Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincaré map that is infinite-dimensional due to delay and...

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Bibliographic Details
Published in:Journal of dynamics and differential equations 2019-12, Vol.31 (4), p.1873-1920
Main Authors: Gurevich, Pavel, Ron, Eyal
Format: Article
Language:English
Subjects:
Applications of Mathematics
Delay
Differential equations
Hysteresis
Linearization
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
Poincare maps
Spectrum analysis
Stability analysis
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Summary:We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincaré map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev–Slobodeckij spaces and explicitly obtain a formal linearization of the Poincaré map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-018-9664-0