Page 635 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts

AMICAL SYSTEMS AND ORDINARY DIFFERENTIAL EQUATIONS AND
APPLICATIONS

Theorem 2. Let coordinates of the attractor index T t be given by one of the equations (3), (4),

(5), (6). Then each trajectory of a dynamic system (1) with an initial state {p1, p2, . . . , pm}
converges to the fixed point {p1∞, p∞2 , . . . , pm∞}

p∞i = lim pti, ∀ i = 1, m,

t→∞

where the coordinates of pi∞ have a view:

p∞ij = τj ∀ i = 1, m, j = 1, n. (7)
W

Stability. The limit state is unstable in cases (3)-(5), however it is stable only in the case (6),

when all limit coordinates are equal to 1 .
n

Application. Such model of dynamic systems can describe the dynamics of real processes.

Attractor index can describe a real external influence on a certain system (for example, in-

formation influence on a society). System behavior can be controlled or described by setting

attractor index which can be exposed to such influence.

References

[1] Koshmanenko, V.D., Satur, O.R. Sure Event Problem in Multicomponent Dynamical
Systems with Attractive Interaction. Journal of Mathematical Sciences, (2020), 249, pp.
629–646. https://doi.org/10.1007/s10958-020-04962-3

[2] Satur O.R. Limit states of multicomponent discrete dynamical systems. Nonlinear Oscil-
lations, (2020), 23, No. 1, pp. 77–89.

[3] Koshmanenko V. D., Spectral Theory of Dynamical Conflict Systems [in Ukrainian].
Naukova Dumka, Kyiv (2016).

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