Singular Perturbations

Shchepakina, Elena; Sobolev, Vladimir; Mortell, Michael P.

doi:10.1007/978-3-319-09570-7

Cited by 72 publications

(18 citation statements)

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“…In [19], it is shown that the dynamics of system (2.2) must be considered on different time scales. Before analysing the dynamical behaviour on the tippedisk in the framework of slow-fast systems, we introduce the basics of singular perturbation theory [24][25][26]. where ε 1 is identified as small fixed perturbation parameter and • := d dt • denotes the derivative with respect to 'slow' time t. The system…”

Section: (B) Singularly Perturbed Dynamicsmentioning

confidence: 99%

Singularly perturbed dynamics of the tippedisk

Sailer

Leine

2021

Proc. R. Soc. A.

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The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disc, being reminiscent of (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow–fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.

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Section: (B) Singularly Perturbed Dynamicsmentioning

confidence: 99%

Singularly perturbed dynamics of the tippedisk

Sailer

Leine

2021

Proc. R. Soc. A.

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“…We now investigate the case 1 in more detail. The equations (63) possess a slow manifold for > 0 sufficiently small [19,16]. To leading order, solutions on the slow manifold satisfy…”

Section: 1mentioning

confidence: 99%

“…with > 0 sufficiently small and A a symmetric positive definite matrix, there exists a smooth manifold M which is exponentially attractive and invariant under the dynamics of (97). See, for example, [19,16] for details. Approximations of the slow manifold can be found by utilising the principle of bounded derivatives, that is,…”

Section: 12mentioning

confidence: 99%

Posterior contraction rates for non-parametric state and drift estimation

Reich¹,

Rozdeba²

2020

Foundations of Data Science

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We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman-Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upon scale separation and a slow manifold approximation.

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“…◻ Proof of Lemma 6.1: We construct P = [P 1 P 2 ...P n ] ∈ R n 2 ×n 2 as an orthogonal matrix, where P 1 = θ1 n 2 ×n , θ > 0, and rewrite P = [P 1 P ∆ ]. We also define a new vectorŝ 1 b ∈ R n 2 asŝ 1 = Pŝ 1 b , then the system (27), which is recalled as the boundary-layer model for (23a), becomes…”

Section: Appendix Bmentioning

confidence: 99%

“…In contrast with the previous work [18], this work considers different convergence rates toward dynamic agreement under the LCA, hereby seeking the NE at each player's own pace. Our work herein is also related to SPS comprised of slow and fast motions, which often occur when we mathematically model some natural processes, for example, biochemistry, fluid dynamics, nonlinear kinetics, and so on [27]. We design algorithms in the form of SPS to seek NE of the game (1).…”

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confidence: 99%

Distributed optimization algorithms for game of power generation in smart grid

Nguyen¹,

Hoang²,

Ahn³

2019

Numerical Algebra, Control &Amp; Optimization

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In this paper, we consider a problem of finding optimal power generation levels for electricity users in Smart Grid (SG) with the purpose of maximizing each user's benefit selfishly. As the starting point, we first develop a generalized model based on the framework of IEEE 118 bus system, then we formulate the problem as an aggregative game, where its Nash Equilibrium (NE) is considered as the collection of optimal levels of generated powers. This paper proposes three distributed optimization strategies in forms of singularly perturbed systems to tackle the problem under limited control authority concern, with rigorous analyses provided by game theory, graph theory, control theory, and convex optimization. Our analysis shows that without constraints in power generation, the first strategy provably exponentially converges to the NE from any initializations. Moreover, under the constraint consideration, we achieve locally exponential convergence result via the other proposed algorithms, one of them is more generalized. Numerical simulations in the IEEE 118 bus system are carried out to verify the correctness of the proposed algorithms.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

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Singular Perturbations

Cited by 72 publications

References 0 publications

Singularly perturbed dynamics of the tippedisk

Singularly perturbed dynamics of the tippedisk

Posterior contraction rates for non-parametric state and drift estimation

Distributed optimization algorithms for game of power generation in smart grid

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