1. 


2. 
 Agliari, Anna, et al.
(författare)

Global bifurcations of basins in a triopoly game
 2002

Ingår i: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering.  02181274. ; 12:10, s. 21752207

Tidskriftsartikel (refereegranskat)abstract
 A Cournot model based on bounded inverse demand function and constant marginal production costs is studied. The case of three producers is considered and the adjustment process reduces to a threedimensional noninvertible map in the output of competitors. The analysis of the dynamical behavior of the map is performed by the "critical curve method", extended to the critical surfaces in 3D. By this method, we explain the different bifurcations in the basins of attraction and in the attracting sets. In particular, given the economic application, feasible trajectories are focused, starting from the simple situation of two identical producers and extending the results to the generic case.


3. 
 Agliari, Anna, et al.
(författare)

The dynamics of a triopoly Cournot game
 2000

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 11:15, s. 25312560

Tidskriftsartikel (refereegranskat)abstract
 This paper reconsiders the Cournot oligopoly (noncooperative) game with isoelastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for noninvertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three firms are identical is focused.


4. 
 Gallegati, Mauro, et al.
(författare)

Hicks’ trade cycle revisited : cycles and bifurcations
 2003

Ingår i: Mathematics and Computers in Simulation.  03784754. ; 63:6, s. 505527

Tidskriftsartikel (refereegranskat)abstract
 In the Trade Cycle, Hicks introduced the idea that endogenous fluctuations could be coupled with a growth process via nonlinear processes. To argue for this hypothesis, Hicks used a piecewiselinear model. This paper shows the need for a reinterpretation of Hicks’ contribution in the light of a more careful mathematical investigation. In particular, it will be shown that only one bound is needed to have non explosive outcome if the equilibrium point is an unstable focus. It will also be shown that when the fixed point is unstable the attracting set has a particular structure: It is a onedimensional closed invariant curve, made up of a finite number of linear pieces, on which the dynamics are either periodic or quasiperiodic. The conditions under which the model produces periodic or quasiperiodic trajectories and the related bifurcations as a function of the main economic parameters are determined.


5. 
 Puu, Tönu, 1936, et al.
(författare)

A Hicksian multiplieraccelerator model with floor determined by capital stock
 2005

Ingår i: Journal of Economic Behavior and Organization.  01672681. ; 56:3, s. 331348

Tidskriftsartikel (refereegranskat)abstract
 This article reconsiders the Hicksian multiplieraccelerator model with the “floor” related to the depreciation on actual capital stock. Through the introduction of the capital variable, a growth trend is created endogenously by the model itself, along with growth rate oscillations around it. The “ceiling” can be dispensed with altogether. As everything is growing in such a model, a variable transformation is introduced to focus relative dynamics of the income growth rate and the actual capital output ratio.


6. 
 Sushko, Iryna, et al.
(författare)

Regular and chaotic growth in a Hicksian Floor/ceiling model
 2010

Ingår i: Journal of Economic Behavior and Organization.  Elsevier.  01672681. ; 75:1, s. 7794

Tidskriftsartikel (refereegranskat)abstract
 In some previous papers the present authors reassembled the Hicksian trade cycle model in a new way. The floor was tied to depreciation on capital, itself the cumulative sum of past net investments, for which the principle of acceleration provided an explanation. Hence no alien elements were needed to include capital, and so close the system. The resulting model created a growth trend along with growth rate cycles, which could be periodic or quasiperiodic. In the current paper, the ceiling, using capital stock as a capacity limit for production, is added. It then turns out that pure growth no longer exists, and chaos and multistability become possible, which they were not in the previous model. A variety of bifurcation scenarios is explored, and a full understanding of the working of the fourpiece, originally threedimensional, piecewise smooth map, is attained, using a reduction to a onedimensional return map.


7. 
 Sushko, Irina, et al.
(författare)

Tongues of periodicity in a family of twodimensional discontinuous maps of real Mobius type
 2004

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 21:2, s. 403412

Tidskriftsartikel (refereegranskat)abstract
 In this paper we consider a twodimensional piecewisesmooth discontinuous map representing the socalled "relative dynamics" of an Hicksian business cycle model. The main features of the dynamics occur in the parameter region in which no fixed points at finite distance exist, but we may have attracting cycles of any periods. The bifurcations associated with the periodicity tongues of the map are studied making use of the firstreturn map on a suitable segment of the phase plane. The bifurcation curves bounding the periodicity tongues in the parameter plane are related with saddlenode and bordercollision bifurcations of the firstreturn map. Moreover, the particular "sausages structure" of the bifurcation tongues is also explained.


8. 
 Tramontana, Fabio, et al.
(författare)

Cournot duopoly when the competitors operate multiple production plants
 2009

Ingår i: Journal of Economic Dynamics and Control.  01651889. ; 33:1, s. 250265

Tidskriftsartikel (refereegranskat)abstract
 This article considers a Cournot duopoly under an isoelastic demand function and cost functions with builtin capacity limits. The special feature is that each firm is assumed to operate multiple plants, which can be run alone or in combination. Each firm has two plants with different capacity limits, so each has three cost options, the third being to run both plants, dividing the load according to the principle of equal marginal costs. As a consequence, the marginal cost functions come in three disjoint pieces, so the reaction functions, derived on basis of global profit maximization, may also consist of disjoint pieces. This is reflected in a particular bifurcation structure, due to bordercollision bifurcations and to particular basin boundaries, related to the discontinuities. It is shown that stable cycles may coexist, and the nonexistence of unstable cycles constitutes a new property. We also compare the coexistent short periodic solutions in terms of the resulting real profits.


9. 
 Tramontana, Fabio, et al.
(författare)

Global bifurcations in a piecewisesmooth Cournot duopoly game
 2010

Ingår i: Chaos, Solitons & Fractals.  Oxford : PergamonElsevier Science Ltd.  09600779. ; 43:12, s. 1524

Tidskriftsartikel (refereegranskat)abstract
 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 [2]. The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the NeimarkSacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties differ 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. (C) 2010 Elsevier Ltd. All rights reserved.


10. 
 Tramontana, Fabio, et al.
(författare)

Mathematical properties of a discontinuous CournotStackelberg model
 2011

Ingår i: Chaos, Solitons & Fractals.  Oxford : PergamonElsevier Science Ltd.  09600779. ; 44:13, s. 5870

Tidskriftsartikel (refereegranskat)abstract
 The object of this work is to perform the global analysis of a recent duopoly model which couples the two points of view of Cournot and Stackelberg [17,18]. 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 investigated 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 differ 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.

