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On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series. (English) Zbl 1364.37164
Summary: Using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems. A goodness-of-fit statistics is developed for a linear statistical model of the observations, based on results for central limit theorems for deterministic dynamical systems, and used to detect linear response breakdown. We apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series, the magnitude of the perturbation and on the choice of the observable. We find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed. Even for simple systems such as the logistic map, one needs of the order of \(10^6\) observations to reliably detect the breakdown with a confidence level of 95%; if less observations are available one may be falsely led to conclude that linear response theory is valid. The amount of data required is larger the smaller the applied perturbation. For judiciously chosen observables the necessary amount of data can be drastically reduced, but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems. Furthermore we explore the use of the fluctuation-dissipation theorem (FDT) in cases with limited data length or coarse-graining of observations. The FDT, if applied naively to a system without linear response, is shown to be very sensitive to the details of the sampling method, resulting in erroneous predictions of the response.

37M10 Time series analysis of dynamical systems
37E05 Dynamical systems involving maps of the interval
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[1] Kubo, R., The fluctuation-dissipation theorem, Rep. Progr. Phys., 29, 1, 255, (1966) · Zbl 0163.23102
[2] Balescu, R., Equilibrium and non-equilibrium statistical mechanics, (1975), John Wiley & Sons New York · Zbl 0984.82500
[3] Zwanzig, R., Nonequilibrium statistical mechanics, (2001), Oxford University Press Oxford · Zbl 1267.82001
[4] Marconi, U. M.B.; Puglisi, A.; Rondoni, L.; Vulpiani, A., Fluctuation-dissipation: response theory in statistical physics, Phys. Rep., 461, 4-6, 111-195, (2008)
[5] Ruelle, D., Differentiation of SRB states, Comm. Math. Phys., 187, 1, 227-241, (1997) · Zbl 0895.58045
[6] Ruelle, D., General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A, 245, 3-4, 220-224, (1998) · Zbl 0940.82035
[7] Ruelle, D., A review of linear response theory for general differentiable dynamical systems, Nonlinearity, 22, 4, 855-870, (2009) · Zbl 1158.37305
[8] Ruelle, D., Structure and f-dependence of the a.c.i.m. for a unimodal map f of misiurewicz type, Comm. Math. Phys., 287, 3, 1039-1070, (2009) · Zbl 1202.37008
[9] Ershov, S. V., Is a perturbation theory for dynamical chaos possible?, Phys. Lett. A, 177, 3, 180-185, (1993)
[10] Baladi, V.; Smania, D., Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21, 4, 677-711, (2008) · Zbl 1140.37008
[11] Baladi, V.; Smania, D., Alternative proofs of linear response for piecewise expanding unimodal maps, Ergodic Theory Dynam. Systems, 30, 01, 1-20, (2010) · Zbl 1230.37030
[12] V. Baladi, Linear response, or else, in: ICM Seoul 2014, Proceedings, Vol. III, 2014, pp. 525-545. · Zbl 1373.37074
[13] Baladi, V.; Benedicks, M.; Schnellmann, D., Whitney-Hölder continuity of the SRB measure for transversal families of smooth unimodal maps, Invent. Math., 201, 3, 773-844, (2015) · Zbl 1359.37049
[14] A. De Lima, D. Smania, Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps, 2015. arXiv:1503.01423 [math.DS]. · Zbl 1387.37025
[15] Reick, C. H., Linear response of the Lorenz system, Phys. Rev. E, 66, (2002)
[16] Cessac, B.; Sepulchre, J.-A., Linear response, susceptibility and resonances in chaotic toy models, Physica D, 225, 1, 13-28, (2007) · Zbl 1114.37023
