Double route to chaos in an heterogeneous triopoly game

We move from a triopoly game with heterogeneous players (E.M.Elabassy et al., 2009. Analysis of nonlinear triopoly game with heterogeneous players. Computers and Mathematics with Applications 57, 488-499). We remove the nonlinearity from the cost function and introduce it in the demand function. We also introduce a different decisional mechanism for one of the three competitors. A double route to complex dynamics is shown to exist, together with the possibility of multistability of different attractors, requiring a global analysis of the dynamical system.Triopoly game; Heterogeneous players; Global analysis

Controlling Chaos Through Local Knowledge

We propose an duopoly game where quantity-setting firms have incomplete information about the demand function. In each time step, they solve a profit maximization problem assuming a linear local approximation of the demand function. In particular, we construct an example using the well known duopoly Puu's model with isoelastic demand function and constant marginal costs. An explicit form of the dynamical system that describes the time evolution of the duopoly game with boundedly rational players is given. The main result is the global stability of the system.Cournot duopoly, incomplete information, isoelastic demand function, time evolution, boundedly rational players.

The dynamics of the NAIRU model with two switching regimes

We consider a model of inflation and unemployment proposed in Ferri et al. (JEBO, 2001), in which the dynamics are described by a discontinuous piecewise linear map, made up of two branches. We shall show that the bounded dynamics may be classified in two cases: we may have either regular dynamics with stable cycles of any period or quasiperiodic trajectories, or only chaotic dynamics (pure chaos in which a unique absolutely continuous invariant ergodic measure exists, and structurally stable),in a rich variety of cyclical chaotic intervals. The main results are the analytical formulation of the border collision bifurcation curves, through which we give a complete picture of the possible outcomes of the model.Phillips curve, Regime switching, NAIRU, Nonlinearities, Discontinuous maps.

A simple financial market model with chartists and fundamentalists: market entry levels and discontinuities

We present a simple financial market model with interacting chartists and fundamentalists. Since some of these speculators only become active when a certain misalignment level has been crossed, the dynamics are driven by a discontinuous piecewise linear map. The model endogenously generates bubbles and crashes and excess volatility for a broad range of parameter values - and thus explains some key phenomena of financial markets. Moreover, we provide a complete analytical study of the model's dynamical system. One of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge.financial market crisis; bull and bear market dynamics; discontinuous piecewise linear maps; border-collision bifurcations; period adding scheme.

New properties of the Cournot duopoly with isoelastic demand and constant unit costs.

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu (1991). The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark-Shacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties di¤er significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.Cournot duopoly, isoelastic demand function, multistability, border-collision bifurcations.

Mathematical Properties of a Combined Cournot-Stackelberg model.

The object of this work is to perform the global analysis of a new duopoly model which couples the two points of view of Cournot and Stackelberg. The Cournot model is assumed with isoelastic demand function and unit costs. The coupling leads to discontinuous reaction functions, whose bifurcations, mainly border collision bifurcations, are investigates as well as the global structure of the basins of attraction. In particular, new properties are shown, associated with the introduction of horizontal branches, which di¤er significantly when the constant value is zero or positive and small. The good behavior of the model with positive constant is proved, leading to stable cycles of any period.Cournot-Stackelberg duopoly, Isoelastic demand function, Discontinuous reaction functions, Multistability, Border-collision bifurcations.

On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders.

We develop a financial market model with heterogeneous interacting agents: market makers adjust prices with respect to excess demand, chartists believe in the persistence of bull and bear markets and fundamentalists bet on mean reversion. Moreover, speculators trade asymmetrically in over and undervalued markets and while some of them determine the size of their orders via linear trading rules others always trade the same amount of assets. The dynamics of our model is driven by a one-dimensional discontinuous map. Despite the simplicity of our model, analytical, graphical and numerical analysis reveals a surprisingly rich set of interesting dynamical behaviors.Financial markets, heterogeneous agents, technical and fundamental analysis, nonlinear dynamics, discontinuous map, bifurcation analysis.

Bifurcation Curves in Discontinuous Maps

Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one or two-dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modied by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree), for the family of maps considered, that depends on five parameters.iterated piecewise linear functions, discrete-time dynamic models, bifurcation curves.

A 'bull and bear' model of interacting financial markets. Part I: dynamics in one and two dimensions

We develop a three-dimensional nonlinear dynamic model in which the stock markets of two countries are linked through the foreign exchange market. Connections are due to the trading activity of heterogeneous speculators. Using analytical and numerical tools, we seek to explore how the coupling of the markets may affect the emergence of 'bull and bear' market dynamics. The dimension of the model can be reduced by restricting investors' trading activity, which enables the dynamic analysis to be performed stepwise, from low-dimensional cases up to the full three-dimensional model. In Part I of our paper, we focus on the one and two-dimensional case.Heterogeneous speculators, bull and bear markets, nonlinear dynamics, homoclinic bifurcations.

A 'bull and bear' model of interacting financial markets. Part II: dynamics in three dimensions

In the fi…rst part of our paper we proposed a three-dimensional nonlinear dynamic model of interacting stock and foreign exchange markets, jointly driven by the speculative activity of heterogeneous investors. We focused, in particular, on the typical 'bull and bear' scenario that emerges from simpli…ed one- and two-dimensional settings. The goal of this part of the paper is to provide a global analysis of the dynamics of the full model. As it turns out, the results we obtained in the fi…rst part may serve as a road map to develop an initial understanding of the much more complicated three-dimensional model.Heterogeneous speculators, bull and bear markets, nonlinear dynamics, homoclinic bifurcations.