摘要
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之间的关系更加复杂,传统的经济学理论已无法准确描述,非线性经济学(或混沌经济学)逐渐成为当代经济学研究的前沿领域,并已取得迅速的进展。本文运用现代非线性动力学理论与随机动力学理论,研究了经济系统中复杂非线性现象的运行规律,并对实际经济序列进行了实证研究,主要完成了以下工作:
1、系统地总结了西方经济周期理论,改进了Hicks提出的消费函数,考虑非线性加速数,同时采用Puu提出的带有立方项的投资函数,建立了动态的非线性经济周期模型;运用现代非线性动力学的分岔理论分析了该模型的动力学行为,发现该系统在不同的参数条件下会发生超临界Hopf分岔、超临界叉形分岔与亚临界叉形分岔的三种分岔形式,通过数值仿真进行了验证,并对分岔产生的实际经济意义进行了分析。
2、首先考虑具有随机形式的自主函数,建立了随机经济周期模型;对该弱阻尼弱激励的拟不可积Hamilton系统,运用基于乘积遍历性定理的Lyapunov指数及一维扩散过程的边界分析,得出该系统的全局稳定性条件;根据系统响应联合概率密度和边缘概率密度以及不同参数条件,研究了该模型的随机Hopf分岔行为,通过数值仿真进行了验证,并对分岔参数进行了讨论分析。
3、首先基于虚假最近邻域概念,同时确定最佳的嵌入维数m与时间延迟τ,对实际经济时间序列进行相空间重构;通过对最大Lyapunov指数、分形维数及测度熵的求解,对实际非线性经济序列进行了实证研究,给出了混沌特性检验的多种有效方法,验证了相空间重构的参数选择的正确性,发现了经济序列中存在混沌现象,并分别求解出经济序列的可预报尺度。
4、根据最大Lyapunov指数的倒数得出混沌预测的最大可预报尺度,建立基于[1/LE]个输入神经元的遗传神经网络,该算法能够自动确定网络拓扑结构,并达到全局优化的效果,将其应用于实际经济时间序列建模预测,并与混沌时间序列常用的预测算法:相空间预测法、局域预测法和普通人工神经网络进行了比较,通过实证计算发现上述混沌时间序列的预测方法都比较准确,平均误差在3%以内,遗传神经网络预测计算结果更为准确,预测平均误差在2%以内。
With the global economy integration, the factors affecting the economy and therelations of the factors become more complex. The traditional economic theorycannot describe it, but nonlinear economics (chaotic economics) has become theresearch focus, and made rapid progress. This dissertation studies the complexnonlinear phenomenon in the economic system with the aid of modern nonlineardynamics and stochastic dynamics theory. The main content is as follows:
1. Summarize the western business cycle theory systemically, based on theHicks' consumption model and Puu's investment model with cubic nonlinearity, setup the dynamic nonlinear business cycle model, considering the nonlinear accelerator.Apply the modern nonlinear dynamics to analysis the dynamic behavior of the model,found that under different parameters three types of bifurcation occured: Hopfbifurcation, supercritical and subcritical furcation bifurcation. Analysis the bifurcationparameters and validate with the numerical simulation.
2. First set up a stochastic business cycle model considering the stochasticindependence function. Gain the complete stable condition with Lyapunov exponentbased on multiplicatibve ergodic theorem and boundary analysis of one-dimensiondiffuse process to this weak damp and weak stimulation Quanti-non-integrateHamiltion system. Study the stochastic Hopf bifurcation behavior on the basis of jointprobability density and marginal probability density under different parameterscondition, and validate with the numerical simulation.
3. Based on the conception of false nearest neighbor, which determines the bestembedding dimension m and time delay τ simultaneously, reconstruct the practicaleconomic time series. Demonstrate the practical economic time series with the maxLyapunov exponent, fractal dimension and Kolmognov entropy, present severalchaotic validation method, validate the accuracy of the parameters selection of phase
reconstruction, find out the chaotic property, and figure out the max predictable scale.4. On the basis of the max predictable scale, which is the reciprocal of the maxLyapunov exponent, set up a genetic neural network of [1/LE] input nerve cells. Thisalgorithm can work out the best topologic structure of the network automatically, andget the global optimization. Demonstrate with the practical economic series, andcompared with the common predictive method of chaotic time series: phaseprediction, local prediction and artifical neural network, and find out that all the fourmethod is satisfactory with the mean error of 3%, genetic neural network offers moreaccurate result of mean error within 2%.
引文
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