一类非线性经济周期模型的复杂性研究
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摘要
经济系统是复杂的开放系统。随着全球经济一体化进程的深入,影响经济运行的因素以及这些因素间的关系变得更加复杂,在传统经济学理论框架下研究经济周期问题已表现出诸多不足。随着非线性科学的发展,非线性经济学逐渐成为当代经济学研究的前沿领域,并已取得较大进展。本文以非线性动力学理论为基础,研究了经济系统中复杂非线性现象及规律,并以天津地区为例对经济周期进行了实证研究,全文主要完成了以下工作:
1.应用非线性动力系统理论,研究了一类非线性经济周期模型在共振条件下一阶和二阶近似解,分析了系统的二阶近似解的振幅与相位之间的关系,系统外在激励幅值、近似解的频率以及与阻尼参数间的关系;揭示了一定参数组合条件下,系统出现环面周期解,拟周期解以及系统进入分岔和混沌的道路。
2.研究了非线性经济周期模型在大激励、非共振条件下的内在复杂特征。采用多尺度法得到了该条件下系统的一阶、二阶近似解;给出了其一阶解的振幅、相位、阻尼参数以及驱动频率间的复杂关系;分析了随着大激励值的变化与系统解的全局演化情况;揭示了不同参数组合条件下,系统进入突变、模糊域和混沌域的路径。同时,探讨了在不同情况下经济周期模型的演化趋势。
3.研究了非线性经济周期模型在超谐共振条件下的一解近似阶及其近似解的振幅、相位、系统内在阻尼参数、驱动幅值和驱动频率之间的复杂关系,并通过分析各参数相互依赖的变化探究了系统的复杂演化过程。
4.综述了螺旋历法规律,并以天津地区为例,对经济周期与螺旋历法间的关系进行了实证分析,为寻找经济周期的变化规律提供了有效的依据。
The economic system is an open complex system. With the deeping of global economy integration, the factors affecting the economy and relations of the factors become more complex and the traditional economic theory is insufficient to study business cycle. With the development of nonlinear science, nonlinear economics is becoming the research focus of modern economics and has made great progress. This dissertation studies the complex nonlinear phenomenon in a business cycle model based on the theory of modern nonlinear dynamics. Also, the empirical analysis with the example of Tianjin is provided. The main content is as follows:
1. A nonlinear business cycle model in resonance is studied via the theory of nonlinear dynamical system. The first and second order approximate solutions and relation between the swing and phase of the second order approximate solutions are analyzed. The swing of extrinsic motivation, frequency of the approximate solutions besides the relations among them and the damping parameters are also analyzed. In addition, the periodic solutions, the approximate period solutions of the system and the access to bifurcation and chaos are revealed.
2. The nonlinear business cycle model in oscillation condition and off-resonance is studied. The first and second order approximate solutions of the system are obtained via the method of multiple scales, the relations among the swing, phase, damping parameters, and driving frequency of the first order approximate solutions are provided. The overall evolution of the system solutions with the varying of the force is also revealed. The access of the system to mutation, fuzzy field and chaos and the evolvement of the business cycle model are discussed in detail.
3. The nonlinear business cycle model in super-harmonic resonance is studied. The first order approximate solutions of the system, the relations among the swing, phase, damping parameters, driving amplitude and driving frequency are discussed. The complex evolutionary process of the system is explored via the condition of the interdependence of the parameters.
4. Helical calendar is introduced, and the relationship between the business cycle and helical calendar is studied with the empirical analysis of Tianjin to provide effective basis for finding the discipline of business cycle.
1.应用非线性动力系统理论,研究了一类非线性经济周期模型在共振条件下一阶和二阶近似解,分析了系统的二阶近似解的振幅与相位之间的关系,系统外在激励幅值、近似解的频率以及与阻尼参数间的关系;揭示了一定参数组合条件下,系统出现环面周期解,拟周期解以及系统进入分岔和混沌的道路。
2.研究了非线性经济周期模型在大激励、非共振条件下的内在复杂特征。采用多尺度法得到了该条件下系统的一阶、二阶近似解;给出了其一阶解的振幅、相位、阻尼参数以及驱动频率间的复杂关系;分析了随着大激励值的变化与系统解的全局演化情况;揭示了不同参数组合条件下,系统进入突变、模糊域和混沌域的路径。同时,探讨了在不同情况下经济周期模型的演化趋势。
3.研究了非线性经济周期模型在超谐共振条件下的一解近似阶及其近似解的振幅、相位、系统内在阻尼参数、驱动幅值和驱动频率之间的复杂关系,并通过分析各参数相互依赖的变化探究了系统的复杂演化过程。
4.综述了螺旋历法规律,并以天津地区为例,对经济周期与螺旋历法间的关系进行了实证分析,为寻找经济周期的变化规律提供了有效的依据。
The economic system is an open complex system. With the deeping of global economy integration, the factors affecting the economy and relations of the factors become more complex and the traditional economic theory is insufficient to study business cycle. With the development of nonlinear science, nonlinear economics is becoming the research focus of modern economics and has made great progress. This dissertation studies the complex nonlinear phenomenon in a business cycle model based on the theory of modern nonlinear dynamics. Also, the empirical analysis with the example of Tianjin is provided. The main content is as follows:
1. A nonlinear business cycle model in resonance is studied via the theory of nonlinear dynamical system. The first and second order approximate solutions and relation between the swing and phase of the second order approximate solutions are analyzed. The swing of extrinsic motivation, frequency of the approximate solutions besides the relations among them and the damping parameters are also analyzed. In addition, the periodic solutions, the approximate period solutions of the system and the access to bifurcation and chaos are revealed.
2. The nonlinear business cycle model in oscillation condition and off-resonance is studied. The first and second order approximate solutions of the system are obtained via the method of multiple scales, the relations among the swing, phase, damping parameters, and driving frequency of the first order approximate solutions are provided. The overall evolution of the system solutions with the varying of the force is also revealed. The access of the system to mutation, fuzzy field and chaos and the evolvement of the business cycle model are discussed in detail.
3. The nonlinear business cycle model in super-harmonic resonance is studied. The first order approximate solutions of the system, the relations among the swing, phase, damping parameters, driving amplitude and driving frequency are discussed. The complex evolutionary process of the system is explored via the condition of the interdependence of the parameters.
4. Helical calendar is introduced, and the relationship between the business cycle and helical calendar is studied with the empirical analysis of Tianjin to provide effective basis for finding the discipline of business cycle.
引文
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