Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

Antown, Fadi; Dragičević, Davor; Froyland, Gary

doi:10.1007/s10955-018-1985-1

Cited by 16 publications

(28 citation statements)

References 34 publications

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“…A similar formulation to (40) incorporating control appears from a state-discretization of the controlled Liouville equation as outlined in [77], resulting in a discrete-state, continuous-time Markov model. Variations of this type of model have been used to study optimal response of Markov chains [7] and to control chaos [19].…”

Section: Perron-frobenius Operatormentioning

confidence: 99%

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Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.

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Section: Perron-frobenius Operatormentioning

confidence: 99%

“…A similar problem has been studied based on the infinitesimal generator [61], but resulting in a convex quadratic program. More recent work [7] considers the perturbation of the stochastic part of the dynamical system and provides a solution in closed form.…”

Section: Control Designmentioning

confidence: 99%

Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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“…More recently, the topic of linear response was also studied in the context of random or extended systems [6,16,18,23,31,40,44]. Optimisation of statistichal properties through linear respone was develope in [1,2,22,30].…”

Section: Introductionmentioning

confidence: 99%

Rigorous Computation of Linear Response for Intermittent Maps

Nisoli¹,

Taylor-Crush²

2022

Preprint

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“…Strong nonlinearity. The deterministic Arnold circle map T τ , : S 1 → S 1 is defined by (1) T τ , (x) := x + 2π ω − 2π sin(2πx) mod 1 .…”

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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Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

Cited by 16 publications

References 34 publications

Data-Driven Approximations of Dynamical Systems Operators for Control

Data-Driven Approximations of Dynamical Systems Operators for Control

Rigorous Computation of Linear Response for Intermittent Maps

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

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