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  • 1.
    Agliari, Anna
    et al.
    Catholic University in Milan, Italy/University of Parma, Italy.
    Gardini, Laura
    University of Parma, Italy/Istituto di Scienze Economiche, University of Urbino, Italy.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    The dynamics of a triopoly Cournot game2000Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 11, nr 15, s. 2531-2560Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic 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 non-invertible 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.

  • 2. Banerjee, Malay
    et al.
    Zhang, Lai
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Influence of discrete delay on pattern formation in a ratio-dependent prey-predator model2014Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 67, s. 73-81Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large. 

  • 3.
    Cánovas, José S.
    et al.
    Dpto. Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Marín, Manuel Ruíz
    Dpto. Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena, Cartagena, Spain.
    A method for studying non-autonomous affine maps: An application to a two-region business cycle model2007Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 34, nr 4, s. 1285-1298Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper is devoted to the study of the dynamics of a non-autonomous affine system. We apply the results to a two-region business cycle model of the Samuelson [Interactions between the multiplier analysis and the principle of acceleration. Rev Econ Statist 1939;21:75–8] multiplier–accelerator type.

  • 4.
    Cánovas, José S.
    et al.
    Departamento Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Ruíz, Manuel
    Departamento Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena, Cartagena, Spain.
    The Cournot–Theocharis problem reconsidered2008Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 37, nr 4, s. 1025-1039Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In 1959 Theocharis [Theocharis RD. On the stability of the Cournot solution on the oligopoly problem. Review of economic Studies 1959;27:133–4] showed that with linear demand and constant marginal costs Cournot equilibrium is destabilized when the competitors become more than three. With three competitors the Cournot equilibrium point becomes neutrally stable, so, even then, any perturbation throws the system into an endless oscillation. Theocharis’s argument was in fact proposed already in 1939 by Palander [Palander T. Konkurrens och marknadsjämvikt vid duopol och oligopol. Ekonomisk Tidskrift 1939;41:124–45, 222–50]. None of these authors considered the global dynamics of the system, which necessarily becomes nonlinear when consideration is taken of the facts that prices, supply quantities, and profits of active firms cannot be negative. In the present paper, we address the global dynamics.

  • 5.
    Cánovas, José S
    et al.
    Departamento Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM). Umeå universitet, Samhällsvetenskapliga fakulteten, Institutionen för nationalekonomi.
    Ruíz, Manuel
    Departamento Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena.
    The Cournot-Theocharis Problem Reconsidered2008Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 37, nr 4, s. 1025-1039Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In 1959 Theocharis [Theocharis RD. On the stability of the Cournot solution on the oligopoly problem. Review of economic Studies 1959;27:133–4] showed that with linear demand and constant marginal costs Cournot equilibrium is destabilized when the competitors become more than three. With three competitors the Cournot equilibrium point becomes neutrally stable, so, even then, any perturbation throws the system into an endless oscillation. Theocharis’s argument was in fact proposed already in 1939 by Palander [Palander T. Konkurrens och marknadsjämvikt vid duopol och oligopol. Ekonomisk Tidskrift 1939;41:124–45, 222–50]. None of these authors considered the global dynamics of the system, which necessarily becomes nonlinear when consideration is taken of the facts that prices, supply quantities, and profits of active firms cannot be negative. In the present paper, we address the global dynamics.

  • 6.
    Kumar, Ankit
    et al.
    Department of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, Himachal Pradesh, India.
    Majee, Sudeb
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Jain, Subit K.
    Department of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, Himachal Pradesh, India.
    CDM: A coupled deformable model for image segmentation with speckle noise and severe intensity inhomogeneity2023Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 172, artikel-id 113551Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Speckle noise and intensity inhomogeneity are always challenging issues in the area of image segmentation, especially when both difficulties appear simultaneously. Consequently, the majority of existing deformable models yield inadequate results in this situation. This paper aims to address these issues by combining a bias correction term and a despeckling term into a single variational level set architecture, which not only corrects the severe intensity inhomogeneity but also performs image despeckling and segmentation simultaneously. Specifically, edge and region synergetic terms are defined to correct the severe intensity inhomogeneity during image segmentation that eradicates their shortcomings as well as retains their advantages. Further, a despeckling term is constructed, which effectively suppresses the noise while enhancing image details. Since the lack of thorough investigation on despeckling of images degraded by Rayleigh noise distribution, which usually appear in ultrasound (US) images, so we included it in the coupled deformable model (CDM). Moreover, region-based terms eradicate complex re-initialization and numerical instability during the level set evolution, as a result, no distance regularization term is needed. Additionally, Schauder's fixed point theorem is used to demonstrate the well-posedness of the present system. Extensive numerical experiments on real and synthetic images with speckle noise and severe inhomogeneous intensity demonstrate that the CDM exhibits superior results compared to classical and some recent deformable models in terms of accuracy, robustness, and various quality measures.

