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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  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 \(\tau \in [0.7,0.8]\).


  1. 1.

    Antown, F., Dragičević, D., Froyland, G.: Optimal linear responses for Markov chains and stochastically perturbed dynamical systems. J. Stat. Phys. 170(6), 1051–1087 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  2. 2.

    Arnold, V.I.: Small denominators. I: Mappings of the circumference onto itself. AMS Transl. Ser. 2(46), 213–284 (1965)

    MATH  Google Scholar 

  3. 3.

    Arnold, V.I.: Cardiac arrhythmias and circle mappings. Chaos 1, 20–24 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  4. 4.

    Arnold, V.I.: Geometrical Methods in the Theory of Differential Equations. Springer, New York (1983)

    Google Scholar 

  5. 5.

    Bahsoun, W., Saussol, B.: Linear response in the intermittent family: differentiation in a weighted \(C^{0}\)-norm. Discret. Cont. Dyn. Syst. 36(12), 6657–6668 (2016)

    MATH  Google Scholar 

  6. 6.

    Bahsoun, W., Ruziboev, M., Saussol, B.: Linear response for random dynamical systems. arXiv:1710.03706

  7. 7.

    Bahsoun, W., Galatolo, S., Nisoli, I., Niu, X.: A rigorous computational approach to linear response. Nonlinearity 31(3), 1073–1109 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  8. 8.

    Bailey, M.P., Drks, G., Sheldonm, A.C.: Circle maps with gaps: understanding the dynamics of the two-process model for sleep–wake regulation. Eur. J. Appl. Math. (2018)

  9. 9.

    Bak, P.: The devil’s staircase. Phys. Today 39(12), 38–45 (1986)

    ADS  Google Scholar 

  10. 10.

    Bak, P., Bruinsma, R.: One-dimensional Ising model and the complete devil’s staircase. Phys. Rev. Lett. 49, 249–251 (1982)

    ADS  MathSciNet  Google Scholar 

  11. 11.

    Baladi, V.: On the susceptibility function of piecewise expanding interval maps. Commun. Math. Phys. 275(3), 839–859 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  12. 12.

    Baladi, V., Todd, M.: Linear response for intermittent maps. Commun. Math. Phys. 347(3), 857–874 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  13. 13.

    Baladi, V., Smania, D.: Linear response formula for piecewise expanding unimodal maps. Nonlinearity 21(4), 677–711 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  14. 14.

    Baladi, V.: Linear response, or else. In: Proceedings of the International Congress of Mathematicians—Seoul 2014, vol. III, pp. 525–545 (2014)

  15. 15.

    Baladi, V., Benedicks, M., Schnellmann, N.: Whitney–Holder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201, 773–844 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  16. 16.

    Baladi, V., Kuna, T., Lucarini, V.: Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables. Nonlinearity 30, 1204–1220 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  17. 17.

    Barnston, A.G., Tippett, M.K., L’Heureux, M., Li, S., DeWitt, D.G.: Skill of real-time seasonal ENSO model predictions during 2002–11: is our capability increasing? Bull. Am. Meteorol. Soc. 93(5), 631–651 (2012)

    ADS  Google Scholar 

  18. 18.

    Batista, A.M., Sandro, E., de Pinto, S., Viana, R.L., Lopes, S.R.: Mode locking in small-world networks of coupled circle maps. Physica A 322, 118–128 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  19. 19.

    Bose, C.J., Murray, R.: The exact rate of approximation in Ulam’s method. Discret. Contin. Dyn. Syst. 7, 219–235 (2001)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Chang, P., Wang, B., Li, T., Ji, L.: Interactions between the seasonal cycle and the Southern Oscillation: frequency entrainment and chaos in intermediate coupled ocean-atmosphere model. Geophys. Res. Lett. 21, 2817–2820 (1994)

    ADS  Google Scholar 

  21. 21.

    Chang, P., Ji, L., Li, T., Flügel, M.: Chaotic dynamics versus stochastic processes in El Niño-Southern Oscillation in coupled ocean-atmosphere models. Physica D 98, 301–320 (1996)

    ADS  MATH  Google Scholar 

  22. 22.

    Chekroun, M.D., Simonnet, E., Ghil, M.: Stochastic climate dynamics: random attractors and time-dependent invariant measures. Physica D 240(21), 1685–1700 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  23. 23.

    Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155(2), 389–449 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  24. 24.

    Ding, J., Wang, Z.: Parallel computation of invariant measures. Ann. Oper. Res. 103, 283–290 (2001)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Eisenman, I., Yu, L., Tziperman, E.: Westerly wind bursts: ENSO tail rather than the dog? J. Climate 18, 5224–5238 (2005)

    ADS  Google Scholar 

  26. 26.

    Feigenbaum, M.J., Kadanoff, L.P., Shenker, S.J.: Quasiperiodicity in dissipative systems: a renormalization group analysis. Physica D 5, 370–386 (1982)

    ADS  MathSciNet  Google Scholar 

  27. 27.

