Generalized Play Hysteresis Operators in Limits of Fast-Slow 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 formdo 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.

On hysteresis-reaction-diffusion systems: Singular fast-reaction limit derivation and nonlinear hysteresis feedback

“…we prove rigorously the limit ε → 0. The proof generalises a related previous result of ODE-hysteresis systems [KM17].…”

“…The hysteresis operator is approximated by an ODE regularisation of the variational inequality (34), similar as in [BK13] or [KM17]. As regularisation parameter we use ε = 10 −4 .…”

“…One main element of our proof is to introduce a projection operator p ε onto the (ε independent) area Ṡε = 0 in the f ε -S ε phase space diagram, cf. [KM17]. In order to appropriately define this projection p ε , we require the uniform in ε bounds on the stock level S ε .…”

Duck traps: two-dimensional critical manifolds in planar systems

“…Amongst others, the coupling of (systems) of ordinary differential equations (ODEs) or partial differential equations (PDEs) and a hysteresis operator are often approximated by the singular limit of fastslow systems, see e.g. [12,14,13]. A particular subclass of interesting scalar hysteresis operators are so-called generalized play operators [2,16].…”

A survey on the blow-up method for fast-slow systems

“…The function f (x, y, ε) is smooth and the critical manifold is given by(3.57) C 0 = (x, y) ∈ R 2 | x ≤ 0 ,which is two dimensional. Similarly, many other examples can be created.In a more relevant context, this situation of codimension 0 critical manifolds has recently been considered in[KM18] motivated by hysteresis operators[KM17]. Piecewise-smooth systems: Canard theory (see Sections 3.2 and 3.3) has been extended to piecewise-linear systems [DGP + 16, DFGK + 18] and in a similar context the blow-up method has been used to analyze piece-wise smooth (fast-slow) systems [KH15a, KH15b, dMdS18, Jef16].…”