Communications in Mathematical Sciences

Linear response theory for statistical ensembles in complex systems with time-periodic forcing

Andrew Majda and Xiaoming Wang

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New linear response formulas for unperturbed chaotic (stochastic) complex dynamical systems with time periodic coefficients are developed here. Such time periodic systems arise naturally in climate change studies due to the seasonal cycle. These response formulas are developed through the mathematical interplay between statistical solutions for the time-periodic dynamical systems and the related skew-product system. This interplay is utilized to develop new systematic quasi-Gaussian and adjoint algorithms for calculating the climate response in such time-periodic systems. These new formulas are found in section 4. New linear response formulas are also developed here for general time-dependent statistical ensembles arising in ensemble prediction including the effects of deterministic model errors, initial ensembles, and model noise perturbations simultaneously. An information theoretic perspective is developed in calculating those model perturbations which yield the largest information deficit for the unperturbed system both for climate response and finite ensemble predictions.

Article information

Commun. Math. Sci., Volume 8, Number 1 (2010), 145-172.

First available in Project Euclid: 23 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology [See mainly 76-XX, especially 76D05, 76F20, 86A05, 86A10] 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10] 86A99: Miscellaneous topics 60H10: Stochastic ordinary differential equations [See also 34F05] 34C28: Complex behavior, chaotic systems [See also 37Dxx] 94A15: Information theory, general [See also 62B10, 81P94]

Linear response theory fluctuation-dissipation theory time periodic coefficients quasi-Gaussian and Gaussian approximation climate response information content relative entropy


Majda, Andrew; Wang, Xiaoming. Linear response theory for statistical ensembles in complex systems with time-periodic forcing. Commun. Math. Sci. 8 (2010), no. 1, 145--172.

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