Harbir Lamba

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.

Chaotic, regular and unbounded behaviour in the elastic impact oscillator

Abstract We study the elastic impact oscillator which is a periodically-forced mass on a spring impacting against a rigid wall. This gives rise to an area-preserving mapping that typically displays chaos at low energies and regular behaviour at high energies. The boundary between the two types of behaviour is complicated by the existence of discontinuities. The high energy behaviour is strongly influenced by resonances and the existence of invariant curves that bound the energy gain of the mass has an interesting dependence upon the system parameters.

Stylized facts from a threshold-based heterogeneous agent model

Abstract.A class of heterogeneous agent models is investigated where investors switch trading position whenever their motivation to do so exceeds some critical threshold. These motivations can be psychological in nature or reflect behaviour suggested by the efficient market hypothesis (EMH). By introducing different propensities into a baseline model that displays EMH behaviour, one can attempt to isolate their effects upon the market dynamics. The simulation results indicate that the introduction of a herding propensity results in excess kurtosis and power-law decay consistent with those observed in actual return distributions, but not in significant long-term volatility correlations. Possible alternatives for introducing such long-term volatility correlations are then identified and discussed.

Dynamical Systems and Adaptive Timestepping in ODE Solvers

Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.This assumption of sufficiently strong finite-time convergence results is then examined for adaptive algorithms that use a pair of explicit Runge-Kutta methods of different order to estimate the one-step error. For arbitrary Runge-Kutta pairs the necessary finite-time convergence results fail to hold on a set of points in the phase space that includes all the equilibria of the ODE. Therefore, in general, the asymptotic convergence results cannot be applied to attractors containing equilibria. However, for a particular class of Runge-Kutta pairs, the finite-time convergence results can be strengthened to include neighbourhoods of equilibrium points for which the Jacobian is invertible.

On the periodically perturbed logistic equation

We study the logistic equation modified by a periodic time dependence. The perturbation introduces bifurcation delays which can be calculated explicitly and we qualitatively explain how the bifurcation diagram deforms as the perturbation is increased.

A mean-field model of investor behaviour

In this note we investigate the ability of a mean-field model distilled from a heterogeneous agent model to simulate stylized facts of financial times series.

On entry and exit in response to aggregate shocks

This note extends the Dixit-Pindyck analysis of investment, in the form of market entry and exit under sunk costs, to the case of heterogeneous sunk costs. The implication is that the market displays full hysteresis, in the form of remanence and dependence on the nondominated extremum values of the aggregate shocks experienced. These implications are illustrated by numerical simulations.

Convergence results for the MATLAB ode23 routine

We prove convergence results on finite time intervals, as the user-defined tolerance τ→0, for a class of adaptive timestepping ODE solvers that includes the ode23 routine supplied in MATLAB Version 4.2. In contrast to existing theories, these convergence results hold with error constants that are uniform in the neighbourhood of equilibria; such uniformity is crucial for the derivation of results concerning the numerical approximation of dynamical systems. For linear problems the error estimates are uniform on compact sets of initial data. The analysis relies upon the identification of explicit embedded Runge-Kutta pairs for which all but the leading order terms of the expansion of the local error estimate areO(∥f(u∥)2).