Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number

  • L. Marangio
  • J. Sedro
  • S. GalatoloEmail author
  • A. Di Garbo
  • M. Ghil


Arnold’s standard circle maps are widely used to study the quasi-periodic route to chaos and other phenomena associated with nonlinear dynamics in the presence of two rationally unrelated periodicities. In particular, the El Niño–Southern Oscillation phenomenon is a crucial component of climate variability on interannual time scales and it is dominated by the seasonal cycle, on the one hand, and an intrinsic oscillatory instability with a period of a few years, on the other. The role of meteorological phenomena on much shorter time scales, such as westerly wind bursts, has also been recognized and modeled as additive noise. We consider herein Arnold maps with additive, uniformly distributed noise. When the map’s nonlinear term, scaled by the parameter \(\epsilon \), is sufficiently small, i.e. \(\epsilon < 1\), the map is known to be a diffeomorphism and the rotation number \(\rho _{\omega }\) is a differentiable function of the driving frequency \(\omega \). We concentrate on the rotation number’s behavior as the nonlinearity becomes large, and show rigorously that \(\rho _{\omega }\) is a differentiable function of \(\omega \), even for \(\epsilon \ge 1\), at every point at which the noise-perturbed map is mixing. We also provide a formula for the derivative of the rotation number. The reasoning relies on linear-response theory and a computer-aided proof. In the diffeomorphism case of \( \epsilon <1\), the rotation number \(\rho _{\omega }\) behaves monotonically with respect to \(\omega \). We show, using again a computer-aided proof, that this is not the case when \(\epsilon \ge 1\) and the map is not a diffeomorphism. This lack of monotonicity for large nonlinearity could be of interest in some applications. For instance, when the devil’s staircase \( \rho =\rho (\omega )\) loses its monotonicity, frequency locking to the same periodicity could occur for non-contiguous parameter values that might even lie relatively far apart from each other.


Linear response Random dynamical system ENSO Rotation number Arnold map 

Mathematics Subject Classification

37H99 37C30 86A10 65G30 



The rigorous computations presented in Sects. 4.2 and 5 were performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comté. JS was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 787304). The present paper is TiPES contribution #4; this project has received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 820970.


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Authors and Affiliations

  1. 1.Femto-ST Institute, Université de Bourgogne Franche-ComtéDijonFrance
  2. 2.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance
  3. 3.Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Sorbonne Université, Université de ParisParisFrance
  4. 4.Dipartimento di MatematicaUniversità di PisaPisaItaly
  5. 5.Consiglio Nazionale delle RicercheIstituto di Biofisica Unitá Operativa di PisaPisaItaly
  6. 6.Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), Ecole Normale Supérieure and PSL Research UniversityParisFrance
  7. 7.Department of Atmospheric and Oceanic SciencesUniversity of California at Los AngelesLos AngelesUSA

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