1. 
 Puu, Tönu, 1936, et al.
(författare)

A business cycle model with cubic nonlinearity
 2004

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 19:3, s. 597612

Tidskriftsartikel (refereegranskat)abstract
 This paper deals with a simple multiplieraccelerator model of the business cycle, including a cubic nonlinearity. The corresponding two dimensional iterative map is represented in terms of its bifurcation diagram in parameter space. A number of bifurcation sequences for attractors and their basins are studied.


2. 
 Puu, Tönu, 1936
(författare)

On the Stability of Cournot Equilibrium when the Number of Competitors Increases
 2008

Ingår i: Journal of Economic Behavior and Organization.  01672681. ; 66:34, s. 445456

Tidskriftsartikel (refereegranskat)abstract
 This article reconsiders whether the Cournot equilibrium really becomes a perfect competition equilibrium when the number of competitors goes to infinity. It has been questioned whether the equilibrium remains stable with an increasing number of firms. Contraindications were given for linear and for isoelastic demand functions. However, marginal costs were then taken as constant, which means adding more potentially infinitesized firms. As we want to compare cases with few large firms to cases with many small firms, the model is tuned so as to incorporate capacity limits, decreasing with an increasing number of firms. Then destabilization is avoided.


3. 
 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.


4. 
 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.


5. 
 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.


6. 


7. 
 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.


8. 
 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.


9. 
 Canovas, Jose S, et al.
(författare)

Detecting Chaos in a Duopoly Model via Symbolic Dynamics
 2010

Ingår i: Discrete and continuous dynamical systems. Series B.  Springfield, USA : American Institute of Mathematical Sciences.  15313492. ; 13:2, s. 269278

Tidskriftsartikel (refereegranskat)abstract
 This paper considers a Cournot duopoly model assuming isoelastic demand and smooth cost functions with builtin capacity limits. When the firms cannot obtain positive profits they are assumed to choose small "standby" outputs rather than closing down, in order to avoid substantial fitting up costs when market conditions turnout more favorable. It is shown that the model provides chaotic behavior. In particular, the system has positive topological entropy and hence the map is chaotic in the LiYorke sense. Moreover, chaos is not only topological but also physically observable.


10. 
 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.

