Relaxation times. Determination of slow relaxations

System relaxation can be represented as a motion from the initial state x towards the co-limit set co(x, k). Relaxation time can be defined as the time of this motion. But here we have several possibilities. 

Let us take some e 0 as the accuracy for the achievement of a limit set. Ti (x, k, e) will be the time for the first entry of the system from its initial state into the e-neighbourhood of co(x, k)(for a given %). After then the system can leave the e-neighbourhood and then enter it again and do this several (even many) times. Finally, the (x, )-motion will constantly remain in this neighbourhood, but this process can take more time than it needed for the first 

At the end of Sect. 2, we agreed that the relaxation of (x, [-motion is the tendency to m(x, k) that is its co-limit set. But in this case another viewpoint is possible. We can treat coT(k) as a set of limit regimes and define relaxation as a motion towards it. Then it is not important which of the co-limit points the (x, [-motion has been approached, either its own [from co(x, )] or foreign [from a (y, k) for y x]. Of importance is the approach to the complete set of limit regimes oT(k). In this connection we introduce three relaxation times [analogous to t12,3 of eqns. (5)]. 

Let us now define what must be considered as a slow relaxation process. To judge whether the relaxation time is large or small, it must be compared with some specified time scale. But the system has no such specified time scale. Besides, with decreasing s, relaxation times can become arbitrarily large even for the motion towards the only stable fixed point. 

For each initial state x and k, s values all relaxation times are finite. But their set can be unrestricted for %eK, xeX (but for fixed e 0). It is in this case that we will speak about slow relaxations. 

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