NDLI: Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kuehn, Christian Münch, Christian
Copyright Year	2017
Abstract	Hysteresis operators appear in many applications, such as elasto-plasticity and micromagnetics, and can be used for a wider class of systems, where rate-independent memory plays a role. A natural approximation for systems of evolution equations with hysteresis operators are fast-slow dynamical systems, which---in their used approximation form---do not involve any memory effects. Hence, viewing differential equations with hysteresis operators in the nonlinearity as a limit of approximating fast-slow dynamics involves subtle limit procedures. In this paper, we give a proof of Netushil's observation that broad classes of planar fast-slow systems with a two-dimensional critical manifold are expected to yield generalized play operators in the singular limit. We provide two proofs of this observation based upon the fast-slow systems paradigm of decomposition into subsystems. One proof strategy employs suitable convergence in function spaces, while the other considers a geometric strategy via local linearization and patching adapted originally from problems in stochastic analysis. We also provide an illustration of our results in the context of oscillations in forced planar nonautonomous fast-slow systems. The study of this example also strongly suggests that new canard-type mechanisms can occur for two-dimensional critical manifolds in planar systems.
Sponsorship	Deutsche Forschungsgemeinschaft. Volkswagen Foundation
Starting Page	1650
Ending Page	1685
Page Count	36
File Format	PDF
DOI	10.1137/17M1110584
e-ISSN	15360040
Issue Number	3
Volume Number	16
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-08-29
Access Restriction	Subscribed
Subject Keyword	hysteresis operator Netushil's observation canard multiple time scale dynamics generalized play fast-slow system Multiple scale methods Hysteresis
Content Type	Text
Resource Type	Article

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