  • 7.
    Negi, Shekhar Singh
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Torra, Vicenç
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Δ-Choquet integral on time scales with applications2022Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 157, artikel-id 111969Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The fundamental purpose of this work is to analyze Δ-Choquet integrals on time scales which is a special case of Choquet integral on abstract fuzzy (non-additive) measure space. We first present a Δ-Choquet integral with respect to non-additive Δ-measure or more precisely a distorted Lebesgue Δ-measure on an arbitrary time scale. Consequently, we come up with a more general integral than the standard Choquet integral of continuous and discrete calculus. Its use can be seen as convenient in economics, decision making, artificial intelligence, and many more. Particularly, in economics, most of the models are dynamic models (continuous and/or discrete), and those can be easily studied on time scales. Further, some basic essential results and properties of the general integral are studied. For instance, we discuss translation, homogeneity, linearity, and many more with respect to the functions and measures of the integral. Then, after that, we present some theorems for computing the integral. The findings agree to unify and extend a number of well-known results reported in the literature to a broader calculus, including continuous, discrete, and quantum calculus, among others. We also evaluate the integral on an invariant under the translation of time scales. Besides, a short note on Δ-Choquet integral with the Caputo-Fabrizio fractional derivative on the time scales is given. The significance of the outcomes is also further enhanced by a variety of interesting examples. Moreover, eventually, we stop findings after discussing an another way to calculate the Δ-Choquet integral on the time scales. To do this, we define Stieltjes distorted types-I and II Lebesgue Δ-measures on time scales which are accomplished with the help of distorted Lebesgue Δ-measure.

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  • 8.
    Nilsson, Anders
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Georgsson, Fredrik
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Projective properties of fractal sets2008Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 35, nr 4, s. 786-794Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, it is shown that a bound on the box dimension of a set in 3D can be established by estimating the box dimension of the discrete image of the projected set i.e. from an image in 2D. It is possible to impose limits on the Hausdorff dimension of the set by estimating the box dimension of the projected set. Furthermore, it is shown how a realistic X-ray projection can be viewed as equivalent to an ideal projection when regarding estimates of fractal dimensions.

  • 9.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    Chaos in business cycles1991Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 1, nr 5, s. 457-473Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The business cycle is studied in terms of the mapping

    Zt = λZt−1 − (λ+1)Z3t−1 − σYt−1

    Yt = Zt−1 + Y−1

    where the variables Y, Z denote income and rate of income change respectively, and λ, σ are two structural parameters. The model produces chaotic or periodic output for income differences. For small σ income acts as a slow feed back causing bifurcations between periodic and chaotic behaviour over the cycle. Typically, transitions between prosperity and depression set in with chaos after which there follows a period halving route to order.

  • 10.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    Chaos in duopoly pricing1991Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 1, nr 6, s. 573-581Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The dynamics of two competing firms in a market is studied in terms of Cournot's duopoly theory. Assuming iso-elastic demand and constant unit production costs the iterative mapping

    for the outputs of the two firms ensues. The two constants are the unit production costs. The fixed point, the Cournot equilibrium, has earlier been assumed to be the only interesting feature of this model. It is, however, shown that the model can produce persistent motion, periodic or chaotic. 

  • 11.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    Complex dynamics in economic and social systems: Introduction1996Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 7, nr 12, s. R7-R7Artikel i tidskrift (Övrigt vetenskapligt)
  • 12.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    Complex dynamics with three oligopolists1996Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 7, nr 12, s. 2075-2081Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The adjustment process by three Cournot oligopolists is studied. An iso-elastic demand function and constant marginal costs are assumed. The system can easily result in chaotic behaviour, and a much richer variety of bifurcations will occur than in the case of duopoly with two agents, discussed before under the same assumptions.

