Michael Grinfeld

The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor

In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn–Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the flow on the global attractor is shown to be semi-conjugate to the flow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction–diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.

A Test for Strong Hysteresis

The mathematical definition of systems withhysteresis, that is nonlinear input-output systemswith memory, is different from the definition usuallyapplied to economic systems. Economic theory andmodelling practice have almost always specified simpledynamic systems with regular leads and lags in theirresponses, corresponding to input-output systems withunit or zero (or at least stable) roots. These modelscannot capture the ‘selective memory’ feature ofhysteretic behaviour, that is, the influence only ofcertain past events (typically, non-dominatedsequences of previous peaks and troughs). There istherefore a difficulty in testing for and validatingeconomic models containing hysteretic behaviour;appropriate empirical tests have not been developed.In particular, the usual unit vs. zero (or stable)root tests used in econometric analysis are unable todetect hysteretic behaviour or to distinguish it frommore conventional economic behaviour. The purpose ofthis paper is to propose a new way of testing forhysteresis, by drawing on some ideas in themathematical/control theory literature and adaptingthem to fit into the economic frameworks with elementsof hysteresis.

Subharmonic ferroresonance in an LCR circuit with hysteresis

We use the Preisach model of magnetic hysteresis to model the inductance in a series LCR circuit. By introducing a hysteresis parameter into the Preisach functions used, we are able to continuously vary the width of the hysteresis loop and thereby investigate the effects of magnetic hysteresis on the circuits behavior. In particular, it is shown that the stability region of a period-3 subharmonic ferroresonant solution can increase significantly as the hysteresis losses are reduced. The bifurcations leading to the appearance and disappearance of this subharmonic solution are examined. Also, strong numerical evidence is provided for the unsuitability of single-valued M-H curves in the analysis of ferroresonant phenomena.

Hysteresis and economics

The goal of this article is to explore the rationale underlying the application of hysteresis to economic models. In particular, we explain why many aspects of real economic systems are hysteretic. The aim is to be explicit about the difficulties encountered when trying to incorporate hysteretic effects into models that can be validated and then used as possible tools for macroeconomic control. The growing appreciation of the ways that memory effects influence the functioning of economic systems is a significant advance in economic thought and, by removing distortions that result from oversimplifying specifications of input-output relations in economics, has the potential to narrow the gap between economic modeling and economic reality. Static hysteresis input-output systems, hysterons, and Preisach models are defined, and the form that macroeconomic models with hysteresis typically take is described. Then some relevant economics background is sketched, and the distinctive nature of models in economics is discussed in detail. In the following central section of the article the results of approximately two decades of hysteresis modeling in economics are summarized.

Non-local dispersal and bistability

The scalar initial value problem [ u_t = ho Du + f(u), ] is a model for dispersal. Here

Steady state solutions of a bi-stable quasi-linear equation with saturating flux

u

Coarsening in an integro-differential model of phase transitions

represents the density at point

Memory driven instability in a diffusion process

x

Stylized facts from a threshold-based heterogeneous agent model

of a compact spatial region

On the periodically perturbed logistic equation

Omega in mathbb{R}^n