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How Does Noise Induce Order?

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Abstract

In this paper we present a general result with an easily checkable condition that ensures a transition from chaotic regime to regular regime in random dynamical systems with additive noise. We show how this result applies to a prototypical family of nonuniformly expanding one dimensional dynamical systems, showing the main mathematical phenomenon behind Noise-induced Order.

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Data availability

The script used to produce the data, the data, and a Jupyter notebook used for the plot in Fig. 1 can be found at https://github.com/orkolorko/UnimodalNIO.

Notes

  1. the value of \(\tilde{\alpha }\) is contained in [2.67834, 2.67835], therefore, our result does not apply to the case \(\alpha =2\), the quadratic family

  2. with respect to an adapted Riemann metric by conjugating the Perron–Frobenius operator by multiplication with a cocycle.

  3. obtained with Julia ValidatedNumerics package.

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Acknowledgements

The author thanks Y. Sato, M. Benedicks, M. Monge and S. Galatolo for introducing the problem, pointing in the right direction and giving many of the tools. The author thanks warmly E. Ghys and A. Blumenthal for reading the paper, posing questions and providing ideas. The author thanks the anonymous referees, whose questions and comments led to a rewrite of the article, which we hope is clearer. This paper is dedicated to W. Tucker in occasion of his 50th birthday. The author thanks the ICTP for the hospitality and was partially supported by CNPq, University of Uppsala and KAW Grant 2013.0315. UFRJ, CAPES (through the programs PROEX and the CAPES-STINT project “Contemporary topics in non uniformly hyperbolic dynamics”). The author is currently under “Afastamento do país para qualificação profissional, apresentação de trabalhos técnico-cientìficos e colaboração institucional do pessoal docente e técnico-administrativo” from UFRJ and is currently a Specially Appointed Associate Professor at Hokkaido University. The author would like to thank Prof. Hiroki Sumi and Kyoto university for their hospitality during the final revision of this article.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos, 149-Edifício do Centro de Tecnologia, Bloco C (Térreo), Cidade Universitária, Rio de Janeiro, Brazil

    Isaia Nisoli

  2. Department of Mathematics, Hokkaido University, N10 W8, Kita-ku, Sapporo, 001-0010, Japan

    Isaia Nisoli

  3. RIES, Hokkaido University, N20 W10, Kita-ku, Sapporo, 001-0020, Japan

    Isaia Nisoli

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Communicated by Peter Balint.

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Nisoli, I. How Does Noise Induce Order?. J Stat Phys 190, 22 (2023). https://doi.org/10.1007/s10955-022-03041-y

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