Ilaria Foroni

Adaptive and statistical expectations in a renewable resource market

Rational expectations models have increasingly been replaced by models with various forms of learning. This paper studies the global dynamics of a model of renewable resource markets due to Hommes and Rosser [Macroecon. Dyn. 5 (2001) 180] under adaptive and statistical learning systems. The statistical learning system is seen to generate greater complexity of the structures of the basins of attraction, especially at higher discount rates. An element of particular interest is that bifurcations generating lobes in the basins arise from particular focal points, associated with prefocal sets at infinity on the Poincare equator in the statistical learning model.

Homoclinic bifurcations in heterogeneous market models

We analyze a class of models representing heterogeneous agents with adaptively rational rules. The models reduce to noninvertible maps of R 2 . We investigate particular kinds of homoclinic bifurcations, related to the noninvertibility of the map. A first one, which leads to a strange repellor and basins of attraction with chaotic structure, is associated with simple attractors. A second one, the homoclinic bifurcation of the saddle fixed point, also associated with the foliation of the plane, causes the sudden transition to a chaotic attractor (with self-similar structure). 2002 Elsevier Science Ltd. All rights reserved.

Original article: Complex dynamics associated with the appearance/disappearance of invariant closed curves

Abstract: In this paper we study some global bifurcations arising in a heterogeneous financial model with fundamentalists and imitators. Such bifurcations which cause the appearance and disappearance of closed invariant curves (attracting or repelling) involve the stable and unstable sets of a saddle cycle with consequent changes in their dynamic behavior. Numerical investigations show that the transition between two qualitatively different regimes are characterized by the occurrence of homoclinic tangles with chaotic dynamics.

Agglomeration dynamics and first nature asymmetries

The standard footloose capital (FC) model, as well as the discrete time version, assumes that all capital units are internationally mobile between two regions. In this paper, we assume that in one of the two regions some of the blueprints/capital units may be immobile because their utilization requires some locally specific natural resource (first nature advantage). Mobile blueprints, instead, can be utilized in both regions. We focus on this asymmetric distribution of immobile firms/capital units, labeled first nature firms. The central question of our paper is how the existence of first nature asymmetry affects agglomerative processes framed in discrete time. This modification of the FC model leads to a one dimensional piecewise smooth map for which we show analytically that border collision bifurcations are pervasive and (even asymmetric) multistability is possible.

Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

Discrete time price adjustment processes may fail to converge and may exhibit periodic or even chaotic behavior. To avoid large price changes, a version of the discrete time tâtonnement process for reaching an equilibrium in a pure exchange economy based on a cautious updating of the prices has been proposed two decades ago. This modification leads to a one dimensional bimodal piecewise smooth map, for which we show analytically that degenerate bifurcations and border collision bifurcations play a fundamental role for the asymptotic behavior of the model.

Heterogeneous Models with Learning and Homoclinic Bifurcations

We analyze a class of models representing heterogeneous agents with adaptively rational rules. The models reduce to non invertible maps of R2. In particular we shall investigate the homoclinic bifurcation of the saddle fixed point, which causes the sudden transition to a chaotic attractor (or strange attractor, with self-similar structure).