  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
non-linearity 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 second approach 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 non-autonomous 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.