  • 13.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    Pattern formation in spatial economics1993Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 3, nr 1, s. 99-129Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A market economy, extended in continuous two-dimensional space, defined by the differential equations:

     

    is studied, where φ denotes the vector field of traded commodities, λ the scalar field of commodity prices, q the scalar field of excess supply over demand, and k the cost of transportation. The generic theory of differential equations is used to topologically characterize those patterns that are structurally stable, and the transitions among them are then studied by the elliptic and hyperbolic umblic catastrophes.

  • 14.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Nationalekonomi.
    The chaotic monopolist1995Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 5, nr 1, s. 35-44Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The search of a profit maximum by a monopolistic firm is studied, given a demand function with variable elasticity of demand. The system has multiple optimal solutions, so the search algorithm may result in chaotic behaviour.

  • 15.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskaplig fakultet, Centrum för regionalvetenskap (CERUM).
    Gardini, Laura
    Department of Economics, University of Urbino, Italy.
    Sushko, Irina
    Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine.
    On the Change of Periodicities in the Hicksian Multiplier-Accelerator Model with a Consumption Floor2006Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 29, nr 3, s. 681-696Artikel i tidskrift (Övrigt vetenskapligt)
    Abstract [en]

    The Hicksian multiplier-accelerator model with “floor” and “ceiling” continues to be the most successful machine generating business cycles. This is, no doubt, due to its capability of explaining both downturn and upswing through one single model. The “ceiling” is due to a full employment constraint, whereas the “floor” is due to a limit to disinvestment when no worn out capital at all is replaced. However, another “floor” to consumption at zero level seems never to have been discussed. Hence, net disinvestments, even if they are bounded downwards, may also give rise to negative consumption, which is absurd. As we will show, the effect of an additional constraint to avoid this is easy to analyze, and results in a change of the periodicities according to a simple rule.

  • 16.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Gardini, Laura
    Department of Economics, University of Urbino, Italy.
    Sushko, Iryna
    Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine.
    On the change of periodicities in the Hicksian multiplier-accelerator model with a consumption floor2006Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 29, nr 3, s. 681-696Artikel i tidskrift (Refereegranskat)
  • 17.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskaplig fakultet, Centrum för regionalvetenskap (CERUM).
    Marin, Manuel Ruiz
    Univ Politecn Cartagena, Dpto Metodos Cuantitativos & Informat, Spain.
    The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints2006Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 28, nr 2, s. 403-413Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This article considers Cournot oligopoly with three competitors, given an iso-elastic demand function, and cost functions that asymptotically go to infinity when capacity limits are approached. It is shown that for some parameter values (capacity limits) the Cournot equilibrium point loses stability through a Neimark-Sacker bifurcation. In addition global bifurcation scenarios are simulated numerically, and some are illustrated through Arnol'd tongues in the parameter plane.

  • 18.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Marín, Manuel Ruíz
    Facultad de C.C. de la Empresa, Dpto. Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena, Cartagena, Spain.
    The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints2006Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 28, nr 2, s. 403-413Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This article considers Cournot oligopoly with three competitors, given an iso-elastic demand function, and cost functions that asymptotically go to infinity when capacity limits are approached. It is shown that for some parameter values (capacity limits) the Cournot equilibrium point loses stability through a Neimark-Sacker bifurcation. In addition global bifurcation scenarios are simulated numerically, and some are illustrated through Arnol’d tongues in the parameter plane.

  • 19.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Norin, Anna
    Umeå universitet, Samhällsvetenskapliga fakulteten, Handelshögskolan vid Umeå universitet, Nationalekonomi.
    Cournot duopoly when the competitors operate under capacity constraints2003Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 18, nr 3, s. 577-592Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 Cobb–Douglas 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.

  • 20.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Norin, Anna
    Umeå universitet, Samhällsvetenskapliga fakulteten, Institutionen för nationalekonomi.
    Cournot duopoly when the competitors operate under capacity constraints2003Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 18, nr 3, s. 577-592Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 Cobb–Douglas 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.

  • 21.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskaplig fakultet, Centrum för regionalvetenskap (CERUM).
    Panchuk, Anastasiia
    Institute of Mathematics, National Academy of Sciences, Kiev, Ukraine.
    Oligopoly and stability2009Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 41, nr 5, s. 2505-2516Artikel i tidskrift (Refereegranskat)
  • 22.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Sushko, Irina
    Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine.
    A business cycle model with cubic nonlinearity2004Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 19, nr 3, s. 597-612Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper deals with a simple multiplier-accelerator 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.

