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Hysteresis operators appear in many applications such as elasto-plasticityand micromagnetics, and can be used for a wider class of systems, whererate-independent memory plays a role. A natural approximation for systems ofevolution 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 thenon-linearity as a limit of approximating fast-slow dynamics involves subtlelimit procedures. In this paper, we give a proof of Netushil's "observation"that broad classes of planar fast-slow systems with a two-dimensional criticalmanifold are expected to yield generalized play operators in the singularlimit. We provide two proofs of this "observation" based upon the fast-slowsystems paradigm of decomposition into subsystems. One proof strategy employssuitable convergence in function spaces, while the second approach considers ageometric strategy via local linearization and patching adapted originally fromproblems in stochastic analysis. We also provide an illustration of our resultsin the context of oscillations in forced planar non-autonomous fast-slowsystems. The study of this example also strongly suggests that new canard-typemechanisms can occur for two-dimensional critical manifolds in planar systems.

Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems