Laura Gardini

Laura Gardini
Università degli Studi di Urbino Carlo Bo · Department of Economics, Society, Politics (DESP)

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298
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Publications

Publications (298)
Article
We develop a behavioral stock market model in which a market maker adjusts stock prices with respect to the orders of chartists, fundamentalists and sentiment traders. We analytically prove that the mere presence of sentiment traders, i.e. traders who optimistically buy stocks in rising markets and pessimistically sell stocks in falling markets, co...
Article
Policymakers around the world impose some form of capital gains taxes to foster the stability of financial markets. Unfortunately, there is no clarity on the effects of capital gains taxes. Based on a stylized behavioral asset-pricing model highlighting the trading activity of extrapolating speculators, we show that policymakers may involuntary des...
Article
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This paper contributes to studying the bifurcations of closed invariant curves in piecewise-smooth maps. Specifically, we discuss a border collision bifurcation of a repelling resonant closed invariant curve (a repelling saddle-node connection) colliding with the border by a point of the repelling cycle. As a result, this cycle becomes attracting a...
Preprint
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A two-dimensional noninvertible map T : (x', y') = (x(1 + at-y), (1-t)y +tx^2) proposed by Lorenz in 1989, depending on the two parameters a and t, is reconsidered. We show the two different bifurcation scenarios occurring for a > 0 and a < 0. Two particular degenerate cases are investigated, at t= 1 and t= 2; describing the related bifurcations as...
Article
Full-text available
Corporate demand for cash is related to a number of firm-specific characteristics, like the presence of transaction costs, information asymmetry in credit markets, uncertainty and risk aversion. The purpose of this paper is to build a dynamic model that describes the potential chaotic effects of the accumulation of cash by firms over a prolonged pe...
Article
Full-text available
We propose a prototype model of market dynamics in which all functional relationships are linear. We take into account three borders, defined by linear functions, that are intrinsic to the economic reasoning: non-negativity of prices; downward rigidity of capacity (depreciation); and a capacity constraint for the production decision. Given the line...
Article
We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two sad...
Article
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We study the discrete dynamical system defined on a subset of $$R^2$$ R 2 given by the iterates of the secant method applied to a real polynomial p . Each simple real root $$\alpha $$ α of p has associated its basin of attraction $${\mathcal {A}}(\alpha )$$ A ( α ) formed by the set of points converging towards the fixed point $$(\alpha ,\alpha )$$...
Article
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In this work, we reconsider the dynamics of a few versions of the classical Samuelson’s multiplier–accelerator model for national economy. First we recall that the classical one with constant governmental expenditure, represented by a linear second-order difference equation, is able to generate oscillations converging to the equilibrium for a wide...
Article
We reconsider the multiplier–accelerator model of business cycles, first introduced by Samuelson and then modified by many authors. The original simple model, besides damped oscillations, also leads to divergent oscillations. To avoid this, we introduce two different types of governmental expenditures leading a two-dimensional continuous piecewise...
Article
We reconsider the well-known conditions which guarantee the roots of a third-degree polynomial to be inside the unit circle. These conditions are important in the stability analysis of equilibria and cycles of three-dimensional systems in discrete time. A simplified set of conditions determine the boundary of the stability region and we prove which...
Article
A particular system of two-dimensional Lotka-Volterra maps, Ta:(x′,y′)=(x(a−x−y),xy), unfolding a map originally proposed by Sharkovsky for a=4, is considered. We show the routes to chaos leading to the dynamics of map T4. For map T4 we show that even if the stable set of the origin O includes a set dense in an invariant area, the only homoclinic p...
Article
We consider a discrete dynamical system, a two-dimensional real map which represents a one-dimensional complex map. Depending on the parameters, its bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annular regions or disks. We show that on such invariant sets the trajectories are always either periodic...
Article
In this work, we consider a family of Lotka–Volterra maps [Formula: see text] for [Formula: see text] and [Formula: see text] which unfold a map originally proposed by Sharkosky for [Formula: see text] and [Formula: see text]. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region...
Article
Full-text available
An asset pricing model with chartists, fundamentalists and trend followers is considered. A market maker adjusts the asset price in the direction of the excess demand at the end of each trading session. An exogenously given fundamental price discriminates between a bull market and a bear market. The buying and selling orders of traders change movin...
Article
We consider the dynamics of a family of two-dimensional piecewise linear maps at the transition between the invertible and non-invertible cases. This leads to a degeneracy consisting of a half plane which is mapped onto a straight line, the critical line LC. In these regimes the ω-limit set of the trajectories must be on the images of some segment...
Preprint
Full-text available
We reconsider the well-known Schur/Samuelson conditions, which guarantee the roots of a third-degree polynomial to be inside the unit circle. These conditions are important in the stability analysis of equilibria and cycles of three-dimensional systems in discrete time. We derive a simplified set of conditions that determine the boundary of the sta...