  • 23.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskaplig fakultet, Centrum för regionalvetenskap (CERUM).
    Sushko, Irina
    Institute of Mathematics, National Academy of Sciences, Ukraine.
    A business cycle model with cubic nonlinearity2004Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 19, nr 3, s. 597-612Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper deals with a simple multiplier-accelerator 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.

  • 24.
    Puu, Tönu
    et al.
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Tramontana, Fabio
    Can Bertrand and Cournot oligopolies be combined?2019Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 125, s. 97-107Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    There have been some recent attempts to combine Cournot and Bertrand duopolies in one single model. Unfortunately, these attempts do not work. A commodity cannot be homogenous and non-homogenous at the same time. It is always the consumers, who decide whether they perceive competing products as identical or as different brands for which they are willing to pay different prices. There is, of course, nothing that forbids the coexistence of both such consumer groups. Neither is there any obstacle for the competing sellers to sell to both markets. Then we only need an old idea from economic theory, i.e., price discrimination, to rectify the logic. By this the challenging combination idea comes on a stable footing. The model also results in some interesting mathematical facts, such as mulistability and coexistence of attractors. 

  • 25.
    Sushko, Irina
    et al.
    National Academy of Sciences Ukraine, Math Inst.
    Gardini, Laura
    University of Urbino, Dept Economics, Urbino.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Centrum för regionalvetenskap (CERUM).
    Tongues of periodicity in a family of two-dimensional discontinuous maps of real Mobius type2004Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 21, nr 2, s. 403-412Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we consider a two-dimensional piecewise-smooth discontinuous map representing the so-called "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 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 saddle-node and border-collision bifurcations of the first-return map. Moreover, the particular "sausages structure" of the bifurcation tongues is also explained.

  • 26.
    Sushko, Irina
    et al.
    Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine.
    Gardini, Laura
    Department of Economics, University of Urbino, Italy.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Tongues of periodicity in a family of two-dimensional discontinuous maps of real Möbius type2004Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 21, nr 2, s. 403-412Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we consider a two-dimensional piecewise-smooth discontinuous map representing the so-called “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 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 saddle-node and border-collision bifurcations of the first-return map. Moreover, the particular “sausages structure” of the bifurcation tongues is also explained.

  • 27.
    Sushko, Irina
    et al.
    Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Gardini, Laura
    Facoltà di Economia, Università degli Studi di Urbino, Urbino, Italy.
    The Hicksian floor–roof model for two regions linked by interregional trade2003Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 18, nr 3, s. 593-612Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Hicksian multiplier–accelerator model with the original floor–roof 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 the assumption of a certain symmetry between the regions. The Neimark–Hopf 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.

  • 28.
    Sushko, Irina
    et al.
    Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskaplig fakultet, Centrum för regionalvetenskap (CERUM).
    Gardini, Laura
    Facoltà di Economia, Università degli Studi di Urbino, Italy.
    The Hicksian floor–roof model for two regions linked by interregional trade2003Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 18, nr 3, s. 593-612Artikel i tidskrift (Övrigt vetenskapligt)
    Abstract [en]

    The Hicksian multiplier–accelerator model with the original floor–roof 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 the assumption of a certain symmetry between the regions. The Neimark–Hopf 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.

  • 29.
    Tramontana, Fabio
    et al.
    Dept of Economics and Quantitative Methods, Universíty of Urbino, Italy.
    Gardini, Laura
    Dept of Economics and Quantitative Methods, Universíty of Urbino, Italy.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Global bifurcations in a piecewise-smooth Cournot duopoly game2010Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 43, nr 1-2, s. 15-24Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 Neimark-Sacker 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.

  • 30.
    Tramontana, Fabio
    et al.
    Dept of Economics and Quantitative methods, University of Urbino, Italy.
    Gardini, Laura
    Dept of Economics and Quantitative methods, University of Urbino, Italy.
    Puu, Tönu
    Umeå universitet, Samhällsvetenskapliga fakulteten, Centrum för regionalvetenskap (CERUM).
    Mathematical properties of a discontinuous Cournot-Stackelberg model2011Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 44, nr 1-3, s. 58-70Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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. 

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