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return-point memory; Preisach operator; oscillator with memory; hysteresis; operator-differential equation; stability of equilibrium; partial stability; slow-fast system; switching line; excitability
We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear as a component of the recently proposed models of population dynamics. We study dynamics of such systems near an equilibrium point. In particular, we show and examine a similarity in the behaviour of trajectories between the system with the Preisach memory operator and a planar slow-fast ordinary differential equation. The nonsmooth Preisach operator introduces a singularity into the system. Furthermore, we classify the robust equilibrium points according to their stability properties. Conditions for stability, instability and partial stability are presented. A robust partially stable point simultaneously attracts many trajectories and repels many trajectories (a behaviour which is not generic for smooth ordinary differential equations). We discuss implications of such local dynamics for the excitability properties of the system.
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