    Galatolo, S., Giulietti, P.: Linear response for dynamical systems with additive noise. Nonlinearity arXiv:1711.04319

  28. 28.

    Galatolo, S.: Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products. J. Éc. Polytech. Math. 5, 377–405 (2018)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Galatolo, S., Monge, M., Nisoli, I.: Existence of noise induced order, a computer aided proof. arXiv:1702.07024

  30. 30.

    Galatolo, S., Pollicott, M.: Controlling the statistical properties of expanding maps. Nonlinearity 30, 2737–2751 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  31. 31.

    Ghil, M.: Hilbert problems for the geosciences in the 21st century. Nonlinear Process. Geophys. 8, 211–222 (2001)

    ADS  Google Scholar 

  32. 32.

    Ghil, M., Allen, M.R., Dettinger, M.D., Ide, K., Kondrashov, D., Mann, M.E., Robertson, A.W., Saunders, A., Tian, Y., Varadi, F., Yiou, P.: Advanced spectral methods for climatic time series. Rev. Geophys. (2002)

  33. 33.

    Ghil, M., Chekroun, M.D., Simonnet, E.: Climate dynamics and fluid mechanics: natural variability and related uncertainties. Physica D 237, 2111–2126 (2008).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Ghil, M., Lucarini, V.: The physics of climate variability and climate change. Rev. Mod. Phys. arXiv:1910.00583

  35. 35.

    Ghil, M., Jiang, N.: Recent forecast skill for the El Niño/Southern Oscillation. Geophys. Res. Lett. 25, 171–174 (1998)

    ADS  Google Scholar 

  36. 36.

    Ghil, M., Robertson, A.W.: Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy. In: Randall, D. (ed.) General Circulation Model Development: Past, Present and Future, pp. 285–325. Academic Press, San Diego (2000)

    Google Scholar 

  37. 37.

    Ghil, M., Zaliapin, I., Thompson, S.: A delay differential model of ENSO variability: parametric instability and the distribution of extremes. Nonlinear Process. Geophys. 15, 417–433 (2008)

    ADS  Google Scholar 

  38. 38.

    Ghil, M.: The wind-driven ocean circulation: applying dynamical systems theory to a climate problem. Discret. Cont. Dyn. Syst. A 37(1), 189–228 (2017)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Glass, L.: Cardiac arrhythmias and circle maps—a classical problem. Chaos 1, 13–19 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  40. 40.

    Hairer, M., Majda, A.J.: A simple framework to justify linear response theory. Nonlinearity 23, 909–922 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  41. 41.

    Held, I.M.: The gap between simulation and understanding in climate modeling. Bull. Am. Meteorol. Soc. 86, 1609–1614 (2005)

    ADS  Google Scholar 

  42. 42.

    Keener, J.P., Glass, L.: Global bifurcations of a periodically forced nonlinear oscillator. J. Math. Biol. 21, 175–190 (1984)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Kondrashov, D., Chekroun, M.D., Yuan, X., Ghil, M.: Data-adaptive harmonic decomposition and stochastic modeling of Arctic sea ice. In: A. Tsonis (ed.) Nonlinear Advances in Geosciences, pp. 179–206. Springer, New York. (2018)

  44. 44.

    Jiang, S., Jin, F.-F., Ghil, M.: Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model. J. Phys. Oceanogr. 25, 764–786 (1995)

    ADS  Google Scholar 

  45. 45.

    Jin, F.-F., Neelin, J.D., Ghil, M.: El Niño on the Devil’s staircase: annual subharmonic steps to chaos. Science 264, 70–72 (1994)

    ADS  Google Scholar 

  46. 46.

    Jin, F.-F., Neelin, J.D., Ghil, M.: El Niñ o/Southern Oscillation and the annual cycle: subharmonic frequency locking and aperiodicity. Physica D 98, 442–465 (1996)

    ADS  MATH  Google Scholar 

  47. 47.

    Kondrashov, D., Chekroun, M.D., Ghil, M.: Data-driven non-Markovian closure models. Physica D 297, 33–55 (2015).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Korepanov, A.: Linear response for intermittent maps with summable and nonsummable decay of correlations. Nonlinearity 29(6), 1735–1754 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  49. 49.

    Kravtsov, S., Kondrashov, D., Ghil, M.: Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences. In: Palmer, T., Williams, P. (eds.) Stochastic Physics and Climate Modelling, pp. 35–72. Cambridge Univ. Press, Cambridge (2009)

    Google Scholar 

  50. 50.

    Lasota, A., Mackey, M.C.: Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge (1986)

    MATH  Google Scholar 

  51. 51.

    Liverani, C.: Invariant Measures and Their Properties: A Functional Analytic Point of View. Dynamical Systems. Part II, pp. 185–237. Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa (2003)

  52. 52.

    Lucarini, V.: Stochastic perturbations to dynamical systems: a response theory approach. J. Stat. Phys. 146(4), 774–786 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  53. 53.

