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In this paper, we enhance the study of the tâtonnement model with cautious price adjustment introduced by Weddepohl in 1995. Specifically, we investigate the discrete time dynamics of the model when the original assumption of equal values for the maximum rate of increase and decrease of the price is relaxed. As a result, its analytic definition is expressed by a bimodal one-dimensional continuous piecewise smooth map which depends on three parameters. As it happens in general for bimodal maps, it is possible to describe the bifurcation structure in some regions of the parameter space of the model using the skew tent map scenario as a normal form. Nonetheless, we show that some border collision bifurcations which play a fundamental role for the asymptotic behavior of the map essentially pertain to its bimodal shape. Among them we highlight the ones that lead, for some specific parameter values, to the coexistence of two chaotic attractors. Moreover, we identify degenerate border collision bifurcations responsible for the peculiar shapes of the chaotic attractors that distinguish the 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.

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.

The new economic geography (NEG) aims to explain long-term patterns in the spatial allocation of industrial activities. It stresses that endogenous economic processes may enlarge small historic differences leading to quite different regional patterns—history matters for the long-term geographical distribution of economic activities. A pivotal element is that productive factors move to another region whenever the anticipated remuneration is higher in that region. Given the long-term nature of NEG analyses and the crucial role of expectations, it is astonishing that most of the existing models assume only naïve or myopic expectations. However, a recent stream of the literature in behavioral and experimental economics shows that agents often use expectational heuristics, such as trend extrapolating and trend reverting rules. We introduce such expectations formation hypotheses into a NEG model formulated in discrete time. This modification leads to a system of two nonlinear difference equations (corresponding, in the language of dynamical systems theory, to a 2-dimensional piecewise smooth map) and thus enriches the possible dynamic patterns: with trend extrapolating (reverting) the symmetric equilibrium is less (more) stable; and it may lose stability only via a flip bifurcation (or also via a Neimark–Sacker bifurcation) giving rise to a period-doubling cascade (or also to quasi-periodic orbits). In both cases, complex behavior is possible; multistability, that is, the coexistence of locally stable equilibria, is pervasive; and border-collision bifurcations are also allowed. In this sense, our analysis corroborates some of the basic insights of the NEG

In this paper we extend the discrete time Footloose Capital model analyzed in Commendatore et al. (Nonlinear Dyn Psychol Life Sci 11(2):267-289, 2007) by introducing "first nature firms", i.e., firms that use locally specific blueprints and, therefore, are immobile. Due to the presence of first nature firms (symmetrically distributed across the regions), the central dynamic map becomes a piecewise differentiable function: in addition to "standard" flip and pitchfork bifurcations also border collision bifurcations are possible and instances of multistability may emerge. Our analysis confirms and extends the results of Commendatore et al. (2007): (1) continuous time formulation hides complex dynamics patterns; (2) asymmetric distributions of industrial activity can be endogenously generated and are path dependent. © 2011 Springer Science+Business Media, LLC.

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.

In this work a simple financial model with fundamentalists and imitators is being considered. In order to describe the price dynamics of the heterogeneous stock market, a synergetic approach is used and some global bifurcations arising in the model are being studied. It is shown that the fundamental equilibrium point P* may be destabilized through a subcritical Neimark-Sacker bifurcation and that two invariant closed curves, one attracting and one repelling, appear when P* is still stable. This particular bifurcation scenario allows us to show some noticeable features of the market that emerge when the imitation effect is emphasized. Among these features are, for instance, the volatility clusters associated with the presence of multistability (i.e. coexistence of attractors) and the hysteresis phenomenon. © 2008 Springer Science+Business Media, LLC.

Empirical evidence has suggested that, facing different trading strategies and complicated decision, the proportions of agents relying on particular strategies may stay at constant level or vary over time. This paper presents a simple "dynamic market fraction" model of two groups of traders, fundamentalists and trend followers, under a market maker scenario. Market mood and evolutionary adaption are characterized by fixed and adaptive switching fraction among two groups, respectively. Using local stability and bifurcation analysis, as well as numerical simulation, the role played by the key parameters in the market behaviour is examined. Particular attention is paid to the impact of the market fraction, determined by the fixed proportions of confident fundamentalists and trend followers, and by the proportion of adaptively rational agents, who adopt different strategies over time depending on realized profits. © 2005 Elsevier Ltd. All rights reserved.

In this work we analyze a financial market model based on the imitative behaviour between n traders. Every agent has an investment attitude which can be influenced by other agents’ opinions, introducing in this way a “contagion process”. We formalize this process using the “synergetic approach”, leading to the construction of a noninvertible discrete dynamical map. Some bifurcations involving the fundamental equilibrium point and the structure of its basins are analyzed.

The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z(0)) each point having no real preimage, and (ii) (denoted by Z(2)) each point having two real preimages. The regions Z(0), Z(2) are separated by the critical curve LC, locus of points having two coincident preimages. Z(2) is made up of two sheets (giving rise to a plane foliation) joining on LC, each being associated with a well-defined inverse of the map. The considered maps family is structurally unstable. For a wide choice of the parameter space it generates a singular foliation in the sense that the region Z(2) is separated into two zones, Z(2)' and Z(2)", inside which the two preimages do not have the same qualitative behavior. Moreover, the boundary between Z(2)' and Z(2)" is made up of points having only one real preimage at finite distance, the second one being at infinity. This situation gives rise to a nonclassical homoclinic bifurcation. The maps of the family have another important feature: their inverses present a denominator which vanish along a line of the plane. This has a great consequence on a chaotic attractor structure, when it exists. The imbedding of the map into a wider structurally stable family, generating regions Z(k) (k = 1, 3 being the number of real preimages), permits to understand the foliation nature when the imbedding parameter cancels leading to the structurally unstable map.

The object of the present work is to study a one-dimensional nonautonomous equation (a Mann iteration) with geometric weights. Its study amounts to that of a two-dimensional autonomous rational map having a vanishing denominator in a non classical case. A map with a vanishing denominator possesses a set of points, called the prefocal set, with particular properties, that is, whose preimages are related to a single point called the focal point. A contact of a basin boundary with a prefocal set is a bifurcation giving rise to a qualitative change of this boundary by creation of loops at the focal point. With respect to classical cases, the particularity of the map family considered here consists in the fact that the prefocal set is at infinite distance, i.e. it is the set of points (x,±) x varying in , the related focal points being at finite distance. The main result is related to the formulation of a one-to-one correspondence between arcs passing through the focal points and the points at infinity of type (x,±). A change of variables facilitates the study by relocating focal points and prefocal sets at finite distance.

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)

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. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.