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摘要翻译：
考虑了具有时滞投资函数的经济增长模型。假设投资是时间分布的，我们可以利用线性链技巧将时滞微分方程系统转化为常微分系统(ODE)的等价系统。时延参数是gamma分布的平均时延。我们将具有分布延迟的系统简化为三维和四维ODES。我们研究了这类系统关于两个参数的Hopf分支：时滞参数和增长率参数。我们从分析和数值研究中得到了结果。在前者的基础上，我们通过Hopf分支得到了极限环解存在性和稳定性的充分判据。在Dana和Malgrange投资函数的数值研究中，我们发现了两个关于速度增长参数的Hopf分支，并发现了经济中存在稳定的长周期循环。我们发现，增长速率参数的容许值范围根据时间延迟和调整速度参数的不同而分为三个区间。首先给出了稳定焦点，然后给出了极限环，最后给出了两个Hopf分支的稳定解。这种行为出现在增长率参数容许值范围的某个中间区间。
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英文标题：
《Bifurcations in economic growth model with distributed time delay
transformed to ODE》
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作者：
Luca Guerrini, Adam Krawiec, Marek Szydlowski
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最新提交年份：
2020
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分类信息：
一级分类：Economics 经济学
二级分类：Theoretical Economics 理论经济学
分类描述：Includes theoretical contributions to Contract Theory, Decision Theory, Game Theory, General Equilibrium, Growth, Learning and Evolution, Macroeconomics, Market and Mechanism Design, and Social Choice.
包括对契约理论、决策理论、博弈论、一般均衡、增长、学习与进化、宏观经济学、市场与机制设计、社会选择的理论贡献。
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一级分类：Economics 经济学
二级分类：General Economics 一般经济学
分类描述：General methodological, applied, and empirical contributions to economics.
对经济学的一般方法、应用和经验贡献。
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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：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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英文摘要：
We consider the model of economic growth with time delayed investment function. Assuming the investment is time distributed we can use the linear chain trick technique to transform delay differential equation system to equivalent system of ordinary differential system (ODE). The time delay parameter is a mean time delay of gamma distribution. We reduce the system with distribution delay to both three and four-dimensional ODEs. We study the Hopf bifurcation in these systems with respect to two parameters: the time delay parameter and the rate of growth parameter. We derive the results from the analytical as well as numerical investigations. From the former we obtain the sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation. In numerical studies with the Dana and Malgrange investment function we found two Hopf bifurcations with respect to the rate growth parameter and detect the existence of stable long-period cycles in the economy. We find that depending on the time delay and adjustment speed parameters the range of admissible values of the rate of growth parameter breaks down into three intervals. First we have stable focus, then the limit cycle and again the stable solution with two Hopf bifurcations. Such behaviour appears for some middle interval of admissible range of values of the rate of growth parameter.
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PDF链接：
https://arxiv.org/pdf/2002.05016
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