We consider the linear and quadratic higher-order terms associated with the response of the statistical properties of a dynamical system to suitable small perturbations. These terms are related to the first and second derivative of the stationary measure with respect to the changes in the system itself, expressing how the statistical properties of the system vary under the perturbation. We show a general framework in which one can obtain rigorous convergence and formulas for these two terms. The framework is flexible enough to be applied both to deterministic and random systems. We give examples of such an application computing linear and quadratic response for Arnold maps with additive noise and deterministic expanding maps.
REFERENCES
1
F.
Anton
, D.
Dragicevic
, and G.
Froyland
, “Optimal linear responses for Markov chains and stochastically perturbed dynamical systems
,” J. Stat. Phys.
170
(6
), 1051
–1087
(2018
). 2
W.
Bahsoun
, S.
Galatolo
, I.
Nisoli
, and X.
Niu
, “A rigorous computational approach to linear response
,” Nonlinearity
31
(3
), 1073
–1109
(2018
). 3
W.
Bahsoun
, M.
Ruziboev
, and B.
Saussol
, “Linear response for random dynamical systems,” arXiv:1710.03706.4
M.
Baiesi
and C.
Maes
, “An update on the nonequilibrium linear response
,” New J. Phys.
15
, 013004
(2013
). 5
V.
Baladi
, “Linear response, or else,” ICM Seoul 2014 talk, arXiv:1408.2937.6
M.
Chekroun
, E.
Simonnet
, and M.
Ghil
, “Stochastic climate dynamics: Random attractors and time-dependent invariant measures
,” Physica D
240
, 1685
–1700
(2011
). 7
S.
Galatolo
, “Statistical properties of dynamics. Introduction to the functional analytic approach,” arXiv:1510.02615.8
S.
Galatolo
, “Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products
,” J. Éc. Pol. Math.
5
, 377
–405
(2018
).9
S.
Galatolo
and M.
Pollicott
, “Controlling the statistical properties of expanding maps
,” Nonlinearity
30
, 2737
–2751
(2017
). 10
S.
Galatolo
and P.
Giulietti
, “A linear response for dynamical systems with additive noise
,” Nonlinearity
32
(6
), 2269
–2301
(2019
). 11
S.
Gouëzel
and C.
Liverani
, “Banach spaces adapted to Anosov systems
,” Ergodic Theory Dyn. Syst.
26
(1
), 189
–217
(2006
).12
M.
Hairer
and A. J.
Majda
, “A simple framework to justify linear response theory
,” Nonlinearity
23
, 909
–922
(2010
). 13
P.
Hänggi
and H.
Thomas
, “Stochastic processes: Time evolution, symmetries and linear response
,” Phys. Rep. V
88
(4
), 207
–319
(1982
). 14
H.
Hennion
, “Sur un théorème spectral et son application aux noyaux lipschitziens
,” Proc. Amer. Math. Soc.
118
(2
), 627
–634
(1993
).15
T.
Kato
, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics (Springer-Verlag, Berlin, 1995), xxii+619 pp., ISBN: 3-540-58661-X 47A55 (46-00 47-00).16
G.
Keller
and C.
Liverani
, “Stability of the spectrum for transfer operators
,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
28
(1
), 141
–152
(1999
).17
B. R.
Kloeckner
, “The linear request problem
,” Proc. Amer. Math. Soc.
146
, 2953
–2962
(2018
). 18
A.
Lasota
and M. C.
Mackey
, Probabilistic Properties of Deterministic Systems
(Cambridge University Press
, 1986
).19
R. S.
MacKay
, “Management of complex dynamical systems
,” Nonlinearity V
31
(2
), R52
–R66
(2018
). 20
L.
Marangio
, J.
Sedro
, S.
Galatolo
, A.
Di Garbo
, and M.
Ghil
, “Arnold maps with noise: Differentiability and non-monotonicity of the rotation number,” J. Stat. Phys.
(published online); arXiv:1904.11744.21
C.
Liverani
, “Invariant measures and their properties. A functional analytic point of view
,” Dyn. Syst. Part II
2003
, 185
–237
. Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa.22
C.
Liverani
, B.
Saussol
, and S.
Vaienti
, “A probabilistic approach to intermittency
,” Ergodic Theory Dyn. Syst.
19
(3
), 671
–685
(1999
). 23
V.
Lucarini
, “Stochastic perturbations to dynamical systems: A response theory approach
,” J. Stat. Phys.
146
(4
), 774
–786
(2012
). 24
V.
Lucarini
, R.
Blender
, C.
Herbert
, S.
Pascale
, and J.
Wouters
, “Mathematical and physical ideas for climate science
,” Rev. Geophys.
52
, 809
–859
, (2014
). 25
V.
Lucarini
, “Revising and extending the linear response theory for statistical mechanical systems: Evaluating observables as predictors and predictands
,” J. Stat. Phys.
173
, 1698
–1721
(2018
). 26
V
Lucarini
and J.
Wouters
, “Response formulae for n-point correlations in statistical mechanical systems and application to a problem of coarse graining
,” J. Phys. A Math. Theor.
50
, 355003
(2017
). 27
M.
Pollicott
and P.
Vytnova
, “Linear response and periodic points
,” Nonlinearity
29
(10
), 3047
–3066
(2016
). 28
D.
Ruelle
, “Nonequilibrium statistical mechanics near equilibrium: Computing higher-order terms
,” Nonlinearity
11
(1
), 5
(1998
). 29
D.
Ruelle
, “Differentiation of SRB states
,” Commun. Math. Phys.
187
, 227
–241
(1997
). 30
J.
Sedro
, “A regularity result for fixed points, with applications to linear response
,” Nonlinearity
31
(4
), 1417
–1441
(2018
). 31
J.
Sedro
, “On regularity loss in dynamical systems
,” Ph.D. thesis (Université Paris-Saclay, 2018).32
J.
Sedro
, see https://arxiv.org/abs/1711.05647 for information about “Regularity of the spectrum for expanding maps.”33
M.
Viana
, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics 145 (Cambridge University Press, 2014).34
C.
Wormell
and G.
Gottwald
, “On the validity of linear response theory in high-dimensional deterministic dynamical systems
,” J. Stat. Phys.
172
(6
), 1479
–1498
(2018
). 35
H.
Zmarrou
and A. J.
Homburg
, “Bifurcations of stationary measures of random diffeomorphisms
,” Ergodic Theory Dyn. Syst.
27
(5
), 1651
–1692
(2007
). 36
Recall that is a Schwartz function if it is a function that satisfies, for any , The set of Schwartz function is traditionally denoted by .
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