门限自回归模型与非线性系统的极限环

本文根据武汉市某区人口死亡率和加拿大山猫数据建立了SETAR模型,并作出了相应系统的极限环。在此基础上指出,SETAR模型可描述一个确定极限环的某一邻域内的一“族”相轨线,此极限环是轨道稳定的。文中还根据现有结果分析指出,对于具有两个门限段的SETAR模型,门限值r表征观测时序{x_t}的均值特性;在一定情况下,延迟步数d可表征极限环的周期T的特性。利用这些特性,不仅可使建模中对r和d的寻优工作量大为减小,而且还从本质上揭示出SETAR模型参数的物理含义,便于分析研究。

Threshold Autoregressive Model and Limit Cycle of a Nonlinear System

Based on the data of the death rate of population in a district of Wuhan and that of Canada's leopard cat, SETAR models have been developed and the limit cycles for both cases worked out respectively. According to the concept of the stability of a nonlinear system, it is pointed out that the long-term forecast by SETAR model corresponds to a limit cycle of the system and the short-term forecast is a dynamic approach to the cycle from an initial state of the system. As the cycles are slightly different from each other owing to their different initial states, it is concluded that the cycles are essentially a group, of phase trajectories located in the neighbourhood of a certain cycle which is orbitally stable.It is also found that the threshold value r in a SETAR model with two threshold sections is the average value x of a time series {xt} and there exists a relation between the delay factor d and the period T of a limit cycle.