Existence of Noise Induced Order, a Computer Aided Proof

Galatolo, Stefano; Monge, Maurizio; Nisoli, Isaia

doi:10.48550/arxiv.1702.07024

Cited by 5 publications

(24 citation statements)

References 26 publications

(59 reference statements)

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“…This assumption is satisfied, for example, if there is an iterate of the transfer operator having a strictly positive kernel, see Corollary 5.7.1 of [35]. In nontrivial cases the assumption can also be verified by a computer-aided proof (see [21], Section 5, [22], [39]). The other assumptions of Theorem 2.2 can be easily verified in interesting classes of systems and perturbations, as we will see in the following.…”

Section: Existence Of Linear Response For the Invariant Measurementioning

confidence: 94%

“…Thus far we have not been specific about the feasible set P ; we take up this issue in this and the succeeding subsections to provide explicit formulae for the optimal responses in both problems (17) and (22). First, we have not required that the perturbed kernel k δ in (9) be nonnegative for δ > 0, however, this is a natural assumption.…”

Section: Explicit Formulae For the Optimal Perturbationsmentioning

confidence: 99%

See 1 more Smart Citation

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

Antown,

Froyland,

Galatolo

2021

Preprint

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We consider optimal control problems for discrete time random dynamical systems, finding the perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an L 2 kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting in which these optimisation problems have unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau-Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.

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Section: Existence Of Linear Response For the Invariant Measurementioning

confidence: 94%

Section: Explicit Formulae For the Optimal Perturbationsmentioning

confidence: 99%

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

Antown,

Froyland,

Galatolo

2021

Preprint

Self Cite

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“…Another example is given by recently developed methods, which numerically calculate the local fluctuations in stochastic systems around deterministic equilibria by using matrix equations [45,46], and for which rigorous numerics would be directly applicable. In a different direction, we mention the recent work [30], where the existence of noise induced order is established, using computer-assisted techniques based on transfer operators.…”

Section: Validated Computations For Stochastic Systems: Possible Appl...mentioning

confidence: 99%

Rigorous validation of stochastic transition paths

Breden

Kuehn

2019

Journal de Mathématiques Pures et Appliquées

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Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments, including a combination of analytical and validated numerical estimates, to prove that the computed solution has a true solution in a suitable neighbourhood. We demonstrate our approach by computing minimum-energy transition paths via invariant manifolds and heteroclinic connections. We illustrate our method in the context of the classical Müller-Brown test potential.

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“…In this paper we will make use of the software and the methods developed in [29] for dynamical systems on the interval with additive noise. 1 The software will be used in this work for two purposes:…”

Section: Introductionmentioning

confidence: 99%

“…The mixing rate is measured by the norm of the iterates of the transfer operator associated with the system, restricted to the space of measures having zero average on the phase space S 1 . Both of these computations are made possible by a kind of finite-element approximation of the transfer operator, used in combination with quantitative functional analytic stability statements that estimate explicitly, and not asymptotically the approximation errors made in the finite element reduction (as explained in [29]). Further details on this matter will be given in Sections 4.2 and 5, where we also show the results of the computer-aided estimates we use herein.…”

Section: Introductionmentioning

confidence: 99%

Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number

et al. 2019

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Arnold's standard circle maps are widely used to study the quasiperiodic route to chaos and other phenomena associated with nonlinear dynamics in the presence of two rationally unrelated periodicities. In particular, the El Niño-Southern Oscillation (ENSO) phenomenon is a crucial component of climate variability on interannual time scales and it is dominated by the seasonal cycle, on the one hand, and an intrinsic oscillatory instability with a period of a few years, on the other. The role of meteorological phenomena on much shorter time scales, such as westerly wind bursts, has also been recognized and modeled as additive noise.We consider herein Arnold maps with additive, uniformly distributed noise. When the map's nonlinear term, scaled by the parameter , is sufficiently small, i.e. < 1, the map is known to be a diffeomorphism and the rotation number ρ ω is a differentiable function of the driving frequency ω.We concentrate on the rotation number's behavior as the nonlinearity becomes large, and show rigorously that ρ ω is a differentiable function of ω, even for ≥ 1, at every point at which the noise-perturbed map is mixing. We also provide a formula for the derivative of the rotation number. The reasoning relies on linear-response theory and a computer-aided proof. In the diffeomorphism case of < 1, the rotation number ρ ω behaves monotonically with respect to ω. We show, using again a computer-aided proof, that this is not the case when ≥ 1 and the map is not a diffeomorphism. This lack of monotonicity for large nonlinearity could be of interest in some applications. For instance, when the devil's staircase ρ = ρ(ω) loses its monotonicity, frequency locking to the same periodicity could occur for non-contiguous parameter values that might even lie relatively far apart from each other.

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Existence of Noise Induced Order, a Computer Aided Proof

Cited by 5 publications

References 26 publications

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

Rigorous validation of stochastic transition paths

Arnold Maps with Noise: Differentiability and Non-monotonicity of the Rotation Number

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