The limit behaviour of dynamic systems

Here we must make a great jump into some generalities of our presentation. To discuss reasons for the slow transition processes in non-linear systems, we do not need the formalism of chemical kinetics. To begin with we need very little a concept about the phase space X and the time shift Tt, 

The T mapping is not given analytically from the beginning. It is determined with the help of the solution of a system of differential equations. In this sense we do not have it, but we can specify and research general properties of T. 

The language applied is poorer and hence is more simple than that of the theory of differential equations. It is the language of topological dynamics [3-7]. Let us introduce the main concepts required here and in what follows. 

The equality (1) means that, after shifting the system for t and then for t, we will obtain a system shifted for t + t. The function Tt(x) is continuous relative to the totality of arguments t and x. 

For x e X, the function that puts each t e [0, oo] in correspondence with the point Tt(x) is called x-motion. For certain xeX this function can also be extended to the negative value of t, i.e. if for certain yeX and t 0 the condition Tt(y) = x holds, then we will assume that T t(x) = y. In what follows we will use this extension without additional explanation. To avoid misunderstanding, it must be remembered that T t is not determined throughout the whole X (in contrast to %, t 0). x-Motion will be called a whole if Tt(x)eX is determined throughout the whole time axis te(— oo, oo). 

---

For t3, tj3 slow relaxations, the necessary and sufficient conditions have been obtained in terms of the limit behaviour of dynamic systems. Note that the (x, -motion is called positively Poisson-stable (P+-stable) if xeco(x, k). 

---