英文标题：
《The phase space structure of the oligopoly dynamical system by means of
  Darboux integrability》
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作者：
Adam Krawiec, Tomasz Stachowiak, Marek Szydlowski
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最新提交年份：
2017
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英文摘要：
  We investigate the dynamical complexity of Cournot oligopoly dynamics of three firms by using the qualitative methods of dynamical systems to study the phase structure of this model. The phase space is organized with one-dimensional and two-dimensional invariant submanifolds (for the monopoly and duopoly) and unique stable node (global attractor) in the positive quadrant of the phase space (Cournot equilibrium). We also study the integrability of the system. We demonstrate the effectiveness of the method of the Darboux polynomials in searching for first integrals of the oligopoly. The general method as well as examples of adopting this method are presented. We study Darboux non-integrability of the oligopoly for linear demand functions and find first integrals of this system for special classes of the system, in particular, rational integrals can be found for a quite general set of model parameters. We show how first integral can be useful in lowering the dimension of the system using the example of $n$ almost identical firms. This first integral also gives information about the structure of the phase space and the behaviour of trajectories in the neighbourhood of a Nash equilibrium
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中文摘要：
我们利用动力学系统的定性方法研究了三家公司古诺寡头垄断动力学的动力学复杂性，并对该模型的相结构进行了研究。相空间由一维和二维不变子流形（对于垄断和双寡头）以及相空间正象限（古诺平衡）中的唯一稳定节点（全局吸引子）组成。我们还研究了系统的可积性。我们证明了达布多项式方法在搜索寡头垄断的第一积分时的有效性。给出了采用该方法的一般方法和实例。我们研究了线性需求函数寡头垄断的Darboux不可积性，找到了该系统对于特殊类系统的第一次积分，特别是对于一组非常一般的模型参数，可以找到有理积分。我们以n$几乎相同的公司为例，说明了第一积分在降低系统维数方面的作用。第一个积分还提供了有关相空间结构和纳什均衡附近轨迹行为的信息
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Economics        经济学
分类描述：q-fin.EC is an alias for econ.GN. Economics, including micro and macro economics, international economics, theory of the firm, labor economics, and other economic topics outside finance
q-fin.ec是econ.gn的别名。经济学，包括微观和宏观经济学、国际经济学、企业理论、劳动经济学和其他金融以外的经济专题
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一级分类：Mathematics        数学
二级分类：Dynamical Systems        动力系统
分类描述：Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学，力学，经典的少体问题，迭代，复杂动力学，延迟微分方程
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一级分类：Physics        物理学
二级分类：Exactly Solvable and Integrable Systems        精确可解可积系统
分类描述：Exactly solvable systems, integrable PDEs, integrable ODEs, Painleve analysis, integrable discrete maps, solvable lattice models, integrable quantum systems
精确可解系统，可积偏微分方程，可积偏微分方程，Painleve分析，可积离散映射，可解格模型，可积量子系统
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