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.
Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.
Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities both in time and in memory. The main analytical tool is the Kurzweil integral formalism, and the main result proves the well-posedness of the process in the space of right-continuous regulated functions.