1. 
 Agliari, Anna, et al.
(författare)

Global Bifurcations in Duopoly when the Cournot Point is Destabilized through a Subcritical Neimark Bifurcation
 2003

Rapport (övrigt vetenskapligt)abstract
 An adaptive oligopoly model, where the demand function is isoelastic and the competitors operate under constant marginal costs, is considered. The Cournot equilibrium point then loses stability through a subcritical Neimark bifurcation. The present paper focuses some global bifurcations, which precede the Neimark bifurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.


2. 
 Puu, Tönu, 1936, et al.
(författare)

A Business Cycle Model with Cubic Nonlinearity
 2002

Rapport (övrigt vetenskapligt)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.


3. 
 Puu, Tönu, 1936, et al.
(författare)

A Hicksian MultiplierAccelerator Model with Floor Determined by Capital Stock and Tongues of Periodicity in a Family of Twodimensional Discontinuous Maps of Real Möbius Type
 2003

Rapport (övrigt vetenskapligt)abstract
 A Hicksian MultiplierAccelerator ModelThis 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 the end of focusing relative dynamics of the income growth rate and the actual capital output ratio.Tongues of Periodicity in a Family of Twodimensional Discontinuous Maps of Real Möbius TypeIn this paper we consider a twodimensional piecewisesmooth discontinuous map representing the socalled "relative dynmics" 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 first return 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.


4. 
 Puu, Tönu, 1936, et al.
(författare)

Cournot Duopoly when the Competitors Operate under Capacity Constraints
 2002

Rapport (övrigt vetenskapligt)abstract
 The paper considers Cournot duopoly where the competitors have capacity constraints. An isoelastic demand function, which always results when consumers maximise utility functions of the CobbDouglas type, is used. It has been demonstrated that isoelastic demand, combined with constant marginal costs, results in complex dynamics. The purpose of the present paper is to reconsider the case, using in stead cost functions with capacity limits. This is a point on which Edgeworth insisted as important. Comparisons between cases of few large and many small competitors cannot be made when firms have constant returns and hence are all infinitely large in potential.


5. 
 Puu, Tönu, 1936
(författare)

On the Genesis of Hexagonal Shapes
 2004

Rapport (övrigt vetenskapligt)abstract
 Hexagonal shapes for market areas have been dominant in spatial economics ever since Christaller observed them and Lösch explained their emergence in terms of optimality. It is observed in the following that, though hexagons are best, the differences to for instance squares in terms of efficiency are negligibly small. As the existence of hexagonal shapes  in the space economy, as well as in beehives or other patterns of the physical world cannot be denied  a different cause for their emergence is proposed. Such an explanation is structural stability, which allows three market areas, but not four or six, to come together in each common vertex. The focus is hence the way market areas are organized in space, rather than their shape.


6. 


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

Business Cycles
 2002

Rapport (övrigt vetenskapligt)abstract
 The Hicksian multiplieraccelerator model with the original floorroof limits to investments is studied for the case of two regions linked by interregional trade. The result is a piecewise linear continuous four dimensional map, which is reduced to three dimensions through the choice of an appropriate distributed consumption lag. The attractors, basins, and bifurcations of the map are studied under assumption of a certain symmetry between the regions. The NeimarkHopf bifurcation for piecewise linear maps is described in detail which gives rise to the appearance of an attracting closed invariant curve homeomorphic to a circle. The structure of resonance regions in the parameter space are investigated.


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

Mathematical properties of a combined CournotStackelberg model
 2010

Rapport (övrigt vetenskapligt)abstract
 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.


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

New properties of the Cournot duopoly with isoelastic demand and constant unit costs
 2010

Rapport (övrigt vetenskapligt)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 (1991). The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the NeimarkShacker 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.