[17] Lucarini, V.; Sarno, S., A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlinear Processes Geophys., 18, 1, 7-28, (2011)
[18] McWilliams, J. C., Irreducible imprecision in atmospheric and oceanic simulations, Proc. Natl. Acad. Sci., 104, 21, 8709-8713, (2007)
[19] Dolgopyat, D., On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155, 2, 389-449, (2004) · Zbl 1059.37021
[20] A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, 2015. arXiv:1508.06571 [math.DS]. · Zbl 1360.37100
[21] Baladi, V.; Todd, M., Linear response for intermittent maps, Comm. Math. Phys., 1-18, (2016)
[22] Leith, C. E., Climate response and fluctuation dissipation, J. Atmos. Sci., 32, 10, 2022-2026, (1975)
[23] Majda, A. J.; Abramov, R.; Gershgorin, B., High skill in low-frequency climate response through fluctuation dissipation theorems despite structural instability, Proc. Natl. Acad. Sci., 107, 2, 581-586, (2010) · Zbl 1205.86025
[24] Abramov, R. V.; Majda, A. J., Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems, Nonlinearity, 20, 12, 2793, (2007) · Zbl 1134.37365
[25] Abramov, R. V.; Majda, A. J., New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems, J. Nonlinear Sci., 18, 3, 303-341, (2008) · Zbl 1151.82364
[26] Cooper, F. C.; Haynes, P. H., Climate sensitivity via a nonparametric fluctuation-dissipation theorem, J. Atmos. Sci., 68, 5, 937-953, (2011)
[27] Cooper, F. C.; Esler, J. G.; Haynes, P. H., Estimation of the local response to a forcing in a high dimensional system using the fluctuation-dissipation theorem, Nonlinear Processes Geophys., 20, 2, 239-248, (2013)
[28] Bell, T. L., Climate sensitivity from fluctuation dissipation: some simple model tests, J. Atmos. Sci., 37, 8, 1700-1707, (1980)
[29] Gritsun, A.; Dymnikov, V., Barotropic atmosphere response to small external actions: theory and numerical experiments, Izv. Akad. Nauk Fiz. Atmos. Okeana Biol., 35, 565-581, (1999)
[30] Abramov, R. V.; Majda, A. J., A new algorithm for low-frequency climate response, J. Atmos. Sci., 66, 2, 286-309, (2009)
[31] Dymnikov, V. P.; Gritsoun, A. S., Climate model attractors: chaos, quasi-regularity and sensitivity to small perturbations of external forcing, Nonlinear Processes Geophys., 8, 4-5, 201-209, (2001)
[32] North, G. R.; Bell, R. E.; Hardin, J. W., Fluctuation dissipation in a general circulation model, Clim. Dynam., 8, 6, 259-264, (1993)
[33] Cionni, I.; Visconti, G.; Sassi, F., Fluctuation dissipation theorem in a general circulation model, Geophys. Res. Lett., 31, 9, L09206, (2004)
[34] Gritsun, A.; Branstator, G.; Dymnikov, V., Construction of the linear response operator of an atmospheric general circulation model to small external forcing, Russian J. Numer. Anal. Math. Modelling, 17, 399-416, (2002) · Zbl 1007.86003
[35] Gritsun, A.; Branstator, G., Climate response using a three-dimensional operator based on the fluctuation-dissipation theorem, J. Atmos. Sci., 64, 7, 2558-2575, (2007)
[36] Gritsun, A.; Branstator, G.; Majda, A., Climate response of linear and quadratic functionals using the fluctuation-dissipation theorem, J. Atmos. Sci., 65, 9, 2824-2829, (2008)
[37] Ring, M. J.; Plumb, R. A., The response of a simplified GCM to axisymmetric forcings: applicability of the fluctuation-dissipation theorem, J. Atmos. Sci., 65, 12, 3880-3898, (2008)
[38] Gritsun, A. S., Construction of response operators to small external forcings for atmospheric general circulation models with time periodic right-hand sides, Izv. Atmos. Ocean. Phys., 46, 6, 748-756, (2010)
[39] Langen, P. L.; Alexeev, V. A., Estimating \(2 \times C O_2\) warming in an aquaplanet GCM using the fluctuation-dissipation theorem, Geophys. Res. Lett., 32, 23, L23708, (2005)
[40] Kirk-Davidoff, D. B., On the diagnosis of climate sensitivity using observations of fluctuations, Atmos. Chem. Phys., 9, 3, 813-822, (2009)
