Taking into account small perturbations and errors of models

Taking into account small perturbations and errors of models 

Any real system is known to suffer constantly from the perturbing effects of its environment. One can hardly build a model accounting for all the perturbations. Besides, as a rule, models account for the internal properties of the system only approximately. It is these two factors that are responsible for the discrepancy between real systems and theoretical models. This discrepancy is different for various objects of modem science. For example, for the objects of planetary mechanics this discrepancy can be very small. On the other hand, in chemical kinetics (particularly in heterogeneous catalysis) it cannot be negligible. Strange as it is, taking into consideration such unpredictable discrepancies between theoretical models and real systems can simplify the situation. Perturbations smooth out some fine details of dynamics. 

In other words, if for some arbitrary point cj)(t0) one would consider its motion to be due to the dynamic system, the discrepancy between this 

There are two traditional approaches to the consideration of perturbed motions. One is the study of the motion in the presence of small continuously acting perturbations [24 30] the other is the investigation of fluctuations caused by small random perturbations [31-34]. Our results were obtained in terms of the former approach but using some ideas of the latter. 

When studying perturbed motions, every point x is juxtaposed not one x-motion but a bundle of e-motions f starting from this point [ / (0) = x] for a given value of the parameter k. Every e-motion 4 (t) is associated with an co-limit set co( / ). It consists of those points yeJt for which one can find such a sequence tt - oo, as - y. Every xeX (initial value) at a given value of the parameter k is associated with the set of co (x, %). It is the combination of those co(4 ) for which / (0) = x and j t) is the e-motion (at a given k). 

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