### Abstract

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 ef-fects 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.

Original language | English (US) |
---|---|

Pages (from-to) | 145-172 |

Number of pages | 28 |

Journal | Communications in Mathematical Sciences |

Volume | 8 |

Issue number | 1 |

State | Published - 2010 |

### Fingerprint

### Keywords

- Climate response
- Fluctuation-dissipation theory
- Information content
- Linear response theory
- Quasi-Gaussian and Gaussian approximation
- Relative entropy
- Time periodic coefficients

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*8*(1), 145-172.

**Linear response theory for statistical ensembles in complex systems with time-periodic forcing.** / Majda, Andrew J.; Wang, Xiaoming.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 8, no. 1, pp. 145-172.

}

TY - JOUR

T1 - Linear response theory for statistical ensembles in complex systems with time-periodic forcing

AU - Majda, Andrew J.

AU - Wang, Xiaoming

PY - 2010

Y1 - 2010

N2 - 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 ef-fects 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.

AB - 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 ef-fects 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.

KW - Climate response

KW - Fluctuation-dissipation theory

KW - Information content

KW - Linear response theory

KW - Quasi-Gaussian and Gaussian approximation

KW - Relative entropy

KW - Time periodic coefficients

UR - http://www.scopus.com/inward/record.url?scp=77949669181&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77949669181&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77949669181

VL - 8

SP - 145

EP - 172

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 1

ER -