Preprint
Full-text available
In this work we reconsider the dynamics of a few versions of the classical Samuelson's multiplier- accelerator model for national economy. First we recall that the classical one with constant governmental expenditure, represented by a linear second-order difference equation, is able to generate oscillations con- verging to the equilibrium for a wid...
Preprint
Full-text available
We study the discrete dynamical system defined on a subset of $R^2$ given by the iterates of the secant method applied to a real polynomial $p$. Each simple real root $\alpha$ of $p$ has associated its basin of attraction $\mathcal A(\alpha)$ formed by the set of points converging towards the fixed point $(\alpha,\alpha)$ of $S$. We denote by $\mat...
Article
Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifur...
Chapter
We consider a learning mechanism where expected values of an economic variable in discrete time are computed in the form of a weighted average that exponentially discounts older data. Also adaptive expectations can be expressed as weighted sums of infinitely many past states, with exponentially decreasing weights, but these are not averages since t...
Experiment Findings
Full-text available
In this work we consider a family of Lotka-Volterra maps (x_0, y_0) = (x(a x y), bxy) for a > 1 and b > 0 which unfold a map originally proposed by Sharkosky for a = 4 and b = 1. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region y < 0. Some properties and bifurcations are des...
Experiment Findings
Full-text available
A particular system of two-dimensional Lotka-Volterra maps, Ta : (x 0 ; y 0) = (x(a x y); xy); unfolding a map originally proposed by Sharkovsky for a = 4; is considered. We show the routes to chaos leading to the dynamics of map T4 : For map T4 we show that even if the stable set of the origin O includes a set dense in an invariant area, the only...
Article
In this work, we consider a continuous two-dimensional piecewise linear family of maps in which the fixed point and other cycles undergo a center bifurcation, the analogue of a Neimark–Sacker bifurcation, which has a particular structure when occurring in piecewise linear maps. Our goal is to determine and characterize the occurrence of a center bi...
Article
After the seminal works by Schelling, several authors have considered models representing binary choices by different kinds of agents or groups of people. The role of the memory in these models is still an open research argument, on which scholars are investigating. The dynamics of binary choices with impulsive agents has been represented, in the r...
Article
We study a four-parameter family of 2D piecewise linear maps with two discontinuity lines. This family is a generalization of the discrete-time version of the fashion cycle model by Matsuyama, which was originally formulated in continuous time. The parameter space of the considered map is characterised by quite a complicated bifurcation structure f...
Article
Full-text available
Recent publications revisit the growth model proposed by Matsuyama (”Growing through cycles”, Econometrica 1999), presenting new economic interpretations of the system as well as new results on its dynamics described by a one-dimensional piecewise smooth map (also called M-map). The goal of the present paper is to give the rigorous proof of some re...
Chapter
In the present chapter we recall some basic concepts and results associated with the local and global bifurcations of attractors and their basins in smooth and nonsmooth noninvertible maps, continuous and discontinuous. Such maps appear to be important both from theoretical and applied points of view. Using numerous examples we show that noninverti...
Article
This paper extends Matsuyama’s endogenous credit cycle model to account for recent findings on therole of credit market sentiments. The benchmark model uses a parsimonious financial friction specifica-tion in the form of a pledgeability parameter, which indicates how much of the revenue borrowers canpledge for credit. We endogenize this parameter b...
Article
Full-text available
The Boros-Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to the convergence of them. In the paper, we study the dynamics of a one-parameter family of maps which unfolds the Boros-Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros...
Preprint
Full-text available
In this work we consider a continuous two-dimensional piecewise linear family of maps in which the fixed point and other cycles undergo a center bifurcation, the analogue of a Neimark-Sacker bifurcation, which has a particular structure when occurring in piecewise linear maps. Our goal is to determine and characterize the occurrence of a center bif...
Article
We reconsider a regime-switching model of credit frictions which has been proposed in a general framework by Matsuyama for the case of Cobb–Douglas production functions. This results in a piecewise linear map with two discontinuity points and all three branches having the same slope. We offer a complete characterization of the bifurcation structure...
Preprint
Full-text available
We reconsider a regime-switching model of credit frictions which has been proposed in a general framework by Matsuyama for the case of Cobb-Douglas production functions. This results in a piecewise linear map with two discontinuity points and all three branches having the same slope. We offer a complete characterization of the bifurcation structure...
Article
Full-text available
In this work we give necessary and sufficient conditions for a discontinuous expanding map f of an interval into itself, made up of N pieces, to be chaotic in the whole interval. For N = 2 we consider the class of expanding Lorenz maps, for N 3 a class of maps whose internal branches are onto, called Baker-like. We give the necessary and sufficient...
Article
The paper proposes an evolutionary version of a Schelling-type dynamic system to model the patterns of residential segregation when two groups of people are involved. The payoff functions of agents are the individual preferences for integration which are empirically grounded. Differently from Schelling's model, where the limited levels of tolerance...
Article
We consider a discrete-time version of the continuous-time fashion cycle model introduced in Matsuyama, 1992. Its dynamics are defined by a 2D discontinuous piecewise linear map depending on three parameters. In the parameter space of the map periodicity, regions associated with attracting cycles of different periods are organized in the period add...