[41] Fuchs, D.; Sherwood, S.; Hernandez, D., An exploration of multivariate fluctuation dissipation operators and their response to sea surface temperature perturbations, J. Atmos. Sci., 72, 1, 472-486, (2014)
[42] Ragone, F.; Lucarini, V.; Lunkeit, F., A new framework for climate sensitivity and prediction: a modelling perspective, Clim. Dynam., 1-13, (2015)
[43] Melbourne, I.; Stuart, A., A note on diffusion limits of chaotic skew-product flows, Nonlinearity, 24, 1361-1367, (2011) · Zbl 1220.37009
[44] Gottwald, G. A.; Melbourne, I., Homogenization for deterministic maps and multiplicative noise, Proc. R. Soc. A, 469, 2156, (2013) · Zbl 1371.34084
[45] D. Kelly, I. Melbourne, Deterministic homogenization for fast-slow systems with chaotic noises, 2014. arXiv:1409.5748 [math.PR]. · Zbl 1373.60101
[46] Hänggi, P., Stochastic processes 2: response theory and fluctuation theorems, Helv. Phys. Acta, 51, 2, 202-219, (1978)
[47] Hairer, M.; Majda, A. J., A simple framework to justify linear response theory, Nonlinearity, 23, 4, 909, (2010) · Zbl 1186.82006
[48] Chekroun, M. D.; Neelin, J. D.; Kondrashov, D.; McWilliams, J. C.; Ghil, M., Rough parameter dependence in climate models and the role of Ruelle-pollicott resonances, Proc. Natl. Acad. Sci., 111, 5, 1684-1690, (2014)
[49] Cooper, F.; Haynes, P., Assessment of the fluctuation-dissipation theorem as an estimator of the tropospheric response to forcing, Q. J. R. Meteorol. Soc., (2013), submitted for publication
[50] Baladi, V., (Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, vol. 16, (2000), World Scientific Publishing Co., Inc. River Edge, NJ) · Zbl 1012.37015
[51] Ding, J.; Du, Q.; Li, T. Y., High order approximation of the Frobenius-Perron operator, Appl. Math. Comput., 53, 2-3, 151-171, (1993) · Zbl 0769.65025
[52] Boyd, J. P., Chebyshev and Fourier spectral methods, (2001), Courier Corporation Mineola, NY · Zbl 0987.65122
[53] Trefethen, L. N., Approximation theory and approximation practice, (2013), Siam Philadelphia, PA · Zbl 1264.41001
[54] I. Melbourne, Fast-slow skew product systems and convergence to stochastic differential equations, 2015. Lecture notes, available at: http://homepages.lboro.ac.uk/ mawb/Melbourne2_notes.pdf.
[55] Collet, P.; Eckmann, J.-P., Concepts and results in chaotic dynamics: A short course, (2007), Springer Science & Business Media Berlin
[56] Box, G. E.P.; Hunter, J. S.; Hunter, W. G., (Statistics for Experimenters: Design, Innovation, and Discovery, Wiley Series in Probability and Statistics, (2005), Wiley-Interscience Hoboken (N.J.)) · Zbl 1082.62063
[57] Lyubich, M., Almost every real quadratic map is either regular or stochastic, Ann. of Math. (2), 156, 1, 1-78, (2002) · Zbl 1160.37356
[58] Avila, A.; Moreira, C. G., Statistical properties of unimodal maps: the quadratic family, Ann. of Math. (2), 161, 2, 831-881, (2005) · Zbl 1078.37029
[59] Jakobson, M. V., Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., 81, 1, 39-88, (1981) · Zbl 0497.58017
[60] Benedicks, M.; Carleson, L., On iterations of \(1 - a x^2\) on \((- 1, 1)\), Ann. of Math. (2), 122, 1, 1-25, (1985) · Zbl 0597.58016
[61] Collet, P.; Eckmann, J.-P., Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3, 1, 13-46, (1983) · Zbl 0532.28014
[62] Keller, G.; Nowicki, T., Spectral theory, zeta functions and the distribution of periodic points for collet-eckmann maps, Comm. Math. Phys., 149, 1, 31-69, (1992) · Zbl 0763.58024
[63] Young, L.-S., What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108, 5-6, 733-754, (2002) · Zbl 1124.37307
[64] Gallavotti, G.; Cohen, E. G.D., Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett., 74, 2694-2697, (1995)
[65] Gallavotti, G.; Cohen, E., Dynamical ensembles in stationary states, J. Stat. Phys., 80, 5-6, 931-970, (1995) · Zbl 1081.82580
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