Ten Issues About Hysteresis

Visintin, Augusto

doi:10.1007/s10440-014-9936-6

Cited by 22 publications

(12 citation statements)

References 17 publications

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“…This is called by Decomposition method 4011 such physical phenomena as "gap", "reley", "hysteresis", "dry friction" etc. [4][5][6][7][8]12,16,17]. System 1…”

Section: Methods Of Investigationmentioning

confidence: 99%

The decomposition method of research into the nonlinear dynamical systems’ space of parameters

Kamachkin¹,

Shamberov²

2015

ams

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The work suggests to use the approach oriented at precise analytical methods of research into nonlinear dynamical systems that have a complex structure of the state-space. This method allows to investigate the behavior of essential multivariable system through the dynamic behaviors of its basic components i.e. subsystems (both linear and non-linear) which don't compose any reserved system and have the more-lower order state-space. Such approach allows bringing out analytically strict proof of existence of the periodical movements both free and force. In the majority of cases the approach allows to fulfill "partition" of parametric-space into the areas of its qualitatively different dynamic behavior.

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“…This is called by Decomposition method 4011 such physical phenomena as "gap", "reley", "hysteresis", "dry friction" etc. [4][5][6][7][8]12,16,17]. System 1…”

Section: Methods Of Investigationmentioning

confidence: 99%

The decomposition method of research into the nonlinear dynamical systems’ space of parameters

Kamachkin¹,

Shamberov²

2015

ams

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“…In this chapter we will survey recent results on reaction-diffusion equations with a hysteretic discontinuity defined at every spatial point. We also refer to [1][2][3] and the more recent surveys by Visintin [4,5] for other types of partial differential equations with hysteresis.…”

Section: Motivationmentioning

confidence: 99%

Recent Advances in Reaction-Diffusion Equations with Non-ideal Relays

Curran

Gurevich

Tikhomirov

2016

Understanding Complex Systems

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We survey recent results on reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time, and also describe its regularity. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanismrattling, which indicates how one should reset the continuous model to make it well posed.

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“…To begin with, the well-posedness was an open question, due to the discontinuous nature of the hysteresis operator. First analytical results were obtained in [3,26] (see also [2,19,27] and a recent survey [28]), where existence of solutions for multi-valued hysteresis was proved. Formal asymptotic expansions of solutions were recently obtained in a special case in [18].…”

Section: Introductionmentioning

confidence: 99%

Spatially discrete reaction–diffusion equations with discontinuous hysteresis

Gurevich¹,

Tikhomirov

2018

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We address the question: Why may reaction-diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order at −1/2 as t → ∞ and explicitly find the rate a.

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Ten Issues About Hysteresis

Cited by 22 publications

References 17 publications

The decomposition method of research into the nonlinear dynamical systems’ space of parameters

The decomposition method of research into the nonlinear dynamical systems’ space of parameters

Recent Advances in Reaction-Diffusion Equations with Non-ideal Relays

Spatially discrete reaction–diffusion equations with discontinuous hysteresis

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