Preprint
Full-text available
The Boros-Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to the convergence of them. In the paper, we study the dynamics of a one-parameter family of maps which unfolds the Boros-Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros...
Article
We investigate the dynamics of a family of one-dimensional linearpower maps. This family has been studied by many authors mainly in the continuous case, associated with Nordmark systems. In the discontinuous case, which is much less studied, the map has vertical and horizontal asymptotes giving rise to new kinds of border collision bifurcations. We...
Preprint
Full-text available
We consider a discrete-time version of the continuous-time fashion cycle model introduced in Matsuyama, 1992. Its dynamics are de…ned by a 2D discontinuous piecewise linear map depending on three parameters. In the parameter space of the map periodicity regions associated with attracting cycles of di¤erent periods are organized in the period adding...
Preprint
Full-text available
The paper proposes an evolutionary version of a Schelling-type dynamic system to model the patterns of residential segregation when two groups of people are involved. The payo¤ functions of agents are the individual preferences for integration which are empirically grounded. Di¤erently from Schelling's model, where the limited levels of tolerance a...
Article
Full-text available
We consider a two-dimensional continuous noninvertible piecewise smooth map, which characterizes the dynamics of innovation activities in the two-country model of trade and product innovation proposed in [7]. This two-dimensional map can be viewed as a coupling of two one-dimensional skew tent maps, each of which characterizes the innovation dynami...
Chapter
We study the dynamics of a one dimensional discontinuous linear-power map. It has a vertical asymptote giving rise to new kinds of border collision bifurcations. We explain the peculiar periods of attracting cycles, appearing due to cascades of alternating smooth and nonsmooth bifurcations. Robust unbounded chaotic attractors are also described.
Chapter
In the Cournot duopoly game with unimodal piecewise-linear reaction functions (tent maps) proposed by Rand (J Math Econ, 5:173–184, 1978) to show the occurrence of robust chaotic dynamics, a maximum production constraint is imposed in order to explore its effects on the long run dynamics. The presence of such constraint causes the replacement of ch...
Article
Full-text available
A dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despit...
Article
Full-text available
The main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensional piecewise monotone maps. We recall some definitions known from the theory of smooth maps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation cle...
Article
We consider a family of one-dimensional continuous piecewise smooth maps with monotone increasing and monotone decreasing branches. It is associated with a credit cycle model introduced by Matsuyama, under the assumption of the Cobb-Douglas production function. We offer a detailed analysis of the dynamics of this family. In particular, using the sk...
Article
We revisit the model of endogenous credit cycles by Matsuyama (2013, Sections 2-4). First, we show that the same dynamical system that generates the equilibrium trajectory is obtained under a much simpler setting. Such a streamlined presentation should help to highlight the mechanism through which financial frictions cause instability and recurrent...
Article
In this work we describe some properties and bifurcations which occur in a family of linear-power maps typical in Nordmark' systems. The continuous case has been investigated by many authors since a few years, while the discontinuous case has been considered only recently. In particular, having a vertical asymptote, it gives rise to new kinds of bi...
Article
Full-text available
We consider a family of one-dimensional discontinuous invertible maps from an application in engineering. It is defined by a linear function and by a hyperbolic function with real exponent. The presence of vertical and horizontal asymptotes of the hyperbolic branch leads to particular codimension-two border collision bifurcation (BCB) such that if...
Article
Full-text available
The goal of the present paper is to collect the results related to dynamics of a one-dimensional piecewise linear map widely known as the skew tent map. These results may be useful for the researchers working on theoretical and applied problems in the field of nonsmooth dynamical systems. In particular, we propose the complete description of the bi...
Article
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In this paper we consider a Schelling-type segregation model with two groups of agents that differ in some aspects, such as religion, political affiliation or color of skin. The first group is identified as the local population, while the second group is identified as the newcomers, whose members want to settle down in the city or country, or more...
Article
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In this work we consider a class of generalized piecewise smooth maps, proposed in the study of engineering models. It is a class of one-dimensional discontinuous maps, with a linear branch and a nonlinear one, characterized by a power function with a term and a vertical asymptote. The bifurcation structures occurring in the family of maps are clas...
Article
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Chaos based communication represents an attractive solution in order to design secure multiple access digital communication systems. In this paper we investigate the use of piece-wise linear chaotic maps as chaotic generators combined, on the receiver side, with Chebyshev Polynomial Kalman Filters in a dual scheme configuration for demodulation pur...
Article
Abstract In this paper we prove the existence of full measure unbounded chaotic attractors which are persistent under parameter perturbation (also called robust). We show that this occurs in a discontinuous piecewise smooth one-dimensional map f, belonging to the family known as Nordmark's map. To prove the result we extend the properties of a full...
Article
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In this work we continue the study of a family of 1D piecewise smooth maps, defined by a linear function and a power function with negative exponent, proposed in engineering studies. The range in which a point on the right side is necessarily mapped to the left side, and chaotic sets can only be unbounded, has been already considered. In this work...