Attractor stability of randomized function systems
DOI:
https://doi.org/10.17308/sait/1995-5499/2024/2/5-14Keywords:
randomized systems of iterated functions, fractal sets, additional fractal, number of symmetries of the system stateAbstract
A model of a system is considered, the phase space of which is represented by an attractor of a randomized system of iterated functions. A distinctive feature of the state space of such a system is that it can be represented by fractal sets. Geometrically, it is shown that this fact corresponds to the presence of a dominant element among all the coordinates of the phase space. A consequence of this feature of the points of the phase space is the ability to set equivalence relations by allocating sets of points with a dominant element to a separate class. It is shown that the division of the phase space of the system into equivalence sets makes it possible to determine the number of symmetries of the states of the system for each of the equivalence classes. At the same time, sets with a dominant element, due to topological features, will have a large number of symmetries compared to other points in this phase space. In this paper, it is proposed to assume that the states of the system with a large number of symmetries have greater stability and vice versa. Using an alternative procedure allows you to build an additional fractal located in the lacuna zone — free from the points of the main fractal. An additional fractal retains all geometric properties, but being composed of points with fewer symmetries, it will be less stable. Obtaining additional fractal sets is proposed to be considered as a phase transition of the system. The paper attempts to find an answer to the question: why superficially similar fractal structures of objects can exhibit different stability.
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