CrossRefGoogle Scholar
Niţică, V. and Pollicott, M.. Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269.CrossRefGoogle Scholar
Ott, E. and Sommerer, J. C.. Blowout bifurcations: the occurrence of riddled basins and on–off intermittency. Phys. Lett. A 188 (1994), 39–47.CrossRefGoogle Scholar
Platt, N., Spiegel, E. A. and Tresser, C.. On–off intermittency: A mechanism for bursting. Phys. Rev. Lett. 70 (1993), 279–282.CrossRefGoogle ScholarPubMed
Schmidt, K.. Cocycles of Ergodic Transformation Groups (Lecture Notes in Mathematics, 1). MacMillan, India, 1977.Google Scholar
Shiryayev, A. N.. Probability. Springer, New York, 1984.CrossRefGoogle Scholar