Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems

TL;DR

This paper demonstrates that planar fast-slow systems with two-dimensional critical manifolds can approximate hysteresis operators, specifically generalized play operators, through subtle limit procedures, revealing new dynamics like canard phenomena.

Contribution

It provides two rigorous proofs that certain fast-slow systems converge to generalized play hysteresis operators in the singular limit, expanding understanding of hysteresis modeling in dynamical systems.

Findings

Fast-slow systems approximate hysteresis operators in the limit.

Two distinct proofs establish the convergence to generalized play operators.

Identification of potential new canard-type mechanisms in planar systems.

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 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.