Newton and Bouligand derivatives of the scalar play and stop operator

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strenghtened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable

Newton and Bouligand derivatives of the scalar play and stop operator

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable

Hopf bifurcations and simple structures of periodic solution sets in systems with the Preisach nonlinearity

We survey a number of recent results and suggest some new ones on periodic solutions of systems with hysteresis. The main focus of this work is the situation when simple one-parameter structures of periodic regimes appear. We consider forced oscillations, cycles of autonomous systems and Hopf bifurcations from the equilibrium and from infinity

On the Mróz Model

We treat the mathematical properties of the one parameter version of the Mróz model for plastic flow. We present continuity results and an energy inequality for the hardening rule and discuss different versions of the flow rule regarding their relation to the second law of thermodynamics

Asymptotic stability of continual sets of periodic solutions to systems with hysteresis

We consider hysteresis perturbations of the system of ODEs which has an asymptotically stable periodic solution $z_*$. It is proved that if the oscillation of the appropriate projection of $z_*$ is smaller than some threshold number defined by the hysteresis nonlinearity, then the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of $z_*$. The main result is the theorem on stability of this continuum

Some analytical properties of the multidimensional continuous Mróz model of plasticity

We study the geometrical structure of memory induced by the continuous multidimensional Mr'oz model of plasticity. The results are used for proving the thermodynamic consistency of the model and composition and inversion formulas for input - memory state - output operators. We also show an example of nonuniqueness of solutions to a simple initial value problem involving the Mr'oz operator

Stochastic Hysteresis and Resonance in a Kinetic Ising System

We study hysteresis for a two-dimensional, spin-1/2, nearest-neighbor, kinetic Ising ferromagnet in an oscillating field, using Monte Carlo simulations and analytical theory. Attention is focused on small systems and weak field amplitudes at a temperature below $T_{c}$. For these restricted parameters, the magnetization switches through random nucleation of a single droplet of spins aligned with the applied field. We analyze the stochastic hysteresis observed in this parameter regime, using time-dependent nucleation theory and the theory of variable-rate Markov processes. The theory enables us to accurately predict the results of extensive Monte Carlo simulations, without the use of any adjustable parameters. The stochastic response is qualitatively different from what is observed, either in mean-field models or in simulations of larger spatially extended systems. We consider the frequency dependence of the probability density for the hysteresis-loop area and show that its average slowly crosses over to a logarithmic decay with frequency and amplitude for asymptotically low frequencies. Both the average loop area and the residence-time distributions for the magnetization show evidence of stochastic resonance. We also demonstrate a connection between the residence-time distributions and the power spectral densities of the magnetization time series. In addition to their significance for the interpretation of recent experiments in condensed-matter physics, including studies of switching in ferromagnetic and ferroelectric nanoparticles and ultrathin films, our results are relevant to the general theory of periodically driven arrays of coupled, bistable systems with stochastic noise.Comment: 35 pages. Submitted to Phys. Rev. E Minor revisions to the text and updated reference

On the Moving Preisach Model

It is shown that the moving model, which is a variant of the Preisach model for hysteresis, possesses the wiping out property and is continuous in C[0,T] under natural assumptions