摘要
探讨传统动力系统与复杂系统之间的关系,并指出某些系统确实具有复杂系统的明显特征。首先研究切换系统,指出这种系统具有1+1≠2 的特征,并进一步研究这种系统的一些特点。其次,考虑一类动力系统的演化模型,指出这种系统改变着自己的环境和形态,但某些性质不变,将其称为遗传性质。这是我们用严格数学理论与动力系统理论研究复杂系统的尝试。
This paper reveals some relationship between the traditional dynamic systems and the complex systems, and shows that there are some dynamic systems, which do propose some significant properties of complex systems. First, a kind of switching systems is investigated. It is revealed that such systems do have the property that 1+1≠2. Some further properties of such systems are presented as well. Secondly, an evolution model of dynamic system is considered. We show that such a system is changing its environment and style etc., while some other properties remain unchanged. We call such properties the genetic properties. The research conducted in this paper is the first step of our effort to investigate the complex systems via rigorous mathematical tools and the dynamic system theory.1 Introductioni In the last two or three decades the science of complexity has been developed rapidly. Some frames for investigating complex systems have been proposed and investigated. A fantastic frame, which has been proved powerful and matching the real world closely, is the 揳gent-emergence' description of the objects and their evolutions [26]. The 搘ater-pest' model reported in [16] is a good realistic model, which shows the evolution varies from 揳gent-emergence' to 揳gent-chaos' via difference 搇ocal dynamic rule' [16]. Roughly speaking, this approach is essentially a simulation-based investigation. S. Zhang has also studied the complex systems for several years [28]. Zhang has investigated the similar structure, symmetric structure, full-information etc. for dynamic systems. We did some similar investigations aimed on revealing and controlling the complexity of dynamic systems [2], [3], [4], [5], [6], [7]. The theory of 揙pen Complex Giant System' was proposed by Qian and his fellows [19]. A method called the 搈atasynthesis' was presented, which proposes a way to solve the synthesis of complex systems from qualitative to quantitative. By the authors?understanding, matasynthesis suggests that the two approaches mentioned in the last paragraph should be 搈erged' into a collaborated process. It is not difficult to find from Qian抯 classical book [18] till his recent book [20] that he has paid much attention to the quantitative approach. We emphasize that quantitative approach should be an important component in the matasynthesis. Particularly, after more than 10 years of study, a general statement about complexity is less attractive, and some deep mathematical approach is an urgent need. Since Newton, differential and difference equations have been used to describe dynamic systems. Can these powerful tools be used to describe a complex system? It has been discussed in this Qingdao Symposium. Most people agreed that a single traditional dynamic model can not be used to describe a complex process. Say, with different parameters the Logistic mapping. 1(1)nnnxaxx+=- may cause some complex phenomena such as chaos. But
出处
《系统仿真学报》
CAS
CSCD
2002年第11期1447-1449,1454,共4页
Journal of System Simulation
基金
Supported partly by National 973 Project 1998020308 of China.
关键词
复杂动力系统
动力系统
发展
演化
数学理论
dynamic system
development
evolution
switching
genetic properties
