Dynamics of systems with Preisach memory near equilibria

Stephen McCarthy; Dmitrii Rachinskii

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 1, page 39-73
  • ISSN: 0862-7959

Abstract

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We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear as a component of the recently proposed models of population dynamics. We study dynamics of such systems near an equilibrium point. In particular, we show and examine a similarity in the behaviour of trajectories between the system with the Preisach memory operator and a planar slow-fast ordinary differential equation. The nonsmooth Preisach operator introduces a singularity into the system. Furthermore, we classify the robust equilibrium points according to their stability properties. Conditions for stability, instability and partial stability are presented. A robust partially stable point simultaneously attracts many trajectories and repels many trajectories (a behaviour which is not generic for smooth ordinary differential equations). We discuss implications of such local dynamics for the excitability properties of the system.

How to cite

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McCarthy, Stephen, and Rachinskii, Dmitrii. "Dynamics of systems with Preisach memory near equilibria." Mathematica Bohemica 139.1 (2014): 39-73. <http://eudml.org/doc/261098>.

@article{McCarthy2014,
abstract = {We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear as a component of the recently proposed models of population dynamics. We study dynamics of such systems near an equilibrium point. In particular, we show and examine a similarity in the behaviour of trajectories between the system with the Preisach memory operator and a planar slow-fast ordinary differential equation. The nonsmooth Preisach operator introduces a singularity into the system. Furthermore, we classify the robust equilibrium points according to their stability properties. Conditions for stability, instability and partial stability are presented. A robust partially stable point simultaneously attracts many trajectories and repels many trajectories (a behaviour which is not generic for smooth ordinary differential equations). We discuss implications of such local dynamics for the excitability properties of the system.},
author = {McCarthy, Stephen, Rachinskii, Dmitrii},
journal = {Mathematica Bohemica},
keywords = {return-point memory; Preisach operator; oscillator with memory; hysteresis; operator-differential equation; stability of equilibrium; partial stability; slow-fast system; switching line; excitability; return-point memory; Preisach operator; oscillator with memory; hysteresis; operator-differential equation; stability of equilibrium; partial stability; slow-fast system; switching line; excitability},
language = {eng},
number = {1},
pages = {39-73},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Dynamics of systems with Preisach memory near equilibria},
url = {http://eudml.org/doc/261098},
volume = {139},
year = {2014},
}

TY - JOUR
AU - McCarthy, Stephen
AU - Rachinskii, Dmitrii
TI - Dynamics of systems with Preisach memory near equilibria
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 1
SP - 39
EP - 73
AB - We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear as a component of the recently proposed models of population dynamics. We study dynamics of such systems near an equilibrium point. In particular, we show and examine a similarity in the behaviour of trajectories between the system with the Preisach memory operator and a planar slow-fast ordinary differential equation. The nonsmooth Preisach operator introduces a singularity into the system. Furthermore, we classify the robust equilibrium points according to their stability properties. Conditions for stability, instability and partial stability are presented. A robust partially stable point simultaneously attracts many trajectories and repels many trajectories (a behaviour which is not generic for smooth ordinary differential equations). We discuss implications of such local dynamics for the excitability properties of the system.
LA - eng
KW - return-point memory; Preisach operator; oscillator with memory; hysteresis; operator-differential equation; stability of equilibrium; partial stability; slow-fast system; switching line; excitability; return-point memory; Preisach operator; oscillator with memory; hysteresis; operator-differential equation; stability of equilibrium; partial stability; slow-fast system; switching line; excitability
UR - http://eudml.org/doc/261098
ER -

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