Central Limit Theorem for the Modulus of Continuity of Averages of Observables on Transversal Families of Piecewise Expanding Unimodal Maps

Lima, Adriano Fonseca; Smania, Daniel

doi:10.1017/s1474748016000177

Cited by 15 publications

(21 citation statements)

References 20 publications

(56 reference statements)

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“…We then consider the case when the microscopic dynamics when viewed in isolation does not obey LRT. The simplest such system is the logistic map as established by Baladi and co-workers [8,9,7,10,18]. For concreteness, we consider perturbations of the modified logistic map that was introduced in [50],…”

Section: Modelmentioning

confidence: 99%

“…Consider Figure 5. Response term E ε Ψ for an uncoupled heat bath scenario for the map (17) where the parameters are sampled from a raised cosine distribution (18). (a) Infinite M limit with confidence intervals (black) and 21-point moving average with confidence intervals (white) from 15 realisations of 6000 iterates for 10 6 parameters a (j) independently selected for each ε.…”

Section: 23mentioning

confidence: 99%

“…Figure 6. Individual response terms E ε ψ(x (j) , y (j) ) with confidence intervals for the map (17) where the parameters selected from the raised cosine distribution (18).…”

Section: 23mentioning

confidence: 99%

“…Success in reliably estimating the response of a physical system, as exemplified by the above applications in the climate sciences, prompted scientists to believe that general chaotic dynamical systems obeyed LRT. This belief was proven wrong by Baladi and co-workers [8,9,7,10,18] who showed that simple dynamical systems such as the logistic map violate LRT and support an invariant measure that changes non-smoothly with respect to the perturbation. This raises the question of how a high-dimensional dynamical system, despite its constituent subsystems typically individually violating LRT, may exhibit linear response.…”

Section: Introductionmentioning

confidence: 99%

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Linear response for macroscopic observables in high-dimensional systems

Wormell

Gottwald

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood.We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional weakly coupled deterministic dynamical systems, where the weak coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system, and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, linear response is induced via self-generated noise.We present examples in the thermodynamic limit where the macroscopic observable satisfies LRT, although the microscopic subsystems individually violate LRT, as well a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent non-trivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

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Section: Modelmentioning

confidence: 99%

Section: 23mentioning

confidence: 99%

“…Figure 6. Individual response terms E ε ψ(x (j) , y (j) ) with confidence intervals for the map (17) where the parameters selected from the raised cosine distribution (18).…”

Section: 23mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear response for macroscopic observables in high-dimensional systems

Wormell

Gottwald

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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show abstract

“…In a separate strand of research mathematicians have tried to obtain rigorous results extending the validity of LRT to deterministic dynamical systems. There was initial success by Ruelle [51,52,53,54] in the case of uniformly hyperbolic Axiom A systems, however the works of Baladi and colleagues undermined hopes that LRT typically holds in dynamical systems [6,7,4,5,15]. They showed that simple dynamical systems such as the logistic map do not obey LRT but rather their invariant measure changes non-smoothly with respect to the perturbation (even considering only chaotic parameter values).…”

Section: Introductionmentioning

confidence: 99%

On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

Wormell

Gottwald

2018

J Stat Phys

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This theoretical work considers the following conundrum: linear response theory is successfully used by scientists in numerous fields, but mathematicians have shown that typical low-dimensional dynamical systems violate the theory's assumptions. Here we provide a proof of concept for the validity of linear response theory in high-dimensional deterministic systems for large-scale observables. We introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large, inhomogeneous collection of unresolved chaotic degrees of freedom. By employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response. We corroborate our result with numerical simulations.

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On the Smooth Dependence of SRB Measures for Partially Hyperbolic Systems

Zhang

2018

Commun. Math. Phys.

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In this paper, we study the differentiability of SRB measures for partially hyperbolic systems.We show that for any s ≥ 1, for any integer ℓ ≥ 2, any sufficiently large r, any ϕ ∈ C r (T, R) such that the map f :exists an open neighbourhood of f in C r (T 2 , T 2 ) such that any map in this neighbourhood has a unique SRB measure with C s−1 density, which depends on the dynamics in a C s fashion.We also construct a C ∞ mostly contracting partially hyperbolic diffeomorphism f : T 3

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Central Limit Theorem for the Modulus of Continuity of Averages of Observables on Transversal Families of Piecewise Expanding Unimodal Maps

Cited by 15 publications

References 20 publications

Linear response for macroscopic observables in high-dimensional systems

Linear response for macroscopic observables in high-dimensional systems

On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

On the Smooth Dependence of SRB Measures for Partially Hyperbolic Systems

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