The Devil is in the details: Spectrum and eigenvalue distribution of the discrete Preisach memory model

Kalmár‐Nagy, Tamás; Amann, Andreas; Kim, Daniel; Рачинский, Дмитрий

doi:10.1016/j.cnsns.2019.04.023

Cited by 8 publications

(7 citation statements)

References 49 publications

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“…This eigenvalue distribution exhibits a "devil's staircase"type self-similarity which is related to the self-similar structure of the mechanistic model (binary tree structure). He et al [32] and Kalmár-Nagy et al [33] reported different classes of self-similar graphs whose adjacency matrices have devil's staircase-type spectrum. In the latter paper, it was also shown that the characteristic polynomial of the adjacency matrix is a product of Chebyshev polynomials.…”

Section: The Eigenvalue Distributionmentioning

confidence: 99%

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

Kalmár‐Nagy

Bak

2019

Nonlinear Dyn

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In Richardson's cascade description of turbulence, large vortices break up to form smaller ones, transferring the kinetic energy of the flow from large to small scales. This energy is dissipated at the smallest scales due to viscosity. We study energy cascade in a phenomenological model of vortex breakdown. The model is a binary tree of spring-connected masses, with dampers acting on the lowest level. The masses and stiffnesses between levels change according to a power law. The different levels represent different scales, enabling the definition of "mass wavenumbers." The eigenvalue distribution of the model exhibits a devil's staircase self-similarity. The energy spectrum of the model (defined as the energy distribution among the different mass wavenumbers) is derived in the asymptotic limit. A decimation procedure is applied to replace the model with an equivalent chain oscillator. For a significant range of the stiffness decay parameter, the energy spectrum is qualitatively similar to the Kolmogorov spectrum of 3D homogeneous, isotropic turbulence and we find the stiffness parameter for which the energy spectrum has the well-known − 5/3 scaling exponent.

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Section: The Eigenvalue Distributionmentioning

confidence: 99%

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

Kalmár‐Nagy

Bak

2019

Nonlinear Dyn

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“…He et al [20] demonstrated that the adjacency matrix for a class of symmetric tree graphs have devil's staircase type spectrum. Kalmár-Nagy et al [21] showed the same for a different type of self-similar graph, as well as that the characteristic polynomial of the adjacency matrix of such a graph is a product of Chebyshev polynomials.…”

Section: The Eigenvalue Distribution Of the Mechanistic Modelmentioning

confidence: 85%

Reproducing the Kolmogorov spectrum of turbulence with a hierarchical linear cascade model

Kalmár-Nagy,

Bak

2018

Preprint

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In Richardson's cascade description of turbulence, large vortices break up to form smaller ones, transferring the kinetic energy towards smaller scales. Energy dissipation occurs at the smallest scales due to viscosity. We study this energy cascade in a phenomenological model of vortex breakdown. The model is a binary tree of decreasing masses connected by softening springs, with dampers acting on the lowest level. The masses and stiffnesses between levels change according to a power law. The different levels represent different scales, enabling the definition of "mass wavenumbers". The eigenvalue distribution of the model exhibits a devil's staircase self-similarity. The energy spectrum of the model (defined as the energy distribution among the different mass wavenumber) is derived in the asymptotic limit. A decimation procedure is applied to replace the model with an equivalent chain oscillator. For a range of stiffness parameter the energy spectrum is qualitatively similar to the Kolmogorov spectrum of 3D homogeneous, isotropic turbulence and we find the stiffness parameter for which the energy spectrum has the well-known −5/3 scaling exponent.

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“…Among the recent works devoted to the study of the Preisach model as applied to objects with the structure of networks and pathways, we note [52,61,62,[207][208][209]. The transition graphs describing the functioning of the discrete Preisach model were considered in relation to the problems of continuum mechanics in [210].…”

Section: Technical Systemsmentioning

confidence: 99%

The Preisach model of hysteresis: fundamentals and applications

Semenov,

Borzunov,

Meleshenko

et al. 2024

Phys. Scr.

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The Preisach model is a well-known model of hysteresis in the modern nonlinear science. This paper provides an overview of works that are focusing on the study of dynamical systems from various areas (physics, economics, biology), where the Preisach model plays a key role in the formalization of hysteresis dependencies. Here we describe the input-output relations of the classical Preisach operator, its basic properties, methods of constructing the output using the demagnetization function formalism, a generalization of the classical Preisach operator for the case of vector input-output relations. Various generalizations of the model are described here in relation to systems containing ferromagnetic and ferroelectric materials. The main attention we pay to experimental works, where the Preisach model has been used for analytic description of the experimentally observed results. Also, we describe a wide range of the technical applications of the Preisach model in such fields as energy storage devices, systems under piezoelectric effect, models of systems with long-term memory. The properties of the Preisach operator in terms of reaction to stochastic external impacts are described and a generalization of the model for the case of the stochastic threshold numbers of its elementary components is given.

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The Devil is in the details: Spectrum and eigenvalue distribution of the discrete Preisach memory model

Cited by 8 publications

References 49 publications

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

Reproducing the Kolmogorov spectrum of turbulence with a hierarchical linear cascade model

The Preisach model of hysteresis: fundamentals and applications

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