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.

MSC codes

  1. fast-slow system
  2. multiple time scale dynamics
  3. hysteresis operator
  4. generalized play
  5. canard
  6. Netushil's observation

MSC codes

  1. 34C55
  2. 34E13

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1650 - 1685
ISSN (online): 1536-0040


Submitted: 5 January 2017
Accepted: 3 July 2017
Published online: 29 August 2017

MSC codes

  1. fast-slow system
  2. multiple time scale dynamics
  3. hysteresis operator
  4. generalized play
  5. canard
  6. Netushil's observation

MSC codes

  1. 34C55
  2. 34E13



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : IGDK 1754

Funding Information

Volkswagen Foundation https://doi.org/10.13039/501100001663

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