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


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.

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  1. 1.

    The code used in this paper is at .

  2. 2.

    Notation If AB are two normed vector spaces and \(T:A\rightarrow B\) we write \(\Vert T\Vert _{A\rightarrow B}:=\sup _{f\in A,\Vert f\Vert _{A}\le 1}\Vert Tf\Vert _{B}\)

  3. 3.

    A Borel map \(T:X\rightarrow X\) is a said to be nonsingular with respect to the Lebesgue measure m if for any measurable N\(m(T^{-1}(N))=0\iff m(N)=0 \).

  4. 4.

    The algorithm and the code used in this work (see Note 1) is almost identical to the one used in [29]. The only important difference is the fact that in our code the convolution on \({\mathbb {S}}^{1}\) is implemented, while in the original work of [29] a reflecting boundaries convolution is considered.

  5. 5.

    The zip file at contains the results of more than 300 computer-aided estimates, including the ones listed in Table 2; these numerical data extend the result of proposition 27 for \(\tau \in [0.7,0.8]\).


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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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Marangio, L., Sedro, J., Galatolo, S. et al. Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number. J Stat Phys 179, 1594–1624 (2020).

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  • Linear response
  • Random dynamical system
  • ENSO
  • Rotation number
  • Arnold map

Mathematics Subject Classification

  • 37H99
  • 37C30
  • 86A10
  • 65G30