    Latif, M., Barnett, T.P., Flügel, M., Graham, N.E., Xu, J.-S., Zebiak, S.E.: A review of ENSO prediction studies. Clim. Dyn. 9, 167–179 (1994)

    Google Scholar 

  54. 54.

    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    ADS  MathSciNet  MATH  Google Scholar 

  55. 55.

    Lorenz, E.N.: The mechanics of vacillation. J. Atmos. Sci. 20, 448–464 (1963)

    ADS  Google Scholar 

  56. 56.

    MacKay, R.S.: Management of complex dynamical systems. Nonlinearity 31(2), 52–64 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  57. 57.

    Murray, R.: Optimal partition choice for invariant measure approximation for one-dimensional maps. Nonlinearity 17, 1623–1644 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  58. 58.

    Neelin, J.D., Battisti, D.S., Hirst, A.C., et al.: ENSO theory. J. Geophys. Res. Oceans 103(C7), 14261–14290 (1998)

    ADS  Google Scholar 

  59. 59.

    Palmer, T., Williams, P. (eds.): Stochastic Physics and Climate Modelling. Cambridge University Press, Cambridge (2009)

  60. 60.

    Penland, C.: A stochastic model of Indo-Pacific sea-surface temperature anomalies. Physica D 98, 534–558 (1996)

    ADS  MATH  Google Scholar 

  61. 61.

    Pierini, S., Ghil, M., Chekroun, M.D.: Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case. J. Climate 29, 4185–4202 (2016)

    ADS  Google Scholar 

  62. 62.

    Philander, S.G.H.: El Niño, La Niña, and the Southern Oscillation. Academic Press, San Diego (1990)

    Google Scholar 

  63. 63.

    Pollicott, M., Vytnova, P.: Linear response and periodic points. Nonlinearity 29(10), 3047–3066 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  64. 64.

    Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  65. 65.

    Saunders, A., Ghil, M.: A Boolean delay equation model of ENSO variability. Physica D 160, 54–78 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  66. 66.

    Schneider, S.H., Dickinson, R.E.: Climate modeling. Rev. Geophys. Space Phys. 12, 447–493 (1974)

    ADS  Google Scholar 

  67. 67.

    Sedro, J.: A regularity result for fixed points, with applications to linear response. arXiv:1705.04078

  68. 68.

    Timmermann, A., Jin, F-F.: A nonlinear mechanism for decadal El Niño amplitude changes. Geophys. Res. Lett. (2002)

  69. 69.

    Tziperman, E., Stone, L., Cane, M., Jarosh, H.: El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science 264, 72–74 (1994)

    ADS  Google Scholar 

  70. 70.

    Tziperman, E., Cane, M.A., Zebiak, S.E.: Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos. J. Atmos. Sci. 50, 293–306 (1995)

    ADS  Google Scholar 

  71. 71.

    Tucker, W.: Validated Numerics A Short Introduction to Rigorous Computations. Princeton Univ. Press, Princeton (2011)

    MATH  Google Scholar 

  72. 72.

    Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publisher, New York (1960)

    MATH  Google Scholar 

  73. 73.

    Verbickas, S.: Westerly Wind Bursts in the Tropical Pacific. Weather 53, 282–284 (1998)

    ADS  Google Scholar 

  74. 74.

    Viana, M.: Lectures on Lyapunov Exponents. Cambridge Studies in Advanced Mathematics, vol. 145. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  75. 75.

    Wiggins, S.: Introduction to Applied Dynamical Systems and Chaos. Springer, New York (2003) (see theorem 21.6.16)

  76. 76.

    Zaliapin, I., Ghil, M.: A delay differential model of ENSO variability. Part 2: phase locking, multiple solutions and dynamics of extrema. Nonlinear Process. Geophys. 17, 123–135 (2010)

    ADS  Google Scholar 

  77. 77.

    Zhang, Z.: On the smooth dependence of SRB measures for partially hyperbolic systems. arXiv:1701.05253

  78. 78.

    Zmarrou, H., Homburg, A.J.: Bifurcations of stationary measures of random diffeomorphisms. Ergodic Theory Dynam. Systems 27(5), 1651–1692 (2007)

    MathSciNet  MATH  Google Scholar 

  79. 79.

    Herman, M.R.: Mesure de Lebesgue et nombre de rotation. In: Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976). Lecture Notes in Math., vol. 597, pp. 271–293. Springer, Berlin (1977)

  80. 80.

    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  81. 81.

    Matsumoto, S.: Derivatives of the rotation number of one parameter families of circle diffeomorphisms. Kodai Mat. J. 35, 115–125 (2012)

    MathSciNet  MATH  Google Scholar 

  82. 82.

    Luque, A., Villanueva, J.: Computation of the derivatives of the rotation number for parametric families of circle diffeomorphisms. Physica D 237, 2599–2615 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to S. Galatolo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Valerio Lucarini.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


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

Mathematics Subject Classification

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