Five Element Structure And Differential Equations Model Of Complex Adaptive System

On the premise that main subjects in complex system are adaptive, and on the basis of "unity of opposites", we first prove that the existence of the completely normal complex adaptive system(CRCAS) and there are only five main elements (main subjects) by using the mathematical axiomatic method. And the interaction of the five elements in this system could satisfy a hypercycle network system-the elements would activate when it is in the neighboring position; and would inhibit when in the alternate position. The results has established the mathematical basis of deductive system for "Yin-Yang Five Elements theory". In that circumstances, the paper has established a scientific basis for the mathematics of Chinese medicine theory. Since we deduce complete "Yin-Yang Five Elements theory" by axiom "unity of opposites" in philosophy, then "Yin-Yang Five Elements theory" is not only a hypothesis, but also a science theory by Popper’s conclusion.Secondly, for CRCAS, we build an event model which will evolve over time by using the hypercycle network system given above and ordinary differential equation set, and popularize the model to n-dimensional system. We observe the tendency of changes over time of CRCAS by using computer simulation and get the stability of equilibrium solution in a low-dimentional system.Finally, for the circulation system which is expressed by semi-linear differential equations, we discuss the regularization of its ill-posed problems. Byδ-function and Moore-Penrose generalized inverse, we give the representation of the generalized Green function of with two-point boundary value condition. And we provide the best approximate solution with the generalized Green function.