Abstract
It is important that large scale complex system has some good properties e.g. variety, flexibility, fault tolerability. Recently, an autonomous decentralized system has been studied for getting such a system. And it was already shown in the previous paper that (1) homogeneity of subsystems, (2) a system which expresses a global order (this system is called “total system”) and (3) a potential function of this system are very important. For (2) and (3), the following two conditions are necessary for the large scale system.
(1) the total system is autonomous.
(2) the total system is a gradient one.
If so, the large scale system can be dealt as a lower order total one, and the behavior is easy to understand because the order depends on the potential function. And the local dynamics of subsystems determine whether these conditions are satisfied or not.
In this paper, the differences of states are chosen as variables of total system. And it shows the necessary and sufficient condition for local dynamics to make the total system to a gradient one. This theorem makes clear the relation between local dynamics and global order. Using these results, it is discussed the fault tolerability of this system as to keep a gradient system. Lastly, some computer simulations demonstrate this theorem.
