基于复杂性理论的经济系统Hopf分岔及其应用研究

发布时间：2018-08-20 19:54

【摘要】：随着社会的发展与进步，我们所面对的经济、金融系统越来越复杂，而导致这种复杂性的根源就是经济系统内部的非线性，这促使学者对复杂经济现象背后的非线性规律的研究，进而揭示经济现象的本质，以指导于实际的经济活动中。本文在学者们研究工作的基础上，采用管理学、经济学及其非线性动力系统的理论和方法，做了以下方面的工作：论文首先综述了非线性经济理论的研究进展，主要包括混沌和分形理论两个领域，列举了经济中重要的非线性模型，深入介绍了混沌、分形两大经典非线性理论在经济领域的应用；在国内外学者研究工作的基础上，研究了一类经济金融系统不同参数组合条件下的一系列非线性特性包括：稳定结点、鞍点、分岔、Hopf分岔、研究了a,b,c不同组合条件下系统进入分岔、混沌的情况下各参数相互依存变化情况，并对其全局复杂性进行分析，研究了系统参数变化所对应的经济运行中的停滞僵化、稳定或不稳定增长、通货膨胀、经济萧条或经济局面失控从而导致严重的社会动荡等的情况；研究了富有弹性条件下，一类复杂经济系统的Hopf分岔，分岔发生的参数演化条件，，研究了Hopf分岔前出现的周期轨道的稳定性问题，并依次研究了系统拓扑结构演化的参数临近值，根据Taken’s估计研究了上述各种情况下，系统的内在复杂性的演化情况；应用复原图(RP)和相干复原图(CRP)方法，结合复杂度分析的ApEn算法，研究了时间序列复杂性及不同特性时序数据的复原图和相干复原图上的特性，研究了Lorenz系统、稳定的汇、鞍点、分岔及Hopf分岔点时序数据的复原图和相干复原图，并研究了其相应的时序数据的复杂度，得到了五种新的复原图和相干复原图；对一类金融系统的内在复杂性进行研究，通过数值模拟，本文发现了导致系统进入混沌的两条路径：通往混沌的Ruelle-Takens路径，以及高阶平衡点分岔接倍周期分岔通向混沌，并在此过程中产生了奇异非混沌吸引子(SNA)；之后采用了时滞参数τ作为分岔参数研究了时滞对金融系统的影响；通过一类经济系统的雅克比矩阵的变化推导出了系统Lyapunov指数的数学表达式，研究其实部随着参数a,b,c的变化的情况，研究了不同参数组合条件下系统进入分岔、混沌的道路，为系统的混沌控制奠定基础。本文的研究成果将对深入分析一类经济金融系统的内部运行状况起到更进一步的促进作用，提出了系统进入混沌的两条新路径，为政府制定控制经济系统的政策提供了依据，具有实际的应用价值。
[Abstract]:With the development and progress of society, the economic and financial system we face is becoming more and more complex, and the source of this complexity is the nonlinearity within the economic system. This urges scholars to study the nonlinear laws behind complex economic phenomena, and then to reveal the essence of economic phenomena, in order to guide the actual economic activities. Based on the research work of scholars, this paper uses the theories and methods of management, economics and nonlinear dynamic systems to do the following work: firstly, this paper summarizes the research progress of nonlinear economic theory. This paper mainly includes two fields: chaos and fractal theory, enumerates the important nonlinear models in economy, and introduces the application of chaos and fractal in the economic field. Based on the research work of scholars at home and abroad, a series of nonlinear characteristics of a class of economic and financial systems with different parameter combinations are studied, including stable node, saddle point, bifurcation and Hopf bifurcation. In this paper, the variation of the interdependence of the system parameters under the condition of bifurcation and chaos is studied, and the global complexity is analyzed. The stagnation and fossilization in the economic operation corresponding to the variation of the system parameters are studied. Stable or unstable growth, inflation, economic depression or runaway economic situation leading to serious social unrest, etc.; the Hopf bifurcation of a class of complex economic systems under elastic conditions is studied. The parameter evolution condition of bifurcation, the stability of periodic orbit before Hopf bifurcation is studied, and the parameter approaching value of topological evolution of system is studied in turn. According to the Taken's estimation, the above conditions are studied. The evolution of the inherent complexity of the system is studied by using the restoration graph (RP) and coherent restoration graph (CRP) methods, combined with the ApEn algorithm of complexity analysis, and the characteristics of the restoration graph and coherent restoration graph of time series complexity and time series data with different characteristics are studied. In this paper, the restoration diagram and coherent restoration diagram of Lorenz system, stable convergence, saddle point, bifurcation and Hopf bifurcation data are studied. The complexity of the corresponding time series data is studied, and five kinds of new restoration graphs and coherent restoration graphs are obtained. The intrinsic complexity of a class of financial systems is studied. By numerical simulation, two paths leading to chaos are found: the Ruelle-Takens path to chaos. The bifurcation of higher order equilibrium points leads to chaos, and the singular nonchaotic attractor (SNA);) is produced in the process. The delay parameter 蟿 is used as the bifurcation parameter to study the influence of time delay on the financial system. The mathematical expression of Lyapunov exponent is derived from the change of Jacobian matrix of a class of economic systems. The change of system Lyapunov exponent with the change of parameter a / b / c is studied, and the bifurcation of the system is studied under the condition of different combination of parameters. The road of chaos lays the foundation for the chaos control of the system. The research results of this paper will further promote the analysis of the internal operation of a kind of economic and financial system, and bring forward two new ways for the system to enter chaos, which provides the basis for the government to formulate the policy of controlling the economic system. It has practical application value.
【学位授予单位】：天津大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：O415.5